Citation
Approximate minimum degree column ordering algorithm

Material Information

Title:
Approximate minimum degree column ordering algorithm
Creator:
Larimore, Stefan I. ( Dissertant )
Davis, Timothy ( Thesis advisor )
Sahni, Sartaj ( Reviewer )
Rajasekaran, Sanguthevar ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
Department of Computer and Information Science and Engineering, University of Florida
Publication Date:
Copyright Date:
1998
Language:
English
Physical Description:
ix, 161 leaves ; 29 cm.

Subjects

Subjects / Keywords:
Algorithms ( jstor )
Approximation ( jstor )
Data models ( jstor )
Factorization ( jstor )
Garbage collection ( jstor )
Heuristics ( jstor )
Index numbers ( jstor )
Matrices ( jstor )
Permutations ( jstor )
Statistics ( jstor )
Computer and Information Science and Engineering thesis, M.E
Linear systems ( lcsh )
Dissertations, Academic -- UF -- Computer and Information Science and Engineering
Matrices -- Computer programs ( lcsh )
Sparse matrices -- Computer programs ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )

Notes

Abstract:
An approximate minimum degree ordering algorithm (COLAMD) is pre-ordering an unsymmetric sparse matrix A prior to numerical factorization is presented. Using the quotient graph representation of ATA and a tight and fast approximate degree, this algorithm is a good alternative to using symmetric codes directly on the explicit representation of ATA. Additionally, COLAMD is shown to be superior to COLMMD, the column minimum degree ordering code in MATLAB.
General Note:
Typescript.
General Note:
Vita.
Thesis:
Thesis (M.E.)--University of Florida, 1998.
Bibliography:
Includes bibliographical references (leaves 158-160).

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
2427501 ( ALEPH )
41875671 ( OCLC )

Downloads

This item has the following downloads:

1998281 ( .pdf )

1998281_Page_079.txt

1998281_Page_056.txt

1998281_Page_055.txt

1998281_Page_070.txt

1998281_Page_125.txt

1998281_Page_063.txt

1998281_Page_099.txt

1998281_Page_014.txt

1998281_Page_005.txt

1998281_Page_159.txt

1998281_Page_080.txt

1998281_Page_078.txt

1998281_Page_162.txt

1998281_Page_130.txt

1998281_Page_139.txt

1998281_Page_030.txt

1998281_Page_007.txt

1998281_Page_065.txt

1998281_Page_118.txt

1998281_Page_115.txt

1998281_Page_012.txt

1998281_Page_160.txt

1998281_Page_121.txt

1998281_Page_168.txt

1998281_Page_091.txt

1998281_Page_076.txt

1998281_Page_104.txt

1998281_Page_096.txt

1998281_Page_127.txt

1998281_Page_164.txt

1998281_Page_082.txt

1998281_Page_002.txt

1998281_Page_054.txt

1998281_Page_129.txt

1998281_Page_016.txt

1998281_Page_069.txt

1998281_Page_052.txt

1998281_Page_062.txt

1998281_Page_161.txt

1998281_Page_144.txt

1998281_Page_094.txt

1998281_Page_122.txt

1998281_Page_025.txt

1998281_Page_039.txt

1998281_Page_086.txt

EQLZN0KFI_6YMYZG_xml.txt

1998281_Page_141.txt

1998281_Page_008.txt

1998281_Page_044.txt

1998281_Page_073.txt

1998281_Page_013.txt

1998281_Page_170.txt

1998281_Page_098.txt

1998281_Page_026.txt

1998281_Page_081.txt

1998281_Page_166.txt

1998281_Page_071.txt

1998281_Page_034.txt

1998281_Page_061.txt

1998281_Page_011.txt

1998281_Page_066.txt

1998281_Page_017.txt

1998281_Page_106.txt

1998281_Page_084.txt

1998281_Page_019.txt

1998281_Page_085.txt

1998281_Page_040.txt

1998281_Page_171.txt

1998281_Page_157.txt

1998281_Page_041.txt

1998281_Page_153.txt

1998281_Page_135.txt

1998281_Page_018.txt

1998281_Page_004.txt

1998281_Page_043.txt

1998281_Page_163.txt

1998281_Page_038.txt

1998281_Page_095.txt

1998281_Page_103.txt

1998281_Page_023.txt

1998281_Page_145.txt

1998281_Page_059.txt

1998281_Page_107.txt

1998281_Page_124.txt

1998281_Page_046.txt

1998281_pdf.txt

1998281_Page_035.txt

1998281_Page_131.txt

1998281_Page_058.txt

1998281_Page_020.txt

1998281_Page_147.txt

1998281_Page_112.txt

1998281_Page_083.txt

1998281_Page_057.txt

1998281_Page_088.txt

1998281_Page_021.txt

1998281_Page_156.txt

1998281_Page_029.txt

1998281_Page_152.txt

1998281_Page_123.txt

1998281_Page_045.txt

1998281_Page_048.txt

1998281_Page_064.txt

1998281_Page_119.txt

1998281_Page_113.txt

1998281_Page_134.txt

1998281_Page_150.txt

1998281_Page_138.txt

1998281_Page_031.txt

1998281_Page_060.txt

1998281_Page_154.txt

1998281_Page_158.txt

1998281_Page_114.txt

1998281_Page_110.txt

1998281_Page_128.txt

1998281_Page_151.txt

1998281_Page_116.txt

1998281_Page_090.txt

1998281_Page_097.txt

1998281_Page_165.txt

1998281_Page_068.txt

1998281_Page_050.txt

1998281_Page_137.txt

1998281_Page_074.txt

1998281_Page_140.txt

1998281_Page_146.txt

1998281_Page_105.txt

1998281_Page_117.txt

1998281_Page_009.txt

1998281_Page_142.txt

1998281_Page_072.txt

1998281_Page_051.txt

1998281_Page_028.txt

1998281_Page_006.txt

1998281_Page_148.txt

1998281_Page_143.txt

1998281_Page_101.txt

1998281_Page_047.txt

1998281_Page_033.txt

1998281_Page_108.txt

1998281_Page_003.txt

1998281_Page_102.txt

1998281_Page_167.txt

1998281_Page_037.txt

1998281_Page_032.txt

1998281_Page_136.txt

1998281_Page_053.txt

1998281_Page_120.txt

1998281_Page_001.txt

1998281_Page_155.txt

1998281_Page_049.txt

1998281_Page_077.txt

1998281_Page_024.txt

1998281_Page_132.txt

1998281_Page_027.txt

1998281_Page_149.txt

1998281_Page_092.txt

1998281_Page_100.txt

1998281_Page_015.txt

1998281_Page_093.txt

1998281_Page_089.txt

1998281_Page_036.txt

1998281_Page_067.txt

1998281_Page_111.txt

1998281_Page_010.txt

1998281_Page_126.txt

1998281_Page_087.txt

1998281_Page_042.txt

1998281_Page_133.txt

1998281_Page_109.txt

1998281_Page_075.txt

1998281_Page_022.txt

1998281_Page_169.txt


Full Text












AN APPROXIMATE MINIMUM DEGREE COLUMN ORDERING ALGORITHM


By

STEFAN I. LARIMORE
















A TIIHEIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
i: I \TER OF ENGINEERING


UNIVERSITY OF FLORIDA


1998

















For my father, who always encouraged my scholarly pursuits.
















ACKNOWLEDGEMENTS


I would like to thank Dr. Tim Davis for offering me the research position which

eventually led to this thesis. He has always been willing to discuss ideas and has provided

tremendous help in the development of both this thesis and in the research project. He has

been very patient when I have been busy yet always found time to provide assistance to

me despite his busy schedule.

I would also like to thank Dr. John Gilbert and Dr. Esmond Ng who have provided

me with valuable insight and explanations during my discussions with them.

Additionally, I am grateful to Dr. Sartaj Sahni and Dr. Sanguthevar R.i.-i 1..L...

for joining my committee and for the inspiration they gave me in the field of algorithms

during my course work.

Lastly but not least, I would also like to thank my friends and family who have

supported me during my seven years at college. Without the joy of their companionship, I

could not have finished. I am particularly grateful to my last roommates who have extended

me great hospitality during the last few months.

















TABLE OF CONTENTS




ACKNOWLEDGEMENTS ................... .......... iii

LIST OF r!' \LES ...... ...... ......... ........ vi

LIST OF FIGUTRES ..................... ............. vii

ABSTRACT ................................ ... ... ix

CHAPTER

1 INTRODUCTION ................................ 1

1.1 Literature Review .............................. 1
1.1.1 Graph I I..iiy Basics ........................ 2
1.1.2 Global Approaches ......................... 2
1.1.3 Local Approaches .......................... 6
1.2 Overview of COLAMD ........................... 8
1.3 Organization of I rl -i-. .......................... 8

2 REC('E I WORK ON MINIMUM DEGREE .......... .......... 9

2.1 : i..--. Elil iii.ll, i- ................... .......... 10
2.2 Supernodes .................................. 10
2.3 Quotient Graph : 1.i..1. i ....... ............ ....... 11
2.4 Incomplete Degree Update ................... ..... 12
2.5 ('Clii Absorption ................... ....... 12
2.6 i il:iEi ,. El ii ii.,. i .i. ............................ 12
2.7 External Degree ............... . . . . . .... 13
2.8 A A Orderings ............... . . . . . ...... 13
2.9 Approximate Degrees ............ . . . . . .... 15
2.10 Approximate iiniiiijl Deficiency ....... . . . . ... 16

3 COLAMD PRESENTED... ................... . . ..18

3.1 Features of COLAMD ............ . . . . . ...... 18
3.2 I Ih Greedy Heuristic ............ . . . . . .... 19
3.3 I Ir Formal Algorithm Description .... . . . . .. 20

4 IMPLEMENTATION DETAILS .................. ..... .. 27

4.1 Data Structures ............... . . . . . ...... 27
4.1.1 Row Structure ......... . . . . . . ... 28
4.1.2 Column Structure and Parent Tree . . . . . . 29










4.1.3 Index Array ....... ....... ............. 30
4.1.4 I 11 Degree List and Hash Table . . . . . . ..... 30
4.2 I I- Algorithms ................ . . . . . ... 30
4.2.1 : i., ii Routine ............. . . . . . ... 31
4.2.2 InitRows_Cols ............. . . . . . ... 31
4.2.3 Init_Scoring .................. .......... 31
4.2.4 Find_Ordering ............. . . . . . ... 32
4.2.5 GarbageCollection .................. ... .. 35
4.2.6 Detect_Super_Cols .................. ....... 35
4.2.7 Or(d. I_ l . . . . ... ;i

5 PERFORMANCE I I ............. . . . . . . 37

5.1 11, Square Unsymmetric (RUA) : i.,t .- . ................ 38
5.1.1 Result Tables ............. . . . . . ... 38
5.1.2 Comparison Plots .................. ..... . 63
5.1.3 Discussion ............... . . . . . ... 70
5.2 Il Rectangular (RRA) i.,. ......... . . . .... 72
5.2.1 Results Tables ............. . . . . . ... 73
5.2.2 Comparison Plots .................. ..... . 90
5.2.3 Discussion .............. . . . . . ... 96
5.3 I I, Symmetric (SYM) : i.,Ii,, .... ............. .... .97
5.3.1 Results Tables ............. . . . . . ... 98
5.3.2 Comparison Plots .................. ..... . 117
5.3.3 Discussion .............. . . . . . ... 123

6 CONCLUSIONS .................. . . . . . .... 127


APPENDIX ................... ................... . 128

REFERENCES ................... ......... ........ 159

BIOGRAPHICAL SKETCH ............... . . . . . .... 162
















LIST OF r 1.\LES





5.1 General information for the RUA matrices. .... . . . . . ......

5.2 Ordering times in seconds for the RUA matrices. . . . . . ... 46

5.3 |L + UI for the RUA matrices. ................ . ... .. .. 52

5.4 Flop counts in millions for the LU factorization of A. . . . . ... 58

5.5 RUA matrices for which AMDBAR orderings required more storage than
the factorization. ................. . . . . . ... 64

5.6 General information for the RRA matrices. .. . . . . . ...... 75

5.7 Ordering times in seconds for the RRA matrices. . . . . . ... 79

5.8 |L| for the RRA matrices. ............ . . . . . ..... 83

5.9 Flop counts in millions for the ('I! 1! -1:. factorization of A'A . . ... 87

5.10 General information for the SYM matrices. . . . . . . ...... 100

5.11 Ordering times in seconds for the SYM matrices. . . . . . .... 104

5.12 |L| for the SYM matrices. .................. ....... .. 109

5.13 Flop counts in millions for the ('I!,, -I!: factorization of A. . . . ... 113

5.14 Effect of dense row/column removal on ordering time and flop counts for the
Gupta matrices. ................. . . . . . ..... 125
















LIST OF FIGUT:IE


1.1 Block tri-diagonal form .............. . . . . . ..... 3

1.2 Doubly-bordered block form ............ . . . . . ..... 3

1.3 Doubly-bordered block form and its ('!l. --!I:. factor . . . . . .. 4


2.1 1 I. basic minimum degree algorithm. ................... .. 9

2.2 Relation between A and AA. .................. . . .... 15


3.1 Initial degrees comparison: Log2(F1,I/. LAAID_AI /F1,1'. LAAID4.) . . 21


5.1 Plotting order of the RUA matrices: by best flop count. . . . . ... 39

5.2 RUA matrices: Plot of Log2(TimecoLAl AID/ I* COLMAID) . . . .. 65

5.3 RUA matrices: Plot of Log2(TimeCOLAID/ I AMDBAR) . ......... .

5.4 RUA matrices: Plot of Log2(IL + UICOLAMAD/L + UICOLAAIMD) ..... 67

5.5 RUA matrices: Plot of Log2(IL + UICOLAMAD/L + UIAAIDBAR) ...... . 68

5.6 RUA matrices: Plot of Log2(FlpI. LLAMD/FI''/. OLMMAID) . . . . . 69

5.7 RUA matrices: Plot of Log2(FpI. ILAMD/FlIOAMDBAR) . . . .. . 70

5.8 Plotting order of the RRA matrices: by best flop count. . . . . ... 74

5.9 RRA matrices: Plot of Log2(TimeCOLAMAID/ I COLAMAID) . . . .. 91

5.10 RRA matrices: Plot of Log2(TimeCOLAMAIDI AMDBAR) . . . .. . 92

5.11 RRA matrices: Plot of Log2(ILICOLAMAID/ILCOLAMAID) . . . . .. 93

5.12 RRA matrices: Plot of Log2(ILICOLAMlD/ILIAMDBAR) . . . . ... .. 94

5.13 RRA matrices: Plot of Log2(Fl.,I. OLAMD/FI'./. ,LMAID) . . . . .. 95

5.14 RRA matrices: Plot of Log2(F'ii,. ,OLAMD/Flop AMDBAR) . . . .. . 96










5.15 Plotting order of the SYM matrices: by best flop count. . . . . ... ..99

5.16 SYM matrices: Plot of Log2(TimeSYAlAAID/ ,I YAIAIAID) . . . 118

5.17 SYM matrices: Plot of Log2(TimeSYMAMDID/ I AMDBAR) . . . .. .119

5.18 SYM matrices: Plot of Log2(IL SYAMAAD/iL SYAAMMAID) . . . . .. 120

5.19 SYM matrices: Plot of Log2(IL SYMAMAAD/iL AMDBAR) . . . . .. 121

5.20 SYM matrices: Plot of Log2(FlopsYAIAAID/FlopSYMAIAIAID) . . . .. 122

5.21 SYM matrices: Plot of Log2(FlopsYAIAAID/FlopAAIDBAR) . . . .. 123

5.22 : li.,Ii :. guptal, from Anshul Gupta ................... . ..124
















Abstract of I l -i-i Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of : f.-, of Engineering





AN APPROXIMATE MINIMUM DEGREE COLUMN ORDERING ALGORITHM

By

Stefan I. Larimore

December 1998


C('.,n ijn.,ir Dr. Timothy A. Davis
i., i. i Department: Computer and Information Science and Engineering

An approximate minimum degree column ordering algorithm (COLAMD) for pre-

ordering an unsymmetric sparse matrix A prior to numerical factorization is presented.

Using the quotient graph representation of A A and a tight and fast approximate degree,

this algorithm is a good alternative to using symmetric codes directly on the explicit rep-

resentation of A'A. Additionally, COLAMD is shown to be superior to COLMMD, the

column minimum degree ordering code in MATLAB.
















CHAPTER 1
INTRODUCTION


When solving large sparse unsymmetric linear systems of the form Ax = b, the

storage requirements for the numerical factors of A are often much greater than for A

itself. 1 Ir i !i -! it is common to precede numerical factorization with a reordering of

A with the goal of minimizing fill-in. I I!i- is helpful for both a partial pivoting LU

factorization of a square matrix A or for a ('Irl. -!:;, factorization of A A for either a

square or over-determined matrix A. Additionally, row permutations for LU factorization

are generally dictated by numerical stability considerations. I Ij, !, 1 !- the goal of this

preordering is to find a column permutation matrix Q such that the factors of AQ are

minimized. Fiijli;j, the optimal order is NP complete i1 I I !r 1,!, heuristics are used.

It has been shown that a good symmetric order of A A is generally a good column ordering

on A [19, 23, 3, 27]. Additionally, A A can be represented implicitly by A using quotient

graphs [24].

1.1 Literature Review

I Ir i have been several approaches to finding a good a priori ordering for the

factorization of a matrix. I Ih -, approaches can be divided into two groups, global and

local approaches. Global approaches try to permute the matrix into some form which

can globally bound the fill-in. Local approaches generally iterate through a simulated

factorization of the matrix making fill minimizing decisions at each elimination step. I Ih.

following is a brief introduction to graph representation of matrices, as many terms used

subsequently are from graph theory. After that, some global and local approaches are

introduced. For a more in depth discussion of some of these approaches see references L

and [li










1.1.1 Graph I I !, i,. Basics

Sparse matrix ordering algorithms traditionally have been based upon graph theory.

Given a symmetric matrix with no zeros on the diagonal, we can create a undirected graph

G = (V, E). Let V be the set of vertices or nodes in the graph such that V = {1, 2,..., n}

where n is the dimension of the matrix. Let E be the set of edges in the graph. I Ir edge

(i, j) is an element of E if aij 7 0 and i 7 j. We let Adjp be the set of nodes which are

connected by an edge to p. I Ir degree, dp of node p is equal to the number of ;,li.i... i l

nodes to it, or dp = '._11/. When performing one step of ('!,.l -!:; factorization upon

graph G with pivot p, edges are added to G such that all nodes ;.-i.l.. i l to p are fully

connected. N\ :.t, all edges ;. ,i .... to p and p itself are then removed from G. We can

write the resulting graph as Gp. Also, given a symmetric structured matrix A with a non-

zero diagonal, let G+(A) be the graph of its ('I!l -I:, factor. I IP- graph G+(A) = (V, E+)

contains the original edges in E plus the edges created during elimination.

A matrix can be symmetrically reordered simply by renumbering the vertices of

the corresponding graph. 'I Ih bandwidth of a matrix relative to its ordering is defined as

max{|vi v21 : (Vi,V2) E E(G)}. N.i by reordering a matrix it is possible to reduce the

bandwidth of the matrix.

Another important graph result is called the level structure [22]. I I!-i is defined as

L(G) and is composed of level sets, L1, L2,...Lk such that each vertex in Li can only be

;1li... ii to vertices in Li1_, Li+i, and Li. 'I 11 width of a level structure is the max{|Iil C

L(G)}. 'I I, depth of a level structure is equal to k.
1.1.2 Global Approaches

Organizing matrices into special forms is one way to reduce fill-in during factoriza-

tion. I Ii i, are several forms which can be exploited to minimize fill-in. I Il -. include

the block tri-diagonal, doubly-bordered block diagonal, bordered block triangular form,

and several other forms .; see Fi ii- 1.1 and 1.2. Some of these forms, such as the










doubly-bordered block form, can be exploited in that fill-in need only be stored in the di-

agonal blocks. Computational tricks allow for fill-in outside the diagonal blocks to be done

implicitly.


Fin-,iln 1.1: Block tri-diagonal form


Finl_, 1.2: Doubly-bordered block form


Several of the global methods include some sort of bandwidth minimization phase.

I Ir. i are several algorithms for minimizing bandwidth of a structurally symmetric matrix.

One of the first was due to Cuthill and : 1i lKi [4]. I In!- algorithm generates level structures

starting with vertices of low degree. Letting L1 be a set containing a vertex of low degree,

the algorithm creates L2 containing all vertices ; .1j.... iit to the starting vertex in L1. L3

is created and contains all vertices ;'li.... lit to vertices in L2 but not in L1 and so on.

I In!- continues until all vertices have been placed into level sets. For each level structure

of minimal width, a numbering is given to the vertices. In!- is done such that vertices in

a given level which are I.1.... it to the lowest numbered vertices from the previous level

are ordered first, and in order of increasing degree. Once all qualifying level structures

have been ordered, the ordering which results in the smallest bandwidth is selected as









the final ordering. I I!-' algorithm does produce narrow band structures, however, if the

matrix has many vertices of -. \.-''" degree, we can see that this algorithm would go through

many iterations. One final note about this algorithm is due to George [13]. He found that

reversing the Cuthill : 1. liK ordering resulted in the same bandwidth but a smaller profile.

I I!,- is known as the Reverse Ct ll !ii-i 1i li 'f ordering or RCM. Liu and Sherman [34] show

that RCM is better than the original Cil i !ii-: 1 lfi ,- algorithm in terms of storage and

computation complexity.

In an attempt to improve on the run-time of the RCM algorithm, Gibbs Poole and

Stockmeyer [22] describe an algorithm which generates similar quality orderings as RCM

but does so much faster. I I!j- algorithm starts with an arbitrary vertex of minimum degree

and generates a level structure starting with this vertex. Let d be its depth. Level structures

are then generated for each vertex in the last level set in order of increasing degree. If one

of these level structures has a depth greater than d, then d is set to this new depth and the

procedure is repeated. Once the depth has been maximized in this manner, a final level

structure is generated and the vertices are numbered as in the RCM algorithm.

Another approach to minimizing fill-in is the one-way dissection approach given by

George [15]. 1 Ihr matrix structure used by this algorithm is the doubly-bordered block

diagonal form. In this approach, a level structure is generated as above. Once this level

structure L(G) has been generated, a separator is formed from roughly equally spaced level

sets in the level structure. I Ir level sets not included in the separator are ordered first.

Additionally, within each of these sets, the vertices are ordered using the RCM algorithm.

Fin..ll;, the separator is ordered last. What results is exactly the doubly-bordered block

diagonal form. I h!-' form can be written generally as in Finiiil 1.3.

A (Al A12 L = L 2
A A A22 WT L22
12 ) 12


Fin_,i 1.3: Doubly-bordered block form and its ('! !l -I:.; factor










Since, All and A22 are banded, we can be assured that fill-in here will be small.

However, with W12 we have so such guarantees. Fortunately, it suffices to have only L11,

L22, and A12 for solving the system [15].

Dissecting a matrix can be further done onto the diagonal sub-matrices produced in

a one-way dissection, as above. George [14] showed this dissection can be done recursively

until a sufficiently small threshold has been reached. I hi- is called nested-dissection. I I.

space and computational bounds for nested dissection are lower than for one-way dissection.

It has been found that for large matrices, this method is better than one-way dissection.

However, for some classes of problems, namely matrices generated from highly rectangular

structures, nested dissection performs poorly.

For unsymmetric matrices, another global approach called, reinversion with the

preassigned pivot procedure, has been developed by Hellerman Rarick [29]. I-i method

first starts by ordering pivots with one row entry to the upper left. Pivots with one column

entry are order to the lower right. What remains is a sub matrix with at least two entries in

each column and row. I I,- sub matrix is further subdivided and sub-blocks ordered with

the attempt of producing "-i'!:. '' columns.

Recent work in global methods centers primarily on graph partitioning techniques.

Similar to George's nested dissection, these methods try to find small vertex sets which

divide the graph of the matrix into well-balanced connected subsets. Several software

packages exist which use graph partitioning techniques. I Ir -, include CHACO [30], the

Watson Graph Partitioning Package WGPP [26], and the METIS package [31].

1 Ir -I codes perform partitioning by first trying to coarsen the graph of the matrix.

Coarsening treats the original graph as a weighted graph with all nodes and edges having

a weight of one. I Ir ij, groups of connected nodes are collapsed into one yielding a weight

equal to the sum of the constituent node weights. Any duplicate edges to outside nodes

from the collapsed node are also collapsed into one edge with the weight summed. Once

the graph is sufficiently collapsed, the partitioning begins. '1 Ir partitioning can be into










an arbitrary number of sets depending on the algorithm and the problem. Lastly, the

graph is uncoarsened and the initial partition is refined, normally using variations of the

Kernighan-Lin heuristic [32].

Another recent technique by Ashcraft and Liu is called multisection ordering [2].

I I!-i algorithm first finds a good separator which divides the graph into two or more subsets.

I Ii separator is not the key to this algorithm. I 1 !i 11 !. ,, any graph partitioning algorithm

from above can provide the separator. Once a separator is found, each partition then is

ordering using what is referred to as a constrained minimum degree algorithm. I I. con-

strained minimum degree algorithm includes the separator nodes in the degree calculations

for the partition nodes. I h- multisector is also ordered using standard minimum degree.

I Ii partition and separator orderings are what improve the quality for problems which

graph partitioning orderings typically do poorly. I Ii !. 1!i using multisector orderings

has been shown to produce good results for a wide range of problem types.
1.1.3 Local Approaches

In contrast to global approaches, local methods pick one pivot at a time, updating

the matrix or its graph after each choice. Since these are local minimization strategies,

we are not guaranteed that the fill-in is globally bound. However, recent advancements in

time and space complexity have made them very popular. 'Ili following describes some

local heuristics for ordering a matrix. One of the first heuristics for ordering a matrix was

due to : .i.. l:..itz [35]. I I!,- heuristic, used on unsymmetric matrices, actually provided a

column and row ordering and was based upon both the pattern and numerical values of the

matrix. Given that for the kth step of factorization, ri(k) is the degree of row i and c (k)

is the degree of column j, the pivot element is chosen such that (ri(k) 1)(cj(k) 1) is

minimized and where the numerical value of the pivot is acceptable for numerical stability.

With this choice of pivot, the fill-in potential for this step is minimized. I Ihp major problem

with the l.i i.: ... itz strategy is its reliance on numerical values and the heavy search costs

associated with finding an acceptable pivot. Additionally, two matrices of identical non-zero










pattern but with different numerical values would have two distinct orders. It is sometimes

desirable to get an ordering based solely on the non-zero pattern so that a pivot ordering

can be used on multiple matrices all with the same non-zero pattern, but with different

numerical values.

A derivative of the l.,: 1.i v.itz algorithm is the Tinney and Walker algorithm L.;',

later named the minimum degree algorithm. 1 I,' algorithm is the symmetric counterpart

to : 1. !i: \.'i tz. As it is used for symmetric positive definite matrices, pivot choices do not

need to be checked for numerical stability. I Ir. I 1 .I only the non-zero pattern is needed.

I Ii- heuristic chooses the pivot element from the row (or column) of minimum degree.

Since diagonal pivoting is used, the matrix remains symmetric, which also implies that the

pivot element is in the column (or row) of minimum degree. Like : 1.,i : i: \.itz, this algorithm

locally bounds the fill-in at each step.

i l research in local approach preordering algorithms has been on either modifica-

tions to the f: i.:. v.-i tz or to the minimum degree algorithms. I Irl -, works have primarily

been concerned with reducing the memory and time complexity of these algorithms. One

example is the work of Zlatev [41] who reduced the search space for the : l.! i:, \. itz heuristic

by limiting the pivot search to rows of low degree. Recent work has been done to reduce

the cost of calculating degrees in the minimum degree algorithm by using approximate

degrees [5, 6, 25]. In other cases, attempts have been made to improve the quality of

heuristics. Tinney and Walker .;', proposed a minimum deficiency version of the minimum

degree algorithm. I I!j- actually minimizes real fill-in at each stage. However, despite slight

quality improvements, the time costs are prohibitively high. Recent work on minimum

deficiency by Rothberg and Eisenstadt I.;i, use approximate deficiencies to dramatically

improve runtimes. I Ir y also show improved quality as compared to the minimum degree

algorithm.










1.2 Overview of COLAMD

COLAMD falls under the category of a local approach. I Ih motivation behind

COLAMD is that we want a good ATA-based ordering of the unsymmetric matrix A

without the cost associated with explicitly forming A A. Additionally, we can lose some

information about A by forming A A. I I !, 1,!.i ,, working directly on A can yield a better

quality ordering.

COLAMD includes several features of earlier minimum degree codes. I Il -, include

the quotient graph representation, supernode detection, mass elimination, clique absorption

and approximate degrees. I Il -, and other features of COLAMD are all discussed in the

following chapter.

1.3 Organization of' I l. -i-

I I, remainder of this thesis is organized as follows: ('I!,.1i. 2 describes the lat-

est developments in the minimum degree algorithm. In ('I!,.i i r 3 formally presents the

COLAMD algorithm. ('I!,.i1t i 4 describes the implementation details. Both the data struc-

tures and algorithms are discussed. In ('Cl.,ii i 5, the performance results are shown and

discussed. ('C,.iti i 6 gives a conclusion of the results and proposes some ideas for further

research.
















CHAPTER 2
RE(CIN I WORK ON MINIMUM DEGREE


I Ii i have been many improvements to the minimum degree algorithm since the

first implementation by Tinney and Walker [;'. George and Liu [17] present the evolution

of the minimum degree algorithm. I I. original algorithm was based upon the elimination

graph model. I I!-, model is an explicit graph representation of the matrix at each step of

the elimination. Let this graph be G. At each step of elimination the node of minimum

degree is chosen and ordered. Let this node be p. Edges are added to G such that all nodes

;.1 ... i li to p are fully connected. \i :.., all edges :;i1.... iii to p are then removed from the

graph and p is also removed. I Ir resulting graph is represented by Gp. I hi, sequence is

repeated until all nodes have been eliminated. See Fiiii 2.1 for a pseudo-code description

of the algorithm.

I Ir i, are three major problems with this implementation of the minimum degree

algorithm. Since at each step we add and delete edges from the graph, we cannot predict

ahead of time the maximum storage requirements for the algorithm. I h. second problem

associated with the minimum degree algorithm is calculating the degrees. One can see

that degree updates are required for each node :i.... iit to the eliminated node p. It has

been shown that these degree updates are the most computationally intensive part of the



G = given symmetric graph
while G # 0 do
select and order node p of minimum degree in G.
G = Gp
end while



F i i- 2.1: I I. basic minimum degree algorithm.









algorithm [1]. 1 Ih final problem is using the minimum degree algorithm to obtain a good

column ordering for unsymmetric matrices.

I l remainder of this chapter is devoted to discussing various solutions to these

two major problems.

2.1 i1.. E liiiniti., i, i

One solution to the degree update problem is due to George and : 1i Iityre [18].

I l y noted that often times a set of nodes can be ordered together, thus reducing the

number of times the degree update phase must be done. I In!- set can be determined by the

following theorem.


Theorem 1 Let q E Adji (p) then AdjGp (q) = Adj (p) 1 if and only if Adji (q) U {q} =

AdjG(p) U {p}.


N\.. 1 -i that can be grouped together via this theorem are also known as indistin-

guishable nodes. I In- leads us to the next major improvement in the minimum degree

algorithm.

2.2 Supernodes

As 1 Ih .._, In 1 states, two nodes u and v are indistinguishable if and only if AdjG(u)U

{u} = Adjc(v) U {v}. It has also been shown by George and Liu [Ii that if two nodes are

indistinguishable in G then they remain indistinguishable in Gp. Knowing this, indistin-

guishable nodes can be represented as one node in the data structures. I In! is referred to

as a supernode. When this supernode is ordered, all constituent nodes are ordered. I I.

use of supernodes can significantly reduce the amount of maintenance work required dur-

ing elimination. Only the degree of one representative node of the supernode needs to be

maintained.










2.3 Quotient Graph: i. i. i

To solve the problem of the unpredictable storage requirements for ordering, Speelpen-

ning [37] introduced a generalized element approach also known as the quotient graph repre-

sentation. In this approach, each edge in G can be thought of as a clique of size two. G can

be represented as a collection of cliques, each initially of size two. As elimination proceeds,

cliques involved in an elimination step are merged together. Assuming node p is chosen

as the kth pivot and this node is contained in cliques Cp C2,, ..., Cp,, the elimination

procedure can be written as follows:


Fi -t create the new clique C = U' = Cp \p.

N\ :.t, remove the cliques Cpi, Cp,, ..., Cp, from G.


Lastly, add the new clique Ck to G.


I l following theorem shows why the quotient graph approach is superior.


Theorem 2 (George and Liu [17]) Given n cliques Cpl,... C, containing the pivot

node p, the resulting merged clique is given '. i Ck with the property that |Ck < E i | Cpj1.


Since the size of the new clique is strictly smaller than the sum of the constituent

clique sizes, the storage requirements for Gp will never exceed that of G.

Implementations using the quotient graph representation do not need to first gen-

erate a pure clique representation of the matrix. 'I Il original ;,i1i.... i '. representation can

be used and as elimination proceeds, a hybrid representation containing both cliques and

edges results as edges are removed and cliques are added.

Several implementations exist which use the quotient graph representation. 'I i. -

include MA27 [9], the Waterloo Sparse Linear Equations Package (SPARSPAK) [21], the

Yale Sparse : i.lt I:.. Package (YSMP) [12], and more recently MC47 [1].










2.4 Incomplete Degree Update

Another innovation in the evolution of the minimum degree algorithm is the use of

incomplete degree updating, first introduced by Eisenstat, Gursky, Schultz, and "'. I I .i ii

[12]. With incomplete degree update, it may be found that the degrees for a subset of nodes

;,1j... iii to the pivot node need not be computed. I Ir reason for this is that these nodes

are said to be outmatched by other nodes ;,iij.... iii to the pivot. A node v is outmatched

by u in G if Adj (u) U {u} C Adj (v) U {v}. I In!- relationship between u and v holds as

elimination proceeds. N.i .. u will always have a lower degree than node v. I I!, i .! if

a node v is found to be outmatched by u, the degree for v does not need to be updated

until after u has been eliminated. I In! has been shown to significantly improve run-times

by reducing the number of degree update calculations.

2.5 ('!hiii Absorption

Duff and Reid [11] introduced an idea they called element absorption. With the

quotient graph model described above, it is easy to see that if there exists a clique which

is a subset of another, C C C m, we can eliminate the subset clique CI without changing

the structure of G. I In,- is known as absorption where clique Cm absorbs CI. If any such

redundancies are found within the quotient graph of G, they can be removed. Performance

will improve due to less clique maintenance.

2.6 1 lit i i E li i i i

In the effort to further delay degree updates, Liu [33] proposed multiple elimination

as a solution. With this, a set of independent nodes of minimum degree are selected and

eliminated before any degree updates are done. I Ir motivation behind this concept is the

fact that any nodes not in Adj(p) are unaffected by the elimination of p. I I! 1 -!. we

may eliminate any nodes outside Adj(p) before performing degree updates on nodes within

Adj(p). Additionally, the removal of pivots outside Adj(p) may reduce the cost of updating

the degrees for nodes {q : q E Adj(p)}. Assume a set P of independent pivots is found in










G. Once these pivots are eliminated, the set of nodes UpEp Adj(p) must have their degrees

updated. However, if the set P was large, there is a good chance that some of these degree

updates will be cheaper than if each pivot was eliminated one by one.

2.7 External Degree

Another improvement to the minimum degree algorithm also suggested by Liu [33]

is the use of the external degree instead of the true degree. I Ir external degree of a pivot

is the true measure of the resulting clique size after that pivot is eliminated. With the

advent of supernodes, this distinction becomes important. Since the goal of the minimum

degree algorithm is to reduce the size of the resulting clique at each step of elimination,

the external degree makes more sense to use. Liu has found that using the external degree

typically produces better quality orderings.

2.8 A A Orderings

In addition to ordering symmetric matrices, the minimum degree algorithm has

been shown to be an effective column ordering algorithm for unsymmetric matrices. Given

an unsymmetric matrix A, it has been shown that a symmetric order of A'A is a good

column ordering of A. I I following theorems present bounds on the structure of either

LU or QR factorization of A based upon the symmetric factorization of A A. George

and Ng [19] showed the following theorem in reference to LU factorization with partial

pivoting.


Theorem 3 Given an unsymmetric matrix A. Let G+(A) be the structure of the Cl.. .l .7

factors of A A. Let L and U be the LU factors of A obtained ', i partial pivoting. I/!. I

G(L + U) CG (A).


Gilbert [23] also shows the following theorem for an irreducible matrix A undergoing

LU factorization with partial pivoting.










Theorem 4 Given an unsymmetric matrix A which is strong Hall and the factor of its

column intersection graph, G (A) = (V,E+). For i.-'i choice of {i,j} where {i,j} E E+,

there exists values for A such that Uij 7 0.


I In-, shows what Gilbert refers to as a "i'i -t -..-tI i upper bound on the structure

of U. Coleman, Edenbrandt, and Gilbert [3] show a bound for QR factorization.


Theorem 5 If performing QR factorization upon A then G(R) C G+(A).


Hare, Johnson, Olesky, and van den Driessche [27] provided a tighter bound if A is

Strong Hall and full rank.


Theorem 6 If performing QR factorization upon A and A is i ...y Hall, then there are

values for the non-zeroes of A such that G(R) = G+(A)


We can therefore see the value of a symmetric analysis on A'A for both standard

partial pivoting LU factorization and for orthogonal QR factorization. In both cases, the

structure of G+ (A) is an upper bound on fill.

Typically, one can obtain a column ordering for an unsymmetric matrix A by form-

ing ATA explicitly first. A A is then given to a symmetric ordering algorithm, like the

minimum degree algorithm. I Ih resulting symmetric permutation is used as a column

permutation for A. I Ih trouble with this approach that time and space required to ex-

plicitly form ATA may be prohibitive. Additionally, information about the structure of A

is lost when ATA is formed. It has been shown by Heggernes and : i..t-t,-.in [28] that this

structure information can be useful in tie breaking strategies for sparse QR orderings.

With the introduction of the quotient graph model of symmetric Gaussian elimi-

nation there exists an alternative to the explicit formation of A'A. As stated earlier, the

quotient graph representation forms a new clique upon each stage of elimination. I I,-

clique is formed as a union of an original row and one or more previously formed cliques.

However, as Speelpenning [37] originally proposed, we could start with a representation of










a symmetric matrix composed entirely of cliques. 1 Ir following definition of the column

intersection graph shows how this can be easily done for A A.


Definition 1 (Gilbert [24]) Let A be an unsymmetric matrix of dimension m 1.'i n. Let

its directed graph be G(A) = (V,E). /I!. column intersection graph of A is Gn(A) =

(V, E) such that = {1, 2,...m} andE = U L1 {(i,j) : (k, i) E, (k,j) C E} Furthermore,

assuming no numerical cancellation, G(A A) = Gn(A).


In other words, each row in A is a clique in the graph of A A. Finii 2.2 shows

the relationship between a row in A and its corresponding clique in ATA.




A=




Fi:ii l2.2: Relation between A and A A.


I Ii !i I with no initial work, an implicit representation of A A exists for any

quotient graph based algorithm operating on A. I I!-i is the approach used by COLAMD

and COLMMD.

2.9 Approximate Degrees

One of the more recent developments involves reducing the time it takes to actually

perform the degree update computation. Recent work by Gilbert, 1i' i,, and Schreiber

[25]; Davis and Duff i,; and Amestoy, Davis, and Duff [1] make use of an upper bound on

the degree.

I I, degree approximation from Gilbert et al. [25] is used in the COLMMD function

of MATLAB, which is a column ordering algorithm for unsymmetric matrices. As they use

A as an implicit form of A A, each row in A is a clique in A A. 'I I, approximation they

use is simply the sum of the external degrees of the cliques to which the supernode belongs.










I I!j- can be found very quickly. I I drawback of this approach is somewhat lower quality

permutations which result in more fill-in than the use of exact degrees.

Amestoy, Davis, and Duff use a more sophisticated approximation in AMD which

has been shown to yield high quality orderings. I I!-i approximation performs set-differences

between the newly-formed clique C with all other cliques to which all nodes in C belong.

I I,, degree for each node in C is determined as the sum of ICI and the size of the set

differences of each additional clique to which the node belongs. I I!-i is the approximation

also used by COLAMD.

2.10 Approximate : inijiiii, Deficiency

Recent work by Rothberg and Eisenstadt i.;I show that modifications to an ap-

proximate minimum deficiency ordering algorithm are almost as fast and produces better

orderings than a comparable minimum degree algorithm. I Ih minimum deficiency algo-

rithm tries to minimize actual fill-in at each step of elimination instead of minimizing the

bound on fill-in as the minimum degree algorithm does. I Iy propose a cheap approxima-

tion to minimum fill and two new heuristics based upon the minimum fill which perform

well.

Rothberg and Eisenstadt propose a bound on the minimum fill based upon the

quotient graph, G. Let Fill(p) be the exact fill resulting in the selection of node p as pivot.

I I, y note that given a clique, C in G, all pairs of nodes in the clique are already :1i.... lit

I I !i 1!, the expression for approximate fill for node p which is a member of clique C, is

given by: Fillapprox(p) = (d dp)/2 (c2 c)/2 where c = \C \ p, where dp is the degree

of node p. I I y found that letting C be the most recently created clique containing p is

sufficient to obtain a good bound on the fill.

I h- first heuristic they call minimum mean local fill or MMF. I I, heuristic notes

that the elimination of a supernode p means the elimination of |pl original nodes. I I. i. li ,,

if the total fill associated by this selection is Fill(p), then the average fill is Fill(p)/|pl.

I I, y found better results by using Fill(p)/ pi .









SIr second heuristic based upon the minimum fill is called minimum increase in

neighbor degree or MIND. With this heuristic Rothberg and Eisenstadt note that the elim-

ination of supernode p causes the addition of Fill(p) edges and the removal of |adj(p)|

edges. However, since p represents |p| nodes, there are |adj(p)| x |P| edges being re-

moved in the real elimination graph. I, !I ,i!. they let their second heuristic score(p) =

Fill(p) adj(p)l x |pl.

With both the above heuristics, Fillapprox (p) can be used in place of Fill (p) to speed

up run-times. 'I Ih two heuristics then become AMMD and AMIND respectively. Rothberg

and Eisenstadt find that AMMD performs best and compared to Liu's MMD [33], AMMD

requires only '.;' more ordering runtime while generating orderings of A that require .'

less flops to factor.
















CHAPTER 3
COLAMD PRESENTED

3.1 Features of COLAMD

In this section, features in COLAMD will be discussed together with the reasons

they are included.

Since we are ordering ATA implicitly from the structure of A, the obvious choice

for representing ATA is via a quotient graph in which each row of A is a clique in ATA.

For the degree computations, the approximation used in COLAMD is the same used by

Amestoy, Davis, Duff [1] in their AMD code. I In!- approximation is fast and results in high

quality orderings in the symmetric case.

Another feature of the Amestoy, Davis, Duff approximation is it provides for easy

checking of redundant cliques. I Il i- !. o!- element absorption is a --! "' feature of COLAMD

as a by-product of the approximate degree computation.

Supernode detection is implemented along with additional mass elimination. It is

sometimes possible that mass elimination can be performed even though supernodes are

detected. I I!,i is because a node q might not become indistinguishable with the pivot node

p until p is eliminated. To further enhance the quality of the algorithm, Liu's [33] external

degree is used instead of the true degree.

Two features of Liu's MMD are not used in COLAMD. Since the AMD approxima-

tion needs to scan each column ;,1.i ... itj to the pivot, doing incomplete update would save

no work. I I.! 1!., I, degree updates are performed for each affected column. Additionally,

performing multiple elimination would not save any work because calculating the approxi-

mate degree does not require the full scanning of all cliques containing nodes :;li.'... lit to the

pivot. I Il, i 1-!. ,, multiple elimination and incomplete updating are not used in COLAMD.










3.2 Ir Greedy Heuristic

Since COLAMD is essentially a greedy algorithm which chooses at each iteration

the column which minimizes a metric. I I!-i metric is based upon some heuristic which

hopefully yields a good ordering. With this in mind, several heuristics were experimented

with during the early implementations of COLAMD. I three major heuristics were: the

minimum degree algorithm, minimizing the size of the householder update matrix, and

minimizing the worst-case .: ..il :.v.itz score.

As described in Section 2.8, the minimum degree algorithm on ATA provides a

bound on fill-in for LU factorization of A. Using this metric, let the score for column j be

equal to dj = Urow Ecolumnj (' ''.' )I. I I,- is the degree of the union of the rows intersecting

column j. Additionally, it has been shown in [1] that approximate degrees are better than

exact degrees when using this heuristic. I Ih 1h I! ,, using an approximate minimum degree

was a logic choice as one of the test heuristics for this code.

Sfiij iij,..' i the size of the Householder update matrix was thought as another po-

tential candidate for the greedy heuristic. With this metric, let the size of the Householder

update for column j in QR factorization of A be equal to hj. 1 Iii- can be written as:

hj = (dj) ( column |), where dj is the degree of the union of rows from above and icolumnj\

is the sum of original rows intersecting with column j.

'I hr last heuristic that was considered is called the .1/., r-.l 1,, kowitz heuristic. With

this heuristic, the worst case : 1. !:. v.i tz score was minimized during each iteration. Since

with partial pivoting, it is not known which row will be chosen as the pivot, it is assumed

that the row of largest degree will be chosen. I Ir !I 1i!. letting this metric be denoted by

mj, it can be written as follows: mj = (maxrow iEcolumnj "...',' )(lcolumnj\ ).

Our initial experiments with these three heuristics clearly showed that the minimum

degree on A'A was best. One interesting note was that the : i.:-.- i l.'. !i v. tz criteria per-

formed very well for small matrices and very poorly for large ones. It can been conjectured

that the symbolic row merging which occurs after each pivot reduces the accuracy of the










row degrees. For small matrices, enough accuracy remains to produce quality orderings.

However, for large matrices, row degrees become inaccurate too quickly to produce accurate

orderings.

Once it was established that the minimum degree heuristic was to be used, it was

thought that using the tight AMD style degree bounds of Amestoy, Davis, Duff [1] for

both the initial degrees (the degrees of each node in the graph of A A) and updated

degrees was a natural choice. However, quite by accident, it was found that using the looser

MATLAB style bounds of Gilbert, I 1. i, and Schreiber [25] for the initial degrees produced

better orderings in the prototype code. Finaii 3.1 shows the flop count comparison of the

two prototype codes. I yI are labeled COLAMD_M and COLAMD_A. I. M refers to

MATLAB style initial degrees while A refers to AMD style initial degrees. Both codes are

identical except for the calculation of the initial degrees and both use AMD style degrees

for degree updates during the ordering. I Ih comparison is done for all the RUA matrices

described in Ch.'i,'p r 5. Points below the zero line indicate that COLAMD_M produced

an ordering which required less floating point operation to factorize than the COLAMD_A

ordering. Big matrices refer to those which required more than 107 floating point operations

to factorize.

It can be seen that for the big matrices, COLAMDM clearly out performs CO-

LAMD_A. COLAMD_M orderings can be expected to require .' less floating point oper-

ation to factorize than COLAMD_A orderings.

I I!. i I. 1!, because of the quality advantage of COLAMD_M and its simpler initial

degree computation, it was decided that MATLAB style initial degrees would be used in

the final version of COLAMD.

3.3 '1 Formal Algorithm Description

Because COLAMD is using the unsymmetric matrix A to order A A, we have two

sets of terminology which mean the same thing. A column in A is the same as a node in

A'A. Additionally, a row in A is a clique in ATA. From now on, terms will be used which














08

06- 0
0

04- o o 8
0 0 0

02 0 000 0
Soo oo o

-0OO 0 0 0 O 0 0 0
00 0 0 O


-0 4 O O 0


-0 O O 0

-08 -
0
50 100 150 200 250
50 100 150 200 250


Ratio


Matrix number
St atist.ifes.


Group Min Mean Max Median -'. I Dev. Covariance
ALL -1.13 -0.07 0.66 -0.00 0.23 0.05
BIG -1.13 -0.16 0.66 -0.11 0.30 0.09


Fil ui 3.1: Initial degrees comparison: Log2(F', '. ALAMD_Al/FI'.,P. LAMAID4)


refer to the original unsymmetric matrix A. I I 1 !,, supernodes become supercolumns.

I I. term row will be used instead of clique. Also, the term column will be used instead

of node. However, the reader should keep in mind to parallels between the symmetric and

unsymmetric terms.

'I quotient graph of A'A can now be described as a set of rows and columns of

A. I I!,' is represented in COLAMD by the sets Ri and Cj.



Ri = {j : aij # 0} C R



Cj = {i : aij # 0} C


R is called the row struct of A and, of course, each Ri is a clique in A A. C is

known as the column struct. Cj represents node j in the graph of ATA.









Columns are divided into two groups. 1 Ir first group are the principal columns.

A principal column names the supercolumn to which it is a member. 1 Ir second type is

a non-principal column. A non-principal column is a member of a supercolumn. However,

it does not represent the supercolumn in the data structure. For example, let column i be

principal and j be a non-principal column belonging to the same supercolumn as i. I Ir. i ,

the supercolumn i is denoted as i = {i,j}. Let |il denote the set size of supercolumn i.

Additionally, let IRil = E jRR IJl-

Supercolumn detection is performed after the degree approximation phase is com-

plete. During the degree approximation, a hash function for each column is calculated

based upon its constituent row indices. During supercolumn detection, columns are com-

pared directly with other columns with the same hash value. If two columns are found to be

indistinguishable, one column is chosen arbitrarily as the non-principal column. Assuming

i and j are found to be indistinguishable (that is, C, = Cj) let j = {j, k, 1, m} and i = {i, n}.

If j is chosen as the non-principal column, then supercolumn i = {i, j, k, 1, m, n} and j is

removed from the data structures. N\, lil = 6 following the supercolumn merging. I I!-,

form of supercolumn detection may miss some indistinguishable columns. I I!J- is because

indistinguishability in the elimination graph (UkEc Rk = UkEcjRk) does not imply indis-

tinguishability in the quotient graph (Cj = Cj). However, the reverse is true and since this

method is faster than complete supercolumn detection, it is used in COLAMD.

Let Adjc(j) be the set of columns ;:ili. i t to column j. Based on the row and

column representation, this can be written as follows.


Adjc(j)= ( U I \{}


Let dj = IAdja (j) be the exact external degree of column j. I Ir- exact degree is too

costly to compute. For the initial node degrees COLAMD uses the degree approximation of

Gilbert,: il.1 i and Schreiber [25]. I I!,- has been demonstrated in Section 3.2 to produce

better quality ordering than using the tighter degree of Amestoy, Davis and Duff.









Let dj = Ei-EC (iRil 1) < dj be the initial approximate column degrees based

upon the approximation in [25]. Let p be the pivot column (the column of smallest degree

dp). Since p is principal, all its constituent columns must be ordered now. I In!- is the mass

elimination step. At this point, a new row r is formed such that Rr = AdjG(p). Let this

row be known as the pivot row. After forming Rr, all rows i E Cp are deleted from the

data structures and the indices to these rows are removed from each {Cj : j E Rr}-. \I.

as stated in I hI ri ij 2, IRrl takes up strictly less space than E ic, IRil. I. 1.!, this

algorithm can operate in an amount of memory that is bounded by the size of the initial

quotient graph.

At this point, all columns j E Rr have their degrees updated using the Amestoy,

Davis, and Duff approximation. I1 hr approximate degree computation takes place in two

phases. I Ir following formula describes the approximate degree:


dj = Rr, + \ IR\Rr,- jl
iECj

I Il first phase is the calculation of the row set differences, w (i) = Ri\ Rr |. Initially,

w(i) = IRil for each i E UjER,Cj. I Ir set difference of each row which contains a column

also in the pivot row is performed by traversing the set Cj for each column j E Rr. As

each row i E Cj is encountered, the supercolumn set size |j| is subtracted from w(i). \Ni.I ,

|j| > 1. If w(i) becomes equal to 0, the row i is a subset of the pivot row and therefore may

be removed from the data structures. I I !- is referred to as I--i -- absorption.

I Ih second phase of the degree approximation is the summing of the row set dif-

ferences, w(i)'s. For each column j Rr, let the degree, dj = EjCCj w(i). N.tl the

dj's are not the final degrees since the degree of the pivot row Rr was not included in

them. At this point, if any temporary degrees equal zero, further mass elimination can

be done. I In! situation may occur despite supercolumn detection because as mentioned

above, some indistinguishable columns may be missed. Assuming column q is found to be

indistinguishable with the pivot column and eliminated at this point, then Rr = Rr \ q.









As mentioned before, supercolumn detection is performed now. \N .1i supercolumn

detection may reduce the number of principal columns in Rr but will not affect IRr -

I I. last phase of the algorithm is performing the final degree computation. I I!- is

a simple matter of adding the degree of the pivot row, IRr| to dj for all remaining principal

columns and subtracting the supercolumn set size. dj = dj + IRr\ Ijl. 'I l supercolumn

set size jl is subtracted to obtain the external degree. Lastly, the pivot row r must be

added to each {Cj : j Rr}.

Algorithm 1 shows the general outline of COLAMD.

Assuming a matrix A with its columns already ordered by COLAMD is given.

Additionally, if the ('Cr. -I:.. factor, U, of A'A is also provided, the time complexity for

COLAMD can be bounded by the following where Ui is the ith row in U and Aj is the jth

column in A.


0 E ( i A|))
i=1 JEUi

I I-', can also be expressed as the time needed to compute AU If A contains a

full row, U will be dense and the time complexity becomes O(n3). If A contains a full

column, U will contain a full row and the time complexity is O(n2). However, even in these

worst case scenarios, supercolumn detection, mass elimination, row in, i [- and i-, I --i.I

absorption will all reduce the actual runtime in the following manner:


Supercolumn detection and mass elimination reduce n.

Row merging reduces IAj .

Aggressive absorption reduces IAj.


I I !. -, the true runtime can be expected to be well below the theoretical upper


bound.















Algorithm 1 (Pseudo-code for COLAMD)
initialization and initial scoring:
for i = 1 to m do
Ri = {j : aij 0}
end for
for j = 1 to n do
Ci = {i: aij # 0}
j= {j}
dj -= EecC(IRil- 1)
end for
order columns:
k=1
r = rn
while k < n do
select and order c such that dc is minimized
k= k + cl
r r + 1
form the merged pivot row:
R, = (Uice Ri) \ c
for each i E Cc do
Ri = 0
end for
compute the row set-differences:
for each j E Rr do
assume w(i) = IRi| for all i
Cj C, \ Cc
for each i E Cj do
if w(i) < 0 then
w(i)= Ri I
end if
w(i) = w(i) |j
,,. i, .. absorption:
if w(i) = 0
C = C\i
Ri = 0
end if
end for
end for





















Algorithm 1 (Continued)
compute approximate degrees:
for each j G Rr do
dj = Eicj w(i)
further mass elimination:
if dj = 0 then
order j
k= k +|j|
Rr = Rr \j

end if
end for
supercolumn detection:
for each pair i and j G R, do
if Ci and Cj are indistinguishable then
i=iUj
Rr = Rr\ j

end if
end for
final degree computation:
for each j E Rr do
dj dj + Rr\ j|
C = Cj U {r}
end for
c= 0
end while
















CHAPTER 4
IMPLEMENTATION DETAILS


I I!-i chapter presents the implementation details of COLAMD including both the

algorithms and data structures used. I Ilj reader is referred to Appendix A, which contains

the complete COLAMD code.

I hl code for COLAMD is separated into four major routines, all called from the

main routine. Ir -- are initrowscols, initscoring, findordering, and orderchildren.

I Ir i, are also two special routines detectsupercols and garbagecollection. In addition,

there are six major data structures maintained during the execution of the algorithm. 'I I

following describes in detail the data structures and algorithms used by COLAMD.

4.1 Data Structures

I Ir i, are six data structures used by COLAMD. I r -i are the row struct, column

struct, index .i, .,. degree list, hash table, and parent tree. In addition, there are five

primary arguments: n_col, n_row, Alen, A, and p. Additionally, the user may pass a

sixth argument, knobs. 'I hl arguments are described below:


nrow: number of rows in the matrix.


n_col: number of columns in the matrix.


A: array of row indices of the matrix plus additional storage.


Alen: length of A in words.


p: array of column start pointers into A. llr row indices of column j are located in

A[p(j),... ,p(j + 1) 1]. I Ir number of entries in the matrix A is nnz = p(n_col).

knobs: array of optional user settings to control dense row and column percentages










I Ir row and column structs contain all information relating to rows and columns

respectively. I Iir index array, which is essentially the forward section of A, contains the row

indices and column indices for the column form and the row form of the matrix respectively.

I row structure, column structure, and index array are allocated from the argument A.

1 Ir degree list is a priority queue structure containing column indices. It is keyed by the

column degree. I I hash table is used for supercolumn detection. It shares the same array

space with the degree list. Special provisions are made to deal with collisions between

the degree list and the hash table. I Il parent tree is actually embedded in the column

struct. It is used by non-principal columns to indicate the principal supercolumn. After

the ordering of all supercolumns, the orderchildren routine uses the parent tree to order

all non-principal columns.

I Il total memory requirements in integers for COLAMD is




2 x (nnz) for A

+((ncol + 1) x 6) for the column struct

+((n_row + 1) x 4) for the row struct

+n_col for the minimum elbow room to guarantee success

+20 for output statistics

COLAMD makes use of shared variables to save space. A shared variable is one

that has multiple names and is used in multiple different contexts. However, the shared

variable takes the space of one variable. A detailed description of each data structure

follows. Additionally, since the implementation of COLAMD is in C, all arrays are zero-

based instead of ones-bases as in the pseudo-code.
4.1.1 Row Structure

I lr row struct contains all information used to describe any row in A. I I, first

is the start index which indicates the start of the row in the index array A. Second, is

the length value which indicates the number of column entries in this row. I Il first










shared variable in the row struct is either the row degree or a pointer value p. I Ir degree

indicates the number of original un-eliminated columns of A that are remaining in the

row. I Ij, pointer p is used during initialization when building the row form of the matrix.

Fin..ll,, we have another shared variable known as either mark or firstcolumn. Mark is

used during various phases to indicate that a given row has been seen. It is also used to

store the row set differences during the approximate degree computation phase, w(i) in

Algorithm 1. Firstcolumn is used for the garbagecollection routine (see Section 4.2.5)

and is used to temporarily store the first column index for the row.
4.1.2 Column Structure and Parent Tree

I Ij, column struct is very similar to the row struct. Like the row struct, there are

start and length which are identical to the row struct's usage. Additionally, there are

a few shared variables. 'I Il first is a next pointer which is used for both the hash table

and degree list data structures. I Irj second shared variable is either thickness or parent.

Before a column is ordered or absorbed by a parent supercolumn, it is itself a supercolumn

with thickness given by thickness. If a column is absorbed into a supercolumn, parent

indicates the parent of the column for the parent tree data structure. I l. third shared

variable is either score or order. Before a column is eliminated, we maintain the column

score here. After elimination, the order of the column is maintained with order. I Il

last shared variable for the column struct is used as the hash value during supercolumn

detection, the previous pointer in the degree list, and for a special purpose called headhash.

All columns in the degree list maintain a previous pointer to indicate the column ahead of

it in the list. Any columns undergoing re-scoring are not in the degree list and therefore can

use this space to record the hash value for supercolumn detection. 'I l- special purpose of

headhash is a result of the fact that the degree list and the hash table are both constructed

on the same array. In the event that a column being put into the hash table collides with

a column at the head of a degree list, the column at the head of a degree list has no need










of its previous pointer and can therefore use this space to point to the hashed column. A

similar degree list is used in MA27 [10].
4.1.3 Index Array

'I Ij index array A is simply an array of size 2 x nnz+n_col. I 11j extra room, n_col

is given as working space and is used by the ordering phase of the algorithm to create new

merged rows. 1 Ir array holds the column form of A, (Cj for all j), followed by the row

form of A, (Ri for all j). If the space following the row form is insufficient to create a new

merged row, garbage collection is performed to reclaim unused space.
4.1.4 I l Degree List and Hash Table

I I, algorithm uses an array of length n_row + 1 for the head array of both the

degree list and the hash table. I Ir -, data structures share the array. I hl shared variable

headhash in the column structure is used to handle collisions between the hash table and

the degree list. I I memory for the this array is given by the user in the form of the column

pointer array. ,I I, degree lists are doubly linked to enable arbitrary adding and removal of

columns. llr hash lists are singly linked.

4.2 I ll Algorithms

I Ir, are four main routines used by COLAMD. I r -I are all called from the

main routine, which is user callable. I l algorithm first initializes all the data structures

and performs input error checking in the initrowscols routine. 'I hl initscoring routine

performs pre-elimination on both null and dense rows and columns. I Ii,- is then followed by

initial column scoring, also in the initscoring routine. I hl findordering routine performs

most of the work. It is here that all the supercolumns are ordered. I l orderchildren

routine simply orders all the non-principal columns absorbed into supercolumns.

In addition, two more routines are called by from the findordering routine to per-

form garbage collection and supercolumn detection. I Ir -I are the garbagecollection and

detectsupercols routines respectively.










4.2.1 1: i.i Routine

I hl user first calls COLAMD and passes the arguments n_col, n_row, Alen, A,

p, and knobs.

After some preliminary input error checking, the main routine then calls the four

main routines in sequence. During the execution of these codes, the input arguments A

and p are destroyed. Additionally, the argument A is overwritten with output statistics for

the user. I l- -i are: the number of dense/null rows found; number of dense/null columns

found; the number of garbage collections performed; and the state of the input argument A.

I Irj input argument A can have one of the following states and still be valid: all row indices

in a column are in sorted order and non-repeating; there are unsorted and/or repeated

row indices in one or more of the columns. 'Ill latter state requires a more intensive

initialization to correct.
4.2.2 Init-RowsCols

I hl initialization section sets up the column and row data structures and also

checks for column pointers and row indices out of range and decreasing column pointers

in p. If one of these is found, the initialization returns an error code. Following this step,

the row form of the matrix is generated. During this section, it is determined whether or

not the columns are jumbled. Columns are jumbled if any of the row indices within any

of the columns are repeated or non-increasing. If the columns are jumbled, an additional

phase must be execute to re-build the column form. I Ii-, section has time complexity

O(n_col + nrow + nnz).
4.2.3 InitScoring

'I hl initial scoring phase is divided into several steps. I Ir -i are dense/null column

and row removal, initial score computation, and column insertion into the degree lists.

I l first step orders any empty columns last. I I!- is done so that in the event that

a singular matrix is given to COLAMD, LU factorization can proceed as far as possible.










i :t.I, dense columns are ordered. 'I Ir are ordered between the null columns and the non-

dense columns. Dense columns are determined either by the default or the user provided

dense column knob setting. Since the row degrees for each row in a dense column need to

be reduced, the time complexity is (|Cj ) for each dense column j removed. I In!- could

result in a total of O(nnz) time if the entire matrix contained dense columns, but then all

work would be complete and the algorithm would be done. El i iij. t ii j dense rows and

columns can significantly decrease run times. Following the dense/null column ordering,

dense and null rows are simply removed from the data structure. Again, dense rows are

determined either by the default or the user provided dense row knob setting.

VN :.I the initial column scores are computed for the remaining columns. For initial

scoring, the loose approximate degree used in [25] is used. It has been shown in Section 3.2

that this is actually better than using the tighter approximate degree of Amestoy Davis,

and Duff [1]. To do this, the algorithm simply traverses each remaining column and sums

up the external row degrees for rows contained in that column. I I. external degree is the

row degree minus the supercolumn thickness, which is equal to one during initial scoring.

During this initial scoring, the columns are pruned of any dead rows which may have been

left from the initial dense and null row elimination. After each column score has been

calculated, any .. .i 1.- ..,I11 columns can be detected and eliminated from the data-structure

and ordered as late as possible. N\ v. l;.-ijill columns are those that contained only dense

rows. I In, entire step has time complexity of e(nnz).

Fi ij..1, all the remaining columns are placed into the degree lists according to their

scores. Ir .1 are place into the lists in natural order to preserve natural order tie-breaking.
4.2.4 FindOrdering

I In! is the main phase of the entire algorithm. It is basically one large loop over

each remaining column in the degree lists. However, for each iteration, there are several

sections which perform specific tasks. 'I Ir first section selects a pivot column of minimum

score from the degree lists and orders it. I ln!- is followed by a section which creates a










new merged pivot row composed of all the rows in the pivot column. A garbage collection

call may be required prior to row merging. Thirdly, set differences with the pivot row are

computed for each row in columns ;.1.... H t to the pivot column. I I!- section also performs

:I.._ --;. absorption. I Ij, forth section does a sum of the set differences. Fifth, further

mass elimination is done. i :\.t, supercolumn detection is done. Fin.l11y, column scores are

finalized for all remaining columns which are ;i1ji .... it to the pivot column. 'I I -i sections

will be described in detail below.

In the first section, the supercolumn with the minimum score is removed from the

degree list and ordered. 'I I!j- is the pivot column. Since this supercolumn may have children

columns, the order counter is incremented by the column thickness. I I!,i lets the children

of this supercolumn be ordered later during the order children phase. I I!-! first section is

the mass elimination step.

I Ij, row merging section works similar to symbolic factorization code of George and

Ng [20]. In this section, the row patterns of all rows in the pivot column are merged into

a new pivot row. I I!j- is done by traversing each row in the pivot column. Each unvisited

column index in the rows are added to the pivot row. As columns are added to the pivot

row, they are also t.i .1 1 as visited to prevent repeated column indices from being inserted.

I I pivot row is built onto the end of the existing rows in A and takes the name of the first

row in the pivot column. After the pivot row formation, all other rows in the pivot column

are eliminated. Before the pivot row is created, a check is performed to ensure that there

is enough space for it. If there is not enough space available after the last row to append

the new pivot row, the garbagecollection routine is called.

I Ij next section of the supercolumn ordering phase computes the size of the row

set differences with the pivot row for all rows ;l.11i. i t i to the pivot row. I I!-i is the core of

the algorithm which results in the approximation of the degree and fast run times. Each

column in the pivot row is scanned. Whenever an unvisited row is found, we initialize the

set difference to its degree and subtract the thickness of the column from the set difference.










If the row has already been seen, we simply subtract the thickness of the column from the

set difference. After all columns have been scanned, it is easy to verify that all the row set

differences will have been calculated. Additionally, if any set differences become equal to

zero, that row is a subset of the pivot row and therefore can be eliminated from the data

structure. I I, i is known as -i,, .-. absorption. Aggressive absorption will speed up the

algorithm and also result in more accurate degree approximations during subsequent pivot

selections.

I I' next section scans each column in the pivot row once again, this time summing

the row set differences and also pruning the columns of ;. i 1.--.1 l, absorbed rows. In

preparation for supercolumn detection, the sum of the row indices hashj = E1EC i for

each column j E Rr, is computed and used in the hash function, hashj = hashj mod n_col.

I l columns are placed into the hash table based on its hash value.

At this point, if any columns have a sum of row set differences equal to zero, they

may be ordered and eliminated immediately. I Ir pivot row's degree must also be reduced

by the thickness of this supercolumn. I I!-, is known as further mass elimination and

columns to which this happens are essentially supernodal with the pivot column. If further

mass elimination were not done, these columns would be ordered next anyway. I I! simply

eliminates some overhead.

V' :%.I, supercolumn detection is performed by calling the detectsupercols routine.

'I hl last section of the supercolumn ordering phase is the final score computation.

Fij.il scores are calculated by simply adding the pivot row degree to the sum of set dif-

ferences and then subtracting the supercolumn thicknesses. I Ihi- results in the external

degree. I I!-i final scoring must wait until this point because of two reasons. Supercolumn

detection may increase supercolumn thicknesses and further mass elimination may reduce

the pivot row degree.










4.2.5 GarbageCollection

Garbage collection is called by the findordering routine when there is insufficient

room to create a new merged row. Garbage collection looks at all live columns in the data

structure and moves them contiguously to the front of A. I !, can be done by looking at

columns 1 through n_col since the physical order of the columns remains unchanged as the

COLAMD proceeds. However, because the physical location of rows does not necessarily

correspond to the row's index, the ones compliment of the row number in each alive row is

stored in the place of the first column index. I Ih first column index is stored temporarily

in the row structure's firstcolumn variable. A is then searched for any negative values.

When found, this marks the start of a row and compaction can start there. Since the user

gives the algorithm at least n_col extra elbow room to create the first pivot row, after

garbage collection, there is guaranteed to be enough space for any subsequent pivot row by

I Ih.i i i 2. I h- time complexity for garbage collection is O(Alen). I Ih garbage collection

strategy used in COLAMD is similar to that used by (MA27) [9] and AMD (MC47) [1].
4.2.6 DetectSuper_Cols

Supercolumn detection is called by the findordering routine following the summing

of row set differences for each column in the pivot row. It is performed by looking at each

column in the pivot row. Given a column, first it is made sure that it has not been absorbed

into another supercolumn. If it is unabsorbed, then we look at the hash list given by the

column's hash number. All columns in this hash list are then pair-wise compared with each

other. Any two indistinguishable columns are merged into one supercolumn. I In!- proceeds

until the hash list has been completely searched. I Ii ni the remaining columns in the pivot

row are checked. N\ it that after the first column in the pivot row has been checked, the

algorithm may find already absorbed columns which it will skip over. I I. worst case time

complexity for supercolumn detection is O(nnz x n_col). But this would only occur if a

large number of columns were in the pivot row and happened to have the same hash value

but were not indistinguishable to each other.










4.2.7 Ordi. '!j;Il.l ii

'I h last phase of COLAMD is the ordering of absorbed children columns. Super-

columns are maintained in a parent tree data structure. Each non-principal column has

a single pointer to the parent column that absorbed it. I I !I 1l!, to order the children,

all columns are scanned. If an unordered child column is encountered, two loops are per-

formed. 1 Irj first traverses the parent pointers until it finds a principal column. \i.1 ,,

this is the supercolumn parent column and it has already been ordered. I hl order of this

supercolumn parent is recorded and the second loop begins. On the second loop, the tree

is collapsed by making the supercolumn parent the parent of all nodes in the path. Each

child on the path is given an order number starting with the supercolumn parent's order.

I I!-i proceeds until an ordered non-principal column or the supercolumn parent is reached.

At this point, the supercolumn parent is reordered to one plus the largest order of all its

children. I I!,- algorithm runs in time complexity O(n_col) because of the collapsing step.
















CHAPTER 5
PERFORMANCE I 'S IS


I Ir following sections present the results of COLAMD on three sets of test matri-

ces: square unsymmetric, rectangular, and symmetric. I Ih -, are abbreviated as the RUA,

RRA, and the SYM matrices respectively following the Harwell-Boeing naming convention.

See http://www.dci. c1rc.ac.uk/Activity/HSL for information on the Harwell Subrou-

tine Library. COLAMD is compared against COLMMD and AMDBAR. I I. version of

COLAMD used in these experiments is 1.1. 1 Ihr only difference between versions 1.0 and

1.1 is the correction of dense row and column dropping in SYMAMD. As mentioned be-

fore, COLMMD is the column minimum degree algorithm used by MATLAB. By default,

it performs supercolumn detection and elemental absorption every three iterations. Ad-

ditionally, it uses an approximate degree described in Section 2.9 which is not as tight as

that used by COLAMD. It also uses multiple elimination. In order to achieve larger sets of

independent pivots, COLMMD creates an elimination set containing columns with degrees

within a threshold of 1.2 d + 1, where d is the minimum degree [25]. AMDBAR is a

symmetric minimum degree algorithm. It is similar to MC47BD in the Harwell Subroutine

Library. MC47BD and AMDBAR are described in the paper by Amestoy, Davis, and Duff

[1]. Features of MC47BD include the use of a quotient graph, the same approximate de-

gree used by COLAMD, supernode detection, and;._- --;,. absorption. AMDBAR is the

same as MC47BD except that it lacks I,--. --;. absorption. MC47BD is only used in the

symmetric experiment.

I Ir, three experiments were done on a Sun Ultra Enterprise 111i 1/5000 with 2 GB

of main memory and 5.5 GB of virtual memory. I I!- machine has eight 248 MHz SUNW,

UltraSPARC-II CPUs (Only one processor was used). All of the experiments were run










through MATLAB using mexfunction interfaces. AMDBAR and MC47BD are written in

FORTRAN, the other codes are in C.

Explicit zeros in the Harwell-Boeing form of a matrix are treated symbolically as

non-zeros in the data structures. However, MATLAB drops explicit zeros. To prevent this,

explicit zeros are replaced with the value 10-200, prior to loading the matrix into MATLAB.

5.1 Ilr Square Unsymmetric (RUA) I .t i. I. -,

Fin.liiji; a good column ordering for LU factorization is the primary goal of CO-

LAMD. For these experiments, COLAMD is tested with 275 unsymmetric test matrices

in University of Florida sparse matrix collection at http://www.cise.ufl.edu/-davis.

I I -, matrices range in size from a few hundred to over a million non-zeros. For perfor-

mance and quality comparison, COLMMD and AMDBAR (on A'A explicitly) are also

used on these matrices.

Each matrix was ordered directly by COLAMD and COLMMD. For AMDBAR, the

symbolic pattern of A'A was first formed (in a FORTRAN mexfunction, not in MATLAB)

and then ordered. 1 hr time to form ATA is included in the ordering time for AMDBAR.

Once the ordering was done, LU factorization was performed by the SuperLU package

using C-BLAS. For information about SuperLU see

http://http.cs.berkeley.edu/ xiaoye/superlu.html as well as reference [7].
5.1.1 Result Tables

Fi nii 5.1 shows the matrices in order of increasing best flop count. I Ir, best flop

count is the lowest floating point operation count found with any of the orderings tested.

I Ir vertical line separates big matrices from the smaller ones. A matrix is referred to as

big if it required a minimum of 107 flops or more to factorize. Table 5.1 lists the matrix

number, name, dimension (n), number of non-zeros in the matrix (|A|), the number of

dense rows and columns, and the best flop count. Best flop count refers to the best floating

point operation count in millions for LU factorization resulting from the three ordering

methods. : i. t i I' are numbered by increasing best flop count. In keeping with the default













12




10








04
S8-












2a)
-J








0 50 100 150 200 250 300
Matrices in increasing flop count order


Finii 5.1: Plotting order of the RUA matrices: by best flop count.



behavior of MATLAB's COLMMD function, a dense row or column is considered I, -, "


if it contains n/2 or more nonzero entries.














Table 5.1: General information for the RUA matrices.

no. name n I|A dense dense best flop
rows cols count
1 bliss 7 15 0.000
2 impcolc 137 411 0.000
3 odepb400 399 399 0.000
4 d-ss 53 149 0.001
5 tubl00 100 396 0.001
6 pores1_ 30 180 0.002
7 d dynl 87 238 0.002
8 d-dyn 87 238 0.002
9 olml00 100 396 0.002
10 impcolb 59 312 0.002
11 west0156 156 371 0.002
12 ex5 27 279 6 6 0.003
13 str_0 363 2454 0.003
14 bwm200 200 796 0.003
15 bfwb62 62 342 0.003
16 west0132 132 414 0.003
17 west0167 167 507 0.004
18 odepa400 400 1201 0.005
19 bfwa62 62 450 0.005
20 impcola 207 572 0.005
21 west0067 67 294 0.006
22 ck104 104 992 0.006
23 steam3 80 928 0.007
24 olm500 500 1996 0.008
25 mhdb416 416 2312 0.010
26 impcole 225 1308 0.011
27 tub1000 1000 3996 0.011
28 ck400 400 2860 0.012
29 west0497 497 1727 0.013
30 Ins_131 131 536 0.013
31 lnsp_131 131 536 0.013
32 ck656 656 3884 0.014
33 impcold 425 1339 0.014
34 bp_0 822 3276 0.016
35 olmOO0 1000 3996 0.017
36 fs_680_1 680 2646 0.019
37 fs_680_3 680 2646 0.019
38 fs_680_2 680 2646 0.019
39 rw136 136 479 0.021
40 str_200 363 3068 0.022
41 gre_115 115 421 0.024
42 str_400 363 3157 0.028
43 bwm2000 2000 7996 0.029
44 lopl63 163 935 0.031
45 olm2000 2000 7996 0.033
46 west0989 989 3537 0.036
47 qh882 882 3354 0.037
48 arcl30 130 1282 1 2 0.041
49 fs_183_4 183 1069 2 0.046













Table 5.1: continued.


n IAI dense dense best flop
rows cols count


fs-183_6
fs-183_1
epb0
poli
b2_ss
pde225
fs-183_3
westl505
str_600
saylrl
shl_O
bdyn
pores_3
bp_200
olm5000
west2021
shl_200
gre_185
west0479
polilarge
gre_216b
mcca
bfwa398
bfvb398
gre_216a
nnc261
add32
shl_400
bp_400
extrl
west0655
utm300
fs_541_1

dw256A
fs-541_2
fs-541_4
fs-541_3
mhda416
rw496
exi
bp_800
bp_1600
mahindas
bp_1200
bfva782
bfvb782
bp_1400
west0381
bp_1000
gre_343


183 1069
183 1069
1794 7764
4008 8188
1089 4228
225 1065
183 1069
1505 5445
363 3279
238 1128
663 1687
1089 4264
532 3474
822 3802
5000 19996
2021 7353
663 1726
185 1005
479 1910
15575 33074
216 876
180 2659
398 3678
398 2910
216 876
261 1500
4960 23884
663 1712
822 4028
2837 11407
655 2854
300 3155
541 4285
512 2500
512 2480
541 4285
541 4285
541 4285
416 8562
496 1859
216 4352
822 4534
822 4841
1258 7682
822 4726
782 7514
782 5982
822 4790
381 2157
822 4661
343 1435


no. name


2 0.046
2 0.046
0.047
0.048
0.059
0.060
2 0.060
0.061
0.063
0.065
0.066
0.068
0.080
0.082
0.083
0.087
0.090
0.095
0.100
0.101
0.102
0.106
0.112
0.120
0.128
0.130
0.133
0.137
0.162
0.175
0.196
0.244
1 0.245
0.257
0.261
1 0.266
1 0.267
1 0.270
0.306
0.317
0.356
0.378
0.381
0.385
0.424
0.432
0.435
0.435
0.439
0.445
0.453













Table 5.1: continued.

no. name n IAI dense dense best flop
rows cols count
101 bp_600 822 4172 0.518
102 ,.,-i 1 -, 317 7327 0.624
103 lnsp_511 511 2796 0.630
104 sherman4 1104 3786 0.635
105 hydrl 5308 23752 0.636
106 ., iH -,i 317 7327 0.637
107 Ins_511 511 2796 0.662
108 .,-1 H- 317 7327 0.697
109 pde900 900 4380 0.762
110 ..,-I -I1 317 7327 0.774
111 steam 240 3762 0.812
112 fs-760_1 760 5976 0.836
113 radfrl 1048 13299 0.860
114 cddel 961 4681 0.869
115 cdde2 961 4681 0.869
116 cdde4 961 4681 0.869
117 cdde6 961 4681 0.869
118 cdde3 961 4681 0.872
119 fs-760_2 760 5976 0.891
120 cdde5 961 4681 0.895
121 nnc666 666 4044 0.929
122 '.. .il' 3083 21216 0.957
123 fs-760-3 760 5976 1.033
124 gematll 4929 33185 1.218
125 shermani 1000 3750 1.248
126 gematl2 4929 33111 1.297
127 gre_512 512 2192 1.392
128 ex32 1159 11343 1.521
129 ex33 1733 22189 1.631
130 ex22 839 22715 1.657
131 add20 2395 17319 1.812
132 hor_131 434 4710 1.954
133 rdb968 968 5632 1.996
134 lhrOl 1477 18592 2.265
135 raefsky5 6316 168658 2.271
136 -,. i, 6747 56196 2.507
137 mcfe 765 24382 2.700
138 pores_2 1224 9613 2.824
139 ex2 441 26839 3.163
140 exlO 2410 54840 3.317
141 steam2 600 13760 3.569
142 exl0hs 2548 57308 3.581
143 dwv1024 2048 10114 3.640
144 nnc1374 1374 8606 4.135
145 lhr02 2954 37206 4.792
146 orsirr_2 886 5970 4.827
147 pde2961 2961 14585 5.363
148 rbsa480 480 17088 5.379
149 ',.,..- 13935 63679 5.739
150 ex21 656 19144 5.831













Table 5.1: continued.


no. name n |A| dense dense best flop
rows cols count
151 swangl 3169 20841 5.977
152 swang2 3169 20841 5.977
153 rbsb480 480 17088 6.077
154 rdist2 3198 56934 6.405
155 ex25 848 24612 6.455
156 shyy41 4720 20042 6.651
157 ex23 1409 43703 6.912
158 orsirrl 1030 6858 7.229
159 ',.,..7I 3268 27836 7.813
160 ,.,..1 3268 27836 7.831
161 utml700b 1700 21509 8.021
162 rdb2048 2048 12032 -.338
163 bayerlO 13436 94926 -.381
164 ,.,i'- 1182 32747 -.709
165 orani678 2529 90158 1 8.846
166 ,.-;i7-17 1182 32747 9.100
167 gre_1107 1107 5664 9.269
168 ex29 2870 23754 9.464
169 ,.,-! l- 1182 32747 9.594
170 ex3 1821 52685 10.580
171 bayer06 3008 27576 10.678
172 rdist3a 2398 61896 10.970
173 jpwh_991 991 6027 11.597
174 ex27 974 40782 11.612
175 -,.,. -i~ 3008 27576 11.858
176 cavity06 1182 32747 12.794
177 rdistl 4134 94408 12.910
178 ,. iI,-. 1182 32747 13.173
179 ex24 2283 48737 13.504
180 watt1_ 1856 11360 15.913
181 sherman5 3312 20793 16.346
182 exl3 2568 75628 17.418
183 ex4 1601 32299 17.434
184 lhr04c 4101 82682 17.647
185 watt_2 1856 11550 17.742
186 ex7 1633 54543 17.953
187 ex6 1651 49533 18.009
188 ex36 3079 53843 18.283
189 -...- ,! 20545 159082 21.311
190 ex37 3565 67591 23.133
191 lnsp3937 3937 25407 25.492
192 Ins_3937 3937 25407 25.622
193 ex15 6867 98671 27.514
194 ex9 3363 99471 27.786
195 ex31 3909 115357 28.318
196 memplus 17758 126150 30.421
197 lhr07c 7337 156508 30.477
198 rw5151 5151 20199 30.900
199 orsreg1_ 2205 14133 33.364
200 ex28 2603 77781 34.753













Table 5.1: continued.

no. name n IAI dense dense best flop
rows cols count
201 '..,. i 57735 277774 36.043
202 exl8 5773 71805 36.610
203 lhrlOc 10672 232633 39.261
204 sherman3 5005 20033 39.620
205 exl9 12005 259879 40.749
206 rdb5000 5000 29600 43.590
207 utm3060 3060 42211 43.892
208 sherman2 1080 23094 47.092
209 ex20 2203 69981 47.808
210 cavityl5 2597 76367 48.929
211 cavityl3 2597 76367 51.100
212 lhrllc 10964 233741 52.567
213 cavityll 2597 76367 52.690
214 ex35 19716 228208 56.490
215 cavityl4 2597 76367 58.608
216 ..,i !-'2 2597 76367 59.814
217 cavitylO 2597 76367 59.879
218 saylr4 3564 22316 62.239
219 garonl 3175 88927 66.293
220 ex8 3096 106344 67.019
221 exl2 3973 80211 67.619
222 onetone2 36057 227628 70.626
223 megl 2904 58142 81.516
224 ex26 2163 94033 82.158
225 lhrl4c 14270 307858 85.262
226 ex14 3251 66775 91.247
227 dw4096 8192 41746 100.257
228 lhrl7c 17576 381975 110.645
229 wangi 2903 19093 120.077
230 wang2 2903 19093 120.077
231 utm5940 5940 83842 125.390
232 cavity25 4562 138187 130.392
233 epbl 14734 95053 130.955
234 cavity23 4562 138187 131.415
235 cavity21 4562 138187 132.275
236 .,i -!', 4562 138187 135.153
237 cavityl7 4562 138187 136.366
238 cavity24 4562 138187 169.134
239 cavity26 4562 138187 169.525
240 ,., I-1, 4562 138187 170.006
241 cavity22 4562 138187 170.007
242 cavity20 4562 138187 170.349
243 cavityl8 4562 138187 170.553
244 lhr34c 35152 764014 239.394
245 .,. F-1 C, 3402 137845 254.506
246 meg4 5860 46842 277.625
247 goodwin 7320 324784 497.528
248 lhr71c 70304 1528092 506.509
249 epb2 25228 175027 620.429
250 -1.--1 i 76480 329762 1020.167











Table 5.1: continued.


no.


name


garon2
ex40
graham 1
raefskyl
raefsky2
rim
epb3
onetonel
olafu
rmal0
venkatOl
venkat25
venkat50
af23560
twotone
raefsky3
raefsky4
ex11
psmigr_2
psmigr_3
psmigr_l
wang4
wang3
bbmat
av41092


n IAI dense dense best flop
rows cols count


13535
7740
9035
3242
3242
22560
84617
36057
16146
46835
62424
62424
62424
23560
120750
21200
19779
16614
3140
3140
3140
26068
26064
38744
41092


390607
458012
335504
294276
294276
1014951
463625
341088
1015156
2374001
1717792
1717792
1717792
484256
1224224
1488768
1328611
1096948
540022
543162
543162
177196
177168
1771722
1683902


1061.332
1075.063
1333.270
1681.522
1681.522
2018.664
2057.583
2519.732
2844.220
4041.353
4456.498
4506.216
4509.489
4515.197
8578.428
8589.957
10991.822
11637.984
13941.055
16137.405
16504.110
27740.799
27758.510
44552.892
50140.185


Table 5.2, shows the ordering time in seconds for each method. In addition, A A

time refers to the needed to form ATA before calling AMDBAR. I I- last column lists

the time taken by SuperLU to factorize the matrix using the column ordering found by

COLAMD (excluding the ordering time itself). I I!-' statistic can be used to compare the

time it takes to order the matrix with the time to factorize the matrix. Ordering times

greater than the LU factorization time are underlined. Ilrl last row of this table shows

how many times the ordering time was greater than the factorization time for each method.














Table 5.2: Ordering times in seconds for the RUA matrices.


COLAMD COLMMD AMDBAR /w
ATA time


ATA time


SuperLU time using
COLAMD ordering


name


bliss
impcolc
odepb400
djss
tub100
pores1_
ddynl
d_dyn
olm100
impcolb
west0156
ex5
str_0
bwm200
bfwb62
west0132
west0167
odepa400
bfwa62
impcol_a
west0067
ck104
steam3
olm500
mhdb416
impcol_e
tub1000
ck400
west0497
Ins_131
lnsp_131
ck656
impcol_d
bp_0
olml000
fs-680_1
fs-680_3
fs-680_2
rw136
str_200
gre_115
str_400
bwm2000
lopl63
olm2000
west0989
qh882
arcl30
fs_183_4
fs_183_6
















no. name

51 fs_183_1
52 epb0
53 poli
54 b2_ss
55 pde225
56 fs_183_3
57 westl505
58 str_600
59 saylrl
60 shl_0
61 bdyn
62 pores_3
63 bp_200
64 olm5000
65 west2021
66 shl_200
67 gre_185
68 west0479
69 polilarge
70 gre_216b
71 mcca
72 bfwa398
73 bfwb398
74 gre_216a
75 nnc261
76 add32
77 shl_400
78 bp_400
79 extrl
80 west0655
81 utm300
82 fs_541_1
83 dw256B
84 dw256A
85 fs_541_2
86 fs_541_4
87 fs_541_3
88 mhda416
89 rw496
90 exi
91 bp_800
92 bp_1600
93 mahindas
94 bp_1200
95 bfwa782
96 bfwb782
97 bp_1400
98 west0381
99 bp_1000
100 gre_343


COLAMD


Table 5.2: continued.

COLMMD AMDBAR /w
ATA time


0.00
0.01
0.02
0.01
0.00
0.00
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0.03
0.02
0.01
0.01
0.01
0.06
0.00
0.00
0.00
0.01
0.01
0.01
0.04
0.00
0.02
0.03
0.01
0.01
0.01
0.01
0.01
0.01
0.02
0.01
0.01
0.01
0.00
0.01
0.01
0.06
0.02
0.02
0.01
0.01
0.01
0.01
0.01


ATA time


SuperLU time using
COLAMD ordering


0.01
0.03
0.03
0.08
0.01
0.01
0.06
0.03
0.01
0.00
0.08
0.03
0.02
3.19
0.09
0.00
0.01
0.02
0.20
0.00
0.01
0.03
0.03
0.01
0.01
0.18
0.00
0.02
0.20
0.03
0.01
0.01
0.02
0.01
0.01
0.02
0.02
0.01
0.03
0.01
0.03
0.04
0.20
0.04
0.10
0.08
0.04
0.03
0.04
0.01


0.01
0.01
0.03
0.02
0.00
0.01
0.01
0.01
0.01
0.01
0.02
0.01
0.05
0.03
0.03
0.01
0.00
0.01
0.43
0.00
0.00
0.01
0.00
0.01
0.00
0.06
0.01
0.05
0.05
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.02
0.01
0.01
0.05
0.05
0.07
0.06
0.02
0.02
0.06
0.01
0.07
0.01


0.00
0.01
0.01
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.02
0.02
0.01
0.01
0.00
0.00
0.13
0.00
0.00
0.00
0.00
0.01
0.00
0.02
0.00
0.01
0.01
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.02
0.00
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.02
0.00
0.02
0.00


0.01
0.03
0.04
0.02
0.01
0.01
0.02
0.01
0.01
0.02
0.02
0.02
0.03
0.09
0.03
0.03
0.01
0.01
0.13
0.01
0.01
0.02
0.01
0.01
0.01
0.07
0.03
0.04
0.06
0.02
0.01
0.02
0.02
0.02
0.02
0.03
0.02
0.05
0.02
0.03
0.05
0.04
0.06
0.05
0.03
0.03
0.05
0.02
0.06
0.02














Table 5.2: continued.


name COLAMD


COLMMD


AMDBAR /w
ATA time


ATA time


----- -----------................. J-_________ _. I


1-1'


bp_600

lnsp_511
sherman4
hydrl
S., I I II!
Ins_511

pde900
1. 1 1 2
steam
fs_760_1
radfrl
cddel
cdde2
cdde4
cdde6
cdde3
fs_760_2
cdde5
nnc666
.I 1I ''
fs_760_3
gemat11
sherman1
gemat12
gre_512
ex32
ex33
ex22
add20
hor_131
rdb968
lhrOl
raefsky5

mcfe
pores_2
ex2
exlO
steam2
exlOhs
dw1024
nnc1374
lhr02
orsirr_2
pde2961
rbsa480

ex2 11
ex21


SuperLU time using
COLAMD ordPrino


0.01
0.01
0.01
0.01
0.09
0.01
0.01
0.01
0.01
0.00
0.00
0.02
0.01
0.02
0.02
0.02
0.01
0.02
0.02
0.02
0.01
0.09
0.02
0.05
0.02
0.05
0.01
0.04
0.06
0.02
0.09
0.01
0.02
0.07
0.14
0.24
0.09
0.03
0.02
0.05
0.02
0.05
0.03
0.02
0.14
0.03
0.04
0.04
0.30
0.02


no.


0.03
0.01
0.02
0.05
1.83
0.01
0.02
0.01
0.04
0.00
0.00
0.07
0.01
0.04
0.05
0.04
0.04
0.04
0.08
0.04
0.03
0.49
0.08
0.25
0.05
0.26
0.02
0.83
1.48
0.03
0.20
0.06
0.06
0.58
0.28
1.53
0.57
0.07
0.05
0.21
0.02
0.18
0.12
0.07
1.26
0.10
0.17
0.05
1.38
0.03


0.05
0.02
0.01
0.02
0.15
0.03
0.01
0.02
0.02
0.03
0.00
0.02
0.02
0.02
0.01
0.01
0.01
0.02
0.03
0.02
0.01
0.12
0.03
0.07
0.02
0.06
0.01
0.03
0.05
0.06
0.34
0.02
0.03
0.12
0.55
0.34
0.14
0.03
0.13
0.15
0.03
0.14
0.03
0.02
0.23
0.02
0.05
0.06
0.44
0.06


0.01
0.02
0.00
0.01
0.02
0.02
0.00
0.01
0.01
0.01
0.00
0.00
0.02
0.00
0.00
0.00
0.00
0.01
0.00
0.01
0.00
0.03
0.01
0.03
0.01
0.03
0.00
0.01
0.03
0.05
0.05
0.01
0.00
0.05
0.46
0.08
0.07
0.01
0.12
0.11
0.02
0.12
0.01
0.00
0.08
0.00
0.01
0.04
0.07
0.04


0.05
0.02
0.03
0.04
0.15
0.02
0.03
0.03
0.04
0.03
0.03
0.11
0.06
0.04
0.04
0.04
0.04
0.04
0.06
0.04
0.04
0.11
0.06
0.13
0.05
0.15
0.06
0.08
0.09
0.06
0.17
0.06
0.07
0.11
0.32
0.36
0.14
0.10
0.08
0.17
0.14
0.17
0.15
0.15
0.23
0.15
0.20
0.17
0.50
0.14














Table 5.2: continued.


COLAMD COLMMD AMDBAR /w
ATA time


swangl
swang2
rbsb480
rdist2
ex25
shyy41
ex23
orsirr_l
' .1 i li

utm1700b
rdb2048
bayerlO

orani678

gre_1107
ex29
. i 1 i II ",
ex3
bayer06
rdist3a
jpwh_991
ex27

cavity06
rdist

ex24
watt1_
sherman5
ex13
ex4
lhr04c
watt_2
ex7
ex6
ex36
-,.. .'Ill1
ex37
lnsp3937
Ins_3937
exl5
ex9
ex31
memplus
lhr07c
rw5151
orsreg1_
ex28


0.53
0.53
0.05
0.08
0.03
0.46
0.08
0.08
0.66
0.66
0.28
0.15
4.79
0.05
24.58
0.05
0.09
2.06
0.05
0.12
0.61
0.07
0.09
0.09
0.59
0.04
0.15
0.04
0.68
0.14
0.08
0.15
0.21
4.82
0.19
0.10
0.08
0.52
7.32
0.14
0.33
0.30
7.25
0.18
2.17
2.94
9.48
0.30
0.20
0.15


ATA time


no. I name


0.06
0.05
0.06
0.15
0.07
0.06
0.14
0.03
0.20
0.21
0.06
0.05
0.65
0.10
4.82
0.10
0.04
0.05
0.10
0.15
0.19
0.21
0.03
0.20
0.19
0.10
0.30
0.10
0.14
0.05
0.06
0.23
0.09
0.66
0.05
0.18
0.15
0.15
1.05
0.23
0.11
0.09
0.20
0.29
0.40
21.53
1.22
0.09
0.05
0.24


SuperLU time using
COLAMD ordering
0.27
0.26
0.16
0.31
0.18
0.29
0.23
0.19
0.43
0.38
0.26
0.25
0.63
0.22
0.49
0.23
0.33
0.33
0.24
0.33
0.40
0.44
0.32
0.28
0.41
0.35
0.69
0.36
0.42
0.43
0.51
0.94
0.46
1.67
0.46
0.47
0.49
0.66
1.47
0.63
0.87
0.86
1.09
2.28
0.83
132.43
1.60
1.02
0.82
0.86














Table 5.2: continued.


COLAMD COLMMD AMDBAR /w
ATA time


,. I 1 1 1 1
exl8
lhrlOc
sherman3
exl9
rdb5000
utm3060
sherman2
ex20
cavityl5
cavityl3
lhrllc
cavityll
ex35
cavityl4
S., r 1 I '
cavityl0
saylr4
garoni
ex8
ex12
onetone2
megl
ex26
lhrl4c
ex14
dw4096
lhrl7c
wangi
wang2
utm5940
cavity25
epbl
cavity23
cavity21
i i i I '
cavity17
cavity24
cavity26
i., i y i,
cavity22
cavity20
cavityl8
lhr34c
I .I. .- i ,
meg4
goodwin
lhr71c
epb2
-1. ir.1


16.90
8.46
15.31
0.48
1.25
0.39
0.68
0.04
0.17
0.10
0.09
8.12
0.09
40.13
0.10
0.10
0.10
0.40
0.25
0.21
0.43
5.41
0.69
0.35
12.29
0.53
0.64
15.22
0.27
0.27
1.56
0.21
0.98
0.19
0.20
0.19
0.21
0.21
0.22
0.24
0.21
0.22
0.22
33.82
0.43
5.96
0.42
77.78
2.62
10.68


ATA time


no.


name


2.08
0.16
1.84
0.08
0.73
0.11
0.12
0.06
0.24
0.25
0.25
1.83
0.24
0.50
0.25
0.25
0.24
0.10
0.26
0.39
0.23
0.75
0.45
0.44
2.43
0.20
0.17
3.08
0.08
0.08
0.24
0.46
0.23
0.44
0.46
0.46
0.44
0.43
0.43
0.44
0.45
0.44
0.45
6.13
0.76
1.18
1.51
12.19
0.61
1.27


SuperLU time using
COLAMD ordering
2.60
1.34
2.60
1.03
1.47
1.12
1.11
1.22
1.13
1.18
1.21
3.17
1.25
2.54
1.37
1.38
1.34
1.49
1.63
3.10
1.76
3.23
2.23
1.69
4.19
2.47
2.68
5.80
2.65
2.59
2.95
2.95
3.49
3.00
3.05
3.03
3.11
4.53
4.38
4.52
4.55
4.42
4.56
162.49
6.28
13.76
44.65
278.45
16.32
26.94











Table 5.2: continued.


COLAMD COLMMD A: !I i:.\i /w
ATA time


garon2
ex40
grahaml
raefskyl
raefsky2
rim
epb3
onetonel
olafu
rmal0
venkatOl
venkat25
venkat50
af23560
twotone
raefsky3
raefsky4
ex11
psmigr_2
psmigr_3
psmigr_l
wang4
wang3
bbmat
av41092


0.61
2.34
0.43
1.06
1.11
1.14
1.05
1.18
4.35
2.37
3.85
3.73
3.80
6.64
5.88
2.36
6.42
7.46
4.97
4.79
4.97
0.88
0.76
26.54
4.98


1.19
13.58
0.58
1.60
1.58
1.43
5.10
17.31
87.36
25.71
9.65
8.71
8.57
115.99
70.82
6.25
21.76
45.04
34.04
33.17
33.82
5.15
5.32
1378.04
64.73


ATA time


1.29
2.61
1.50
2.09
2.11
4.73
1.30
2.21
4.93
12.07
4.42
4.54
4.33
1.72
14.15
7.32
7.45
6.26
20.66
21.07
21.19
0.97
0.96
16.75
128.60


Underline counts 12 94 35 1


no.


Table 5.3, shows the number of non-zeros in the LU factors for each ordering

method. Results larger than 1.2 times the best result for each matrix are underlined.

I I, last row indicates for each method how many times IL + UI was 1.2 times larger than

the best result. In the last column, AT AI refers to the number of non-zeros in both the up-

per and lower triangular halves of ATA after dense rows and columns have been removed.

I In!- gives a good measure of the memory requirements for a symmetric ordering code such


as AMDBAR.


name


1.06
2.22
1.25
1.90
1.92
4.09
0.42
1.04
4.46
10.86
3.57
3.69
3.50
0.86
3.93
6.84
6.58
5.42
17.50
17.89
18.03
0.19
0.17
9.81
114.91


SuperLU time using
COLAMD ordering
25.12
27.21
31.63
63.02
62.36
134.19
54.63
67.84
76.94
120.67
116.21
115.73
116.68
108.35
306.85
254.02
274.11
287.00
350.29
360.89
369.70
761.56
633.63
1191.33
1624.65








52





Table 5.3: |L + U for the RUA matrices.

no. name COLAMD COLMMD AMDBAR IATAI
1 bl1ss 49 44 31 25
2 impcol-c 636 1052 453 1077
3 odepb400 399 399 399 399
4 dss 372 378 354 331
5 tub100 615 616 516 784
6 pores1_ 344 461 351 388
7 ddynl 608 558 664 541
8 d_dyn 608 559 664 541
9 olml0 712 777 711 976
10 impcolb 554 614 542 717
11 west0156 914 860 895 836
12 ex5 351 447 447 177
13 str_0 3081 3127 3092 7123
14 bwm200 1147 1191 1159 1584
15 bfwb62 598 712 740 992
16 west0132 941 1040 964 1252
17 west0167 1120 1228 1231 1545
18 odepa400 2020 2805 2409 2006
19 bfwa62 897 750 900 1340
20 impcol-a 1357 1504 1431 1219
21 west0067 878 918 923 889
22 ck104 1388 1238 1096 1712
23 steam3 1132 1205 1084 1504
24 olm500 3512 3777 3511 4976
25 mhdb416 2756 3124 2756 4224
26 impcole 2466 2331 2271 3983
27 tublO00 6015 5900 5016 7984
28 ck400 3033 3631 3065 5148
29 west0497 3891 3703 4022 9711
30 Ins_131 1638 1588 1652 2167
31 lnsp_131 1627 1740 1597 2167
32 ck656 4057 4901 4089 6940
33 impcol-d 3908 4693 3105 4011
34 bpO 6302 10852 6204 77322
35 olm00O 7012 7527 7011 9976
36 fs-680_1 4257 5338 4267 6178
37 fs_680_3 4257 5338 4267 6178
38 fs_680_2 4257 5338 4267 6178
39 rw136 2136 2256 2208 1418
40 str_200 5147 4986 5037 8331
41 gre_115 2384 2424 2081 1267
42 str_400 6260 5558 5797 8363
43 bwm2000 11639 11708 11644 15984
44 lopl63 3198 3971 3418 3397
45 olm2000 14012 15028 14011 19976
46 west0989 9972 9579 8637 12235
47 qh882 7803 7755 7702 9404
48 arcl30 2861 3045 3927 9676
49 fs_183_4 5665 7990 3624 11175
50 fs_183_6 5690 7990 3632 11175














Table 5.3: continued.

no. name COLAMD COLMM\ D AMDBAR |ATAI
51 fs_183_1 5690 7990 3632 11175
52 epb0 14755 13828 16193 28632
53 poli 17156 18569 18672 62578
54 b2ss 11385 12714 11283 13457
55 pde225 4753 5449 5163 2629
56 fs-183_3 6004 7990 4051 11175
57 .- .11 i 15195 15225 13643 18863
58 str_600 6867 7556 7147 8869
59 saylrl 5141 5760 5392 2788
60 shlO0 12585 9532 13283 5925
61 bdyn 12217 13239 12123 13461
62 pores_3 8950 13277 9501 8356
63 bp_200 17256 24114 11211 88234
64 olm5000 35012 37527 35011 49976
65 west2021 20344 20049 18615 25527
66 shl-200 17847 13618 17960 4153
67 gre_185 5889 5849 5516 4013
68 .-,m 8504 8549 7528 7143
69 polilarge 49231 48968 49859 1286265
70 gre_216b 5781 6764 6849 3016
71 mcca 5299 4930 5081 9820
72 bfwa398 9194 10497 9494 13768
73 bfwb398 9124 9222 9352 11026
74 gre_216a 6558 7919 7341 3016
75 nnc261 7965 8842 8309 5853
76 add32 34400 42226 35670 102422
77 shl_400 15609 20478 15620 3865
78 bp_400 22477 22215 14343 95908
79 extrl 35788 42768 31490 36085
80 .i,,,, 14077 14561 12754 11327
81 utm300 10405 10222 10805 14832
82 fs-541_1 16005 15491 15399 12925
83 I- ,i; 13610 14650 14343 6572
84 (1- ',.\ 13843 15174 14472 6408
85 fs-541_2 17037 16263 16282 12925
86 fs-541_4 17271 16546 16332 12925
87 fs-541_3 17571 16914 16370 12925
88 mhda416 30540 22802 16505 22548
89 rw496 14558 16347 16408 5768
90 exi 15529 16358 11956 19860
91 bp_800 26765 30072 21952 103950
92 bp_1600 24071 25820 22120 106338
93 mahindas 33875 31299 24347 40338
94 bp_1200 26453 31295 23625 109180
95 bfwa782 24465 25465 25530 29148
96 bfwb782 23948 25026 24060 23674
97 bp_1400 29203 34162 23886 109810
98 west0381 17152 17719 16952 13509
99 bp_1000 28880 28853 23132 107374
100 gre_343 14485 17067 14877 5095








54





Table 5.3: continued.

no. name COLAMD COLMMD A:! i:.\1i, ATA
101 bp_600 25291 27964 22877 100466
102 ..,-1i -, 18579 21466 20527 25143
103 lnsp-511 20589 20416 20605 11277
104 sherman4 23935 26541 25424 10346
105 hydrl 81465 86373 75132 90090
106 ., -i- ii 20828 21269 23217 34369
107 Ins_511 20904 22665 21028 11277
108 ,.,-1 1! 19368 21578 21122 25143
109 pde900 31030 36686 33760 11104
110 ,., I -i-2 23232 21466 22771 34369
111 steal 18819 19590 18387 10440
112 fs-760_1 33981 38701 32359 27152
113 radfrl 44393 45457 43979 62520
114 cddel 34013 36733 34745 11877
115 cdde2 34013 36733 34745 11877
116 cdde4 34013 36733 34745 11877
117 cdde6 34013 36733 34745 11877
118 cdde3 34054 36762 34806 11877
119 fs-760_2 35813 40485 33557 27152
120 cdde5 34354 37054 34974 11877
121 nnc666 32717 34730 32942 16518
122 -..,.i'' 69968 76623 73383 69987
123 fs-760_3 38048 42076 36271 27152
124 gematll 91398 91305 86925 83627
125 shermani 32692 36880 37376 9906
126 gematl2 95231 96790 89896 85271
127 gre_512 30030 36128 30392 7960
128 ex32 55069 70034 58734 39075
129 ex33 70366 92954 77149 60783
130 ex22 55571 81918 62548 88349
131 add20 62743 47379 61433 213657
132 hor_131 36655 37943 38489 16574
133 rdb968 52831 59632 55136 16376
134 lhrOl 66424 70458 67701 71487
135 raefsky5 237217 243810 247838 639174
136 '..1. i, 190017 221503 182029 189325
137 mcfe 61788 68288 69828 145741
138 pores_2 57148 66800 57976 24220
139 ex2 51581 64714 55432 82953
140 exlO 118572 162866 143235 193934
141 steam2 63552 65176 62824 39872
142 exIOhs 129868 188391 159644 205230
143 dwv1024 93816 100816 95504 26458
144 nnc1374 98568 104298 96361 36080
145 lhr02 138000 147451 140879 143106
146 orsirr_2 76537 103748 81517 20850
147 pde2961 145291 169779 139729 37397
148 rbsa480 70511 83651 72084 43680
149 '..,- !-'2 305721 361220 348254 264225
150 ex21 95230 103196 89973 78258








55





Table 5.3: continued.

no. name COLAMD COL1MMD AMDBAR iATAi
151 swangl 159239 174819 173953 55439
152 swang2 159239 174819 173953 55439
153 rbsb480 72476 84458 73314 43680
154 rdist2 204353 214773 205009 281638
155 ex25 110226 133484 108428 102326
156 shyy41 172355 204956 189345 53296
157 ex23 146085 262332 206019 188523
158 orsirrl 98266 104431 100798 23532
159 '..i ,7: 168960 170699 167313 131080
160 ,.-. i'I 169108 169815 167331 131080
161 utml700b 132873 159540 139852 91888
162 rdb2048 147732 163872 154408 35336
163 bayerlO 386345 469624 398916 349172
164 ., i 1-' 138578 172770 152432 132218
165 orani678 189137 280892 196900 1858019
166 ,., 1-i7 141088 175106 151631 132218
167 gre_1107 122892 173775 116650 24617
168 ex29 188686 195828 201204 64250
169 .,!1 I' 146583 179586 152825 132218
170 ex3 186188 221961 179357 195319
171 bayer06 188205 220543 203743 129540
172 rdist3a 232751 264494 231164 320386
173 jpwh_991 114617 133856 112813 25141
174 ex27 158434 260697 171444 275714
175 '..1 -i~ 193788 221166 205392 129540
176 cavity06 195827 198924 170066 152964
177 rdistl 359696 334494 367530 491294
178 .. -,,-I-. 190151 193711 173958 152964
179 ex24 252236 270219 259504 236847
180 watt1_ 196228 217882 213683 37964
181 sherman5 208481 246308 220894 89766
182 exl3 393636 400076 288651 279884
183 ex4 226336 285081 252727 164063
184 lhr04c 337444 343627 349635 317755
185 watt_2 208429 262038 212010 53032
186 ex7 227451 334395 248715 259351
187 ex6 270296 348790 256734 210167
188 ex36 330942 402161 294795 246825
189 '-.. .il 698335 748277 694805 534741
190 ex37 282071 350105 286111 318931
191 lnsp3937 375448 436737 359581 101673
192 Ins_3937 379742 449838 361198 101673
193 ex15 586909 790511 733994 266805
194 ex9 754621 547608 402481 373815
195 ex31 458608 888472 592419 593791
196 memplus 4377196 1236504 337222 5121784
197 lhr07c 624099 639312 635510 591387
198 rw5151 386950 461767 370984 64743
199 orsreg1_ 312458 404990 309461 46199
200 ex28 435826 533590 440078 332893








56





Table 5.3: continued.

no. name COLAMD COLMMD AMDBAR IATAI
201 '..1- i 1475487 1502572 1536964 1112589
202 ex18 541920 630967 591726 197145
203 lhrl0c 876305 919693 887950 873494
204 sherman3 363167 435065 375799 59909
205 ex19 898728 1669915 1179316 981383
206 rdb5000 497684 520445 509615 87608
207 utm3060 414242 503145 432463 183900
208 sherman2 289729 326604 305781 102050
209 ex20 482733 589026 463906 321807
210 cavityl5 548519 657176 540928 356209
211 cavityl3 565031 662825 538338 356209
212 lhrllc 964854 1116299 991684 883120
213 cavityll 578056 663739 540052 356209
214 ex35 1375694 1821542 1734003 626212
215 cavityl4 533338 653326 529042 323943
216 ..,- i-!' 536898 658807 526332 323943
217 cavitylO 538325 661064 524146 323943
218 saylr4 527436 660710 532744 74020
219 garonl 622238 678997 696572 325469
220 ex8 837712 712014 596946 510648
221 ex12 746286 906259 876842 414979
222 onetone2 1117503 1080520 1440847 669941
223 megl 519679 525990 663417 353038
224 ex26 632643 1131755 815884 510493
225 lhrl4c 1386569 1473383 1412266 1159196
226 ex14 710596 1011636 885033 347447
227 dw4096 747725 851444 827031 115248
228 lhrl7c 1715351 1766685 1717306 1435272
229 wangl 632073 868435 751829 64165
230 wang2 632073 868435 751829 64165
231 utm5940 979553 1129948 1037466 366860
232 cavity25 1145999 1347320 1210547 644104
233 epbl 1333787 1330392 1317086 246972
234 cavity23 1151199 1356755 1215508 644104
235 cavity21 1156841 1350570 1207901 644104
236 .., i' !' 1178772 1345281 1203062 644104
237 cavityl7 1187870 1350945 1205531 644104
238 cavity24 1335292 1302929 1140809 600318
239 cavity26 1333579 1304347 1141933 600318
240 ,.,-l-i 1338825 1303090 1140943 600318
241 cavity22 1336107 1305118 1142898 600318
242 cavity20 1334587 1298926 1145103 600318
243 cavityl8 1341500 1301357 1144088 600318
244 lhr34c 3555574 3805554 3573572 2870928
245 .,i.F-1 -C 1140523 1531582 1179263 836398
246 meg4 1569915 1238467 1144374 1954494
247 goodwin 5633883 3103015 2733888 1776000
248 lhr71c 7244733 7680840 7271621 5742240
249 epb2 3185544 3547155 3135280 510808
250 -1 -- il 6004927 7576106 6357911 885136








57


Table 5.3: continued.

no. name COLAMD COLMMD A Il :.\i, IATA
251 garon2 5178501 5120230 5127587 1464469
252 ex40 4315377 8537136 6071791 2103566
253 graham 4920087 5344248 4517973 1877147
254 raefskyl 3683048 3236036 2877184 1184340
255 raefsky2 3683048 3236036 2877184 1184340
256 rim 16529429 12530502 9551589 5592480
257 epb3 10151996 10996648 12058221 1345335
258 onetonel 4816029 5593721 4660007 1332757
259 olafu 8569958 9925890 7959855 3388252
260 rmal0 18539696 17544139 15623502 7900917
261 venkatOl 20937280 18739046 17948698 4619968
262 venkat25 21094556 18848464 18101891 4619968
263 venkat50 21106152 18855346 18155863 4619968
264 af23560 12110330 13333424 13130768 2443568
265 twotone 21030267 19623603 21279596 16477870
266 raefsky3 19934416 17528630 14868748 4053376
267 raefsky4 18945706 23751228 17864342 5342569
268 exll 18971639 25935109 18230169 4517874
269 psmigr_2 8643563 8062484 9074980 8549828
270 psmigr_3 8664945 8637893 9156462 8551606
271 psmigrl 8699394 8720579 9172728 8551606
272 wang4 26194516 41519786 24802508 615132
273 wang3 24419818 36879928 26827262 615022
274 bbmat 49833462 88433544 53266851 13303596
275 av41092 43595556 49461532 36514359 27002406
Underline counts 29 97 18




Table 5.4, shows the floating point operation counts or flop counts in millions for

all methods. I Ir flop count is a good measure of the work required to factorize a matrix.








58





Table 5.4: Flop counts in millions for the LU factorization of A.

no. name COLAMD COLMMD AMDBAR
1 bliss 0.000 0.000 0.000
2 impcolc 0.001 0.004 0.000
3 odepb400 0.000 0.000 0.000
4 d(ss 0.001 0.001 0.001
5 tub100 0.002 0.002 0.001
6 pores1_ 0.002 0.004 0.002
7 ddynl 0.002 0.002 0.003
8 d_dyn 0.002 0.002 0.003
9 olml0 0.002 0.002 0.002
10 impcolb 0.003 0.003 0.002
11 west0156 0.003 0.002 0.003
12 ex5 0.003 0.004 0.005
13 str_0 0.003 0.003 0.003
14 bwm200 0.003 0.003 0.003
15 bfwb62 0.003 0.004 0.005
16 west0132 0.003 0.004 0.003
17 west0167 0.004 0.004 0.005
18 odepa400 0.005 0.010 0.006
19 bfwa62 0.008 0.005 0.007
20 impcol-a 0.005 0.006 0.005
21 west0067 0.006 0.006 0.006
22 ck104 0.009 0.007 0.006
23 steam3 0.008 0.009 0.007
24 olm500 0.008 0.010 0.008
25 mhdb416 0.010 0.014 0.010
26 impcole 0.011 0.011 0.011
27 tub1000 0.015 0.015 0.011
28 ck400 0.012 0.018 0.013
29 west0497 0.021 0.013 0.020
30 Ins_131 0.014 0.013 0.014
31 lnsp_131 0.014 0.014 0.013
32 ck656 0.014 0.022 0.015
33 impcol-d 0.026 0.040 0.014
34 bp-0 0.020 0.082 0.016
35 olmOO0 0.017 0.020 0.017
36 fs_680_1 0.019 0.033 0.019
37 fs_680_3 0.019 0.033 0.019
38 fs_680_2 0.019 0.033 0.019
39 rw136 0.021 0.023 0.023
40 str_200 0.026 0.022 0.026
41 gre_115 0.031 0.034 0.024
42 str_400 0.052 0.028 0.041
43 bwm2000 0.029 0.030 0.029
44 lop163 0.031 0.052 0.036
45 olm2000 0.033 0.040 0.033
46 west0989 0.053 0.048 0.036
47 qh882 0.038 0.040 0.037
48 arcl30 0.041 0.050 0.100
49 fs_183_4 0.149 0.329 0.046
50 fs_183_6 0.151 0.329 0.046








59





Table 5.4: continued.

no. name COLAMD COLMMD AMDBAR
51 fs_183_1 0.151 0.329 0.046
52 epb0 0.051 0.047 0.068
53 poli 0.048 0.054 0.054
54 b2_ss 0.059 0.078 0.060
55 pde225 0.060 0.079 0.070
56 fs_183_3 0.165 0.329 0.060
57 westl505 0.076 0.081 0.061
58 str_600 0.063 0.086 0.076
59 saylrl 0.065 0.087 0.072
60 shl_0 0.093 0.066 0.116
61 b-dyn 0.068 0.087 0.070
62 pores_3 0.080 0.195 0.093
63 bp_200 0.270 0.569 0.082
64 olm5000 0.083 0.100 0.083
65 west2021 0.101 0.121 0.087
66 shl_200 0.178 0.090 0.197
67 gre_185 0.109 0.112 0.095
68 west0479 0.125 0.137 0.100
69 polilarge 0.103 0.101 0.107
70 gre_216b 0.102 0.139 0.150
71 mcca 0.110 0.119 0.106
72 bfwa398 0.112 0.160 0.122
73 bfwb398 0.120 0.124 0.129
74 gre_216a 0.128 0.227 0.171
75 nnc261 0.130 0.168 0.150
76 add32 0.133 0.215 0.140
77 shl_400 0.137 0.242 0.153
78 bp_400 0.515 0.402 0.162
79 extrl 0.257 0.428 0.175
80 west0655 0.248 0.249 0.196
81 utm300 0.265 0.244 0.278
82 fs_541_1 0.266 0.247 0.245
83 (I- *..; 0.257 0.278 0.278
84 dw256A 0.261 0.306 0.279
85 fs_541_2 0.292 0.268 0.266
86 fs_541_4 0.298 0.276 0.267
87 fs_541_3 0.309 0.287 0.270
88 mhda416 1.285 0.624 0.306
89 rw496 0.317 0.402 0.393
90 exi 0.605 0.747 0.356
91 bp_800 0.695 0.737 0.378
92 bp_1600 0.414 0.466 0.381
93 mahindas 1.245 0.862 0.385
94 bp_1200 0.573 0.779 0.424
95 bfwa782 0.432 0.493 0.479
96 bfwb782 0.435 0.516 0.440
97 bp_1400 0.739 0.971 0.435
98 west0381 0.491 0.484 0.439
99 bp_1000 0.784 0.613 0.445
100 gre_343 0.453 0.677 0.457














Table 5.4: continued.

no. name COLAMD COLMMD AMDBAR
101 bp_600 0.616 0.746 0.518
102 ., i1r II 0.624 0.938 0.859
103 lnsp_511 0.642 0.630 0.642
104 sherman4 0.635 0.844 0.740
105 hydrl 0.745 0.862 0.636
106 1 ., i, il 0.637 0.746 0.935
107 Ins_511 0.662 0.876 0.671
108 ,., 1 11 0.697 0.994 0.913
109 pde900 0.762 1.124 0.922
110 i .,- II' 0.774 0.775 0.895
111 steal 0.812 1.027 0.856
112 fs_760_1 0.900 1.381 0.836
113 radfrl 0.864 0.945 0.860
114 cddel 0.869 0.997 0.904
115 cdde2 0.869 0.997 0.904
116 cdde4 0.869 0.997 0.904
117 cdde6 0.869 0.997 0.904
118 cdde3 0.872 1.000 0.909
119 fs_760_2 0.986 1.472 0.891
120 cdde5 0.895 1.019 0.922
121 nnc666 0.988 1.057 0.929
122 .. li'' 0.957 1.362 1.194
123 fs_760_3 1.104 1.558 1.033
124 gematll 1.257 1.453 1.218
125 shermani 1.248 1.758 1.817
126 gematl2 1.425 1.691 1.297
127 gre_512 1.392 2.132 1.399
128 ex32 1.521 2.520 1.646
129 ex33 1.631 3.100 2.032
130 ex22 1.657 4.033 2.159
131 add20 5.161 1.812 3.979
132 hor_131 1.954 2.216 2.197
133 rdb968 1.996 2.678 2.097
134 lhrOl 2.265 2.619 2.327
135 raefsky5 2.271 2.422 2.613
136 .- 1 I, 3.040 5.534 2.507
137 mcfe 2.700 4.138 3.973
138 pores_2 2.840 4.360 2.824
139 ex2 3.163 5.203 4.046
140 exlO 3.317 7.159 5.221
141 steam2 3.569 4.268 3.891
142 exl0hs 3.581 8.348 5.784
143 dw1024 3.692 4.231 3.640
144 nnc1374 4.499 4.985 4.135
145 lhr02 4.792 5.623 4.810
146 orsirr_2 4.827 10.371 5.691
147 pde2961 6.277 8.869 5.363
148 rbsa480 5.379 8.918 6.501
149 -,.. .-, li 5.739 11.127 8.162
150 ex21 6.103 8.156 5.831














Table 5.4: continued.

no. name COLAMD COLMMD AMDBAR
151 swangl 5.977 7.666 7.182
152 swang2 5.977 7.666 7.182
153 rbsb480 6.077 8.969 6.606
154 rdist2 6.405 7.278 6.597
155 ex25 6.686 10.535 6.455
156 shyy41 6.651 9.931 8.192
157 ex23 6.912 25.319 14.245
158 orsirrl 7.229 7.967 7.259
159 '.. .li: 7.813 8.763 8.615
160 ,... i I, 7.831 8.646 8.617
161 utm1700b 8.021 12.971 8.682
162 rdb2048 8.338 10.525 8.983
163 bayerlO 8.381 14.566 9.035
164 ., i. 11ii'' 8.709 15.062 10.890
165 orani678 8.846 15.178 9.237
166 .- 1,- II 9.100 16.055 10.934
167 gre_1107 10.634 24.210 9.269
168 ex29 9.464 10.492 11.609
169 1-I1 9.594 16.902 11.110
170 ex3 11.126 16.912 10.580
171 bayer06 10.678 15.770 13.926
172 rdist3a 10.970 15.063 11.100
173 jpwh_991 11.805 17.286 11.597
174 ex27 11.612 42.989 14.655
175 -..i- 11, 11.858 15.871 14.208
176 cavity06 14.508 19.521 12.794
177 rdistl 14.986 12.910 15.763
178 i 1-1 -. 13.939 18.345 13.173
179 ex24 13.504 17.360 16.320
180 watt1_ 15.913 19.762 19.098
181 sherman5 16.346 25.254 19.610
182 ex13 33.872 35.522 17.418
183 ex4 17.434 34.971 25.768
184 lhr04c 17.647 18.086 18.273
185 watt_2 17.742 28.817 18.194
186 ex7 17.953 45.316 23.874
187 ex6 18.009 38.522 19.024
188 ex36 22.353 39.816 18.283
189 -... 111 21.999 26.029 21.311
190 ex37 23.133 44.618 26.558
191 lnsp3937 27.604 40.414 25.492
192 Ins_3937 27.563 45.271 25.622
193 ex15 27.514 62.895 51.991
194 ex9 91.924 56.131 27.786
195 ex31 28.318 140.178 51.391
196 memplus 5597.578 698.493 30.421
197 lhr07c 30.477 31.865 31.319
198 rw5151 32.469 46.158 30.900
199 orsreg1_ 34.913 60.463 33.364
200 ex28 34.753 56.466 37.339














Table 5.4: continued.

no. name COLAMD COLMMD AMDBAR
201 ... 11! 36.043 42.491 41.691
202 ex18 36.610 57.664 48.901
203 lhrl0c 39.261 44.945 39.961
204 sherman3 39.620 61.569 41.395
205 ex19 40.749 188.995 76.382
206 rdb5000 43.789 50.676 43.590
207 utm3060 43.892 73.241 48.795
208 sherman2 47.092 63.282 54.670
209 ex20 47.808 86.128 49.916
210 cavityl5 48.929 97.188 58.017
211 cavityl3 51.100 98.980 57.967
212 Ihrllc 52.567 79.661 56.814
213 cavityll 52.690 100.497 58.822
214 ex35 56.490 130.536 115.766
215 cavityl4 58.608 98.992 64.560
216 .,11 i! 1' 59.814 101.764 64.047
217 cavitylO 59.879 103.037 64.032
218 saylr4 62.256 106.482 62.239
219 garonl 66.293 94.475 103.740
220 ex8 127.019 98.491 67.019
221 exl2 67.619 128.369 119.995
222 onetone2 94.476 70.626 218.565
223 megl 81.516 93.108 144.699
224 ex26 82.158 386.006 177.033
225 lhrl4c 85.262 100.900 90.095
226 ex14 91.247 214.215 156.971
227 dw4096 100.257 124.868 120.427
228 lhrl7c 110.645 120.494 111.559
229 wangl 120.077 242.001 174.339
230 wang2 120.077 242.001 174.339
231 utm5940 125.390 184.034 145.059
232 cavity25 130.392 219.153 178.342
233 epbl 142.720 145.056 130.955
234 cavity23 131.415 221.953 180.009
235 cavity21 132.275 220.989 178.500
236 .,-! I', 135.153 219.879 178.197
237 cavity17 136.366 221.387 179.654
238 cavity24 204.029 218.798 169.134
239 cavity26 203.888 219.703 169.525
240 ,.,-1 i, 204.449 221.411 170.006
241 cavity22 203.959 220.057 170.007
242 cavity20 203.966 218.866 170.349
243 cavityl8 204.729 220.550 170.553
244 lhr34c 239.394 290.165 242.788
245 .i. ,-1 I 1 254.506 495.765 267.861
246 meg4 492.514 319.685 277.625
247 goodwin 1908.819 665.176 497.528
248 lhr71c 506.509 603.954 510.777
249 epb2 646.372 838.714 620.429
250 -11 -ll 1028.677 1573.200 1020.167











Table 5.4: continued.


name
garon2
ex40
graham 1
raefskyl
raefsky2
rim
epb3
onetonel
olafu
rmal0
venkatOl
venkat25
venkat50
af23560
twotone
raefsky3
raefsky4
ex11
psmigr_2
psmigr_3
psmigr_l
wang4
wang3
bbmat
av41092


COLAMD COLMMD A: 11 i!:.


1061.332
1075.063
1333.270
2374.493
2374.493
5553.070
2057.583
2847.909
2844.220
5063.526
4456.498
4506.216
4509.489
4515.197
8848.939
10406.398
10991.822
11901.588
16513.817
16649.557
16808.580
33726.550
27758.510
44552.892
75478.352


1563.413
5369.226
2114.903
2012.165
2012.165
3601.119
2516.863
4173.989
4118.783
5303.292
6016.190
6065.750
6067.632
5349.862
8922.973
12127.949
23530.592
26191.804
13941.055
16137.405
16504.110
83139.633
61973.758
163371.254
89677.234


1505.291
2606.118
1414.170
1681.522
1681.522
2018.664
2745.708
2519.732
3012.960
4041.353
5021.281
5086.083
5119.273
5051.531
8578.428
8589.957
12501.225
11637.984
17545.019
17964.210
18044.172
27740.799
32467.806
51257.577
50140.185


Table 5.5 shows matrices for which the ordering storage of AMDBAR was greater

than the SuperLU factorization storage using the AMDBAR ordering. I In! information

assumes that storage for each real is twice that of each integer. 'I Ir last column list the

ratio of ordering storage to factorization storage.

5.1.2 Comparison Plots


In addition to the tables, plots comparing COLAMD with COLMMD and AMD-

BAR are shown below. Following each plot is a table of ratio statistics and any outlier

matrices. Outlier matrices are those which do not fit onto the plot. : i., i -- which had a

minimum LU factorization flop count greater than 107 are considered big matrices. Another

row of statistics is shown for big matrices as well.

Filii-. 5.2 and 5.3 show plots comparing the ordering times of COLAMD with

the ordering times of COLMMD and AMDBAR. I I. plot shows the Log2 of the ratio of

COLAMD's ordering time over the ordering time of the other method. Points below the












Table 5.5: RUA matrices for which AMDBAR orderings required more storage than the
factorization.


ordering storage
103296
116916
1713652
126351
136507
139680
142975
143795
140743
131964
294783
2342036
6428561


factorization storage
62562
76468
972863
85457
107486
107379
111867
112406
110354
109901
287602
712836
1861012


zero line indicate COLAMD ran faster than the other method. If the ordering time from

Table 5.2 was zero, this was replaced by the smallest time resolution (0.01 seconds) to


compute the ratio.


name
bp_0
bp_200
poliiarge
bp_400
bp_800
bp_1600
bp_1200
bp_1400
bp_1000
bp_600
add20
orani678
memplus


ratio
1.651
1.529
1.761
1.479
1.270
1.301
1.278
1.279
1.275
1.201
1.025
3.286
3.454





































0 0 6 8
1- 0 D 0 0 00 000 00(MO ) 0 0 0 0 D o O O 0
0 0 OC 8o 0 O
S0000 o 0 0 0 O O
oo 6 o ( o
2- oo0o0o 0.oo o o 0 o %
0 00 00 OO 0 0 00 0
SO O 0O O0 0 0
0- 0 00 0 0 0 0 0
0 0 0 0 O O
S0 0 0 0 0
0 0 C 0
0
5
50 100 150 200 250
Matrix number


Group MIin


Ratio Statistics.
Mean Max Median -I.I Dev.


Covariance


ALL -6.79 -1.60 0.32 -1.58 1.39 1.92
BIG -5.70 -2.05 0.32 -1.95 1.29 1.66


Outliers
no. name i -. OLAMID i 0* C AOL A MD Log2 (ratio)
27 tubl000 0.01 0.50 -5.64
43 bwm2000 0.02 2.21 -6.79
45 olm2000 0.01 0.52 -5.70
64 olm5000 0.03 3.19 -6.73
202 ex18 0.26 8.46 -5.02
214 ex35 0.92 40.13 -5.45
274 bbmat 26.54 1378.04 -5.70



Fi nl 5.2: RUA matrices: Plot of Log2(TimecoLAMAID/ I* COLMAID)











































0 0 0 0


OO
0 on


0o 0


0 00 0 0" 0
So o o o o o o o &o
o o o o o o
-1 0 o oo0 ooO 0 0 0 _0 o
0 0 0 0 0
0 0 CO 06) 0 0 0% oo, 0
-2 C 0 0 %0 c 00
0 0 0 0 0 0 C9 0
3 0 0
-3


-4 0

0
5
50 100 150 200 250
Matrix number
Ratio Statistics.
Group Min Mean Max MIedian -. I Dev. Covariance
ALL -4.69 -0.70 1.95 -0.49 0.93 0.87
BIG -4.69 -1.12 1.95 -0.98 0.99 0.98



Finli 5.3: RUA matrices: Plot of Log2(TimecoLAMD/ I "" AMDBAR)








67



Fi i&i, 5.4 and 5.5 show plots indicating the ratios of the sizes of the LU factors


for COLAMD with respect to COLMMD and AMDBAR. Again, the Log2 is taken of the


ratio, therefore, points below the zero line indicate COLAMD produced smaller LU factors


than the other method. I Ir data for these plots is from Table 5.3.


0
0 0


0 0


0 0
0O
00

00


S0
0


o0P


oo o o
o cP o
0 00 0


O 0
0 0
0
0
0

n 0


50 100 150 200 250
Matrix number
Ratio Statistics.
Group Min Mean Max Median II Dev. Covariance
ALL -0.98 -0.15 1.82 -0.13 0.26 0.07
BIG -0.98 -0.16 1.82 -0.19 0.34 0.12


Outliers
no. name |L + UICOLAIID |L + UIC(OLM AD Log2 ratio)
196 mnemplus 4377196 1236504 1.82


Fioni, 5.4: RUA matrices: Plot of Log2(L + UICOLAAID/iL + UICOLA1AD)


0 0
0 0
0
















0 0
0
0
00 0
0


0 0
0


0
0


O0
,-- 0 0


O0

0 0 9


50 100 150 200 250
Matrix number
Ratio Statistics.
Group Min Mean Max Median I- Dev. Covariance
ALL -0.50 0.04 3.70 -0.01 0.31 0.10
BIG -0.49 0.06 3.70 -0.01 0.43 0.19


Outliers
no. name IL -+ JUCOLAAID IL + U IAADBAR Log2(ratio)
197 memplus 4377196 337222 3.70
253 goodwin 5633883 2733888 1.04


Fiill 5.5: RUA matrices: Plot of Log2(IL + UIcoLAAID/iL + UIAAIDBAR)



Fiili -' 5.6 and 5.7, show the ratio of the factorization flop counts for COLAMD

vs. COLMMD and AMDBAR. Like the previous plots, points below the zero line indicate

that COLAMD's ordering produced less flops that the method being compared with. I I.

data for these plots is from Table 5.4. \N.t only the outliers for the big matrices are

included for these plots.



















08k


0 O
O


o0 0
0


Group Min


O O



0 00


0 00




o
80 o
O O
0 0
0 0


SoC

0 0
0
0
o
o
0O
0


0
0

150
Matrix number


Ratio Statistics.
Mean Max Median I


00


0 0


09 o


Sc o
o0 0 (


0


0

f o
0
po o
0 80
000
0 0
O O
o o


Dev.


0


0
o0
0
0


Covariance


ALL -2.32 -0.40 3.00 -0.36 0.59 0.35
BIG -2.32 -0.51 3.00 -0.54 0.70 0.49


Big Outliers
no. name FlopCOLAMD FlopcOLAfD Log (ratio)
174 ex27 11611516 42988596 -1.89
183 ex4 17434000 34970840 -1.00
186 ex7 17952764 45316452 -1.34
187 ex6 18009100 38522408 -1.10
193 exl5 27514288 62894828 -1.19
195 ex31 28318484 140178320 -2.31
196 memplus 5597578240 698492672 3.00
205 exl9 40749216 188995072 -2.21
214 ex35 56490416 130536184 -1.21
224 ex26 82157920 386006080 -2.23
226 exl4 91247400 214215024 -1.23
229 wangl 120077152 242000720 -1.01
230 wang2 120077152 242000720 -1.01
247 goodwin 1908819200 665176320 1.52
252 ex40 1075063168 5369226240 -2.32
267 raefsky4 10991821824 23530592256 -1.10
268 exll 11901588480 26191804416 -1.14
272 wang4 33726550016 83139633152 -1.30
273 wang3 27758510080 61973757952 -1.16
274 bbmat 44552892416 163371253760 -1.87



Fib ,' 5.6: RUA matrices: Plot of Log2(F'lIp. 4LAMD/F' ,I OLMMAID)


00 0

-02- O


-04 -


-06


-08


S000 o 00
0 C C

o 0

0
0

0
o O
o




o o
a c


v




















0 4 0 0
01- 0


02 0 % 0 .


0 0

a


50


0 00
0 0


100 150
Matrix number
R.at.in Statist.ics.


Group Min Mean Max Median I Dev. Covariance
ALL -1.28 0.03 7.52 -0.05 0.68 0.47
BIG -1.28 -0.01 7.52 -0.09 0.89 0.80


name


Big Outliers


FlopCOLAMID


FIOp1AMDBAR


Log2 (ratio)


194 ex9 91924480 27785688 1.73
196 memplus 5597578240 30421034 7.52
214 ex35 56490416 115765832 -1.04
222 onetone2 94476360 218565296 -1.21
224 ex26 82157920 177033136 -1.11
247 goodwin 1908819200 497527936 1.94
252 ex40 1075063168 2606118144 -1.28
256 rim 5553069568 2018664064 1.46


Fibii 5.7: RUA matrices: Plot of Log2(F'!.i,. ,)LAAD/FlopAMIDBAR)



5.1.3 Discussion


From the above results for the unsymmetric square matrices, it can be concluded


that COLAMD performs very well. In both terms of ordering time and ordering quality,


COLAMD is as good or better than the alternatives.

In terms of ordering time, COLAMD is almost always faster than COLMMD. Only


for matrix 208, ,i.. _.-', did COLMMD order faster than COLAMD, and by only one


0
S 00
0


&b8


0


O O










hundredth of a second. Typically, COLAMD is about three times as fast. For the big

matrices, COLAMD is typically four times as fast.

When compared to AMDBAR, COLAMD is again almost always faster. COLAMD

can be expected to be 1.4 times as fast as AMDBAR. Again, for the big matrices, this

factor of improvement becomes greater. For the big matrices, COLAMD is twice as fast

as AMDBAR. For all the matrices, COLAMD is never more than four times slower than

AMDBAR. However, COLAMD was up to twenty-six times faster as AMDBAR. I !I,-

occurred for the largest matrix in the group, number 275, av41092. I I. formation of A A

took 'i I' of the ordering time for AMDBAR for this matrix.

Looking at the number of times a method took longer to order a matrix than it took

to factorize the matrix (bottom of Table 5.2), one can see that this only occurred 12 times

for COLAMD as compared to 94 and 35 times for COLMMD and AMDBAR respectively.

f i., importantly, it never happened to COLAMD on big matrices.

Quality comparisons in Fil:ii, 5.6 show that COLAMD is very competitive. Versus

COLMMD, it generally produces orderings which require 1l 1' less factor storage and 22'-

less flops to factorize. For the big matrices, COLAMD requires 1_'" less factor storage

and :;1' less factorization flops than COLMMD's orderings. Surprisingly, COLMMD's

best ordering required only 12.f5' of the flops than COLAMD's ordering. I I!J- occurred for

matrix 196 memplus. I Ir best COLAMD ordering needed 2 1' the flops of COLMMD for

matrix 252, cx40. In other words, COLMMD's best ordering was better than COLAMD's

best. Despite this fact, it is clear that COLAMD outperforms COLMMD for nearly all

matrices.

COLAMD and AMDBAR ordering qualities are very similar. I Ir average difference

in factor storage and factorization flop counts is negligible for all matrices and also for the

big ones. One observation is that the results from Filiin 5.7 are very l.:..;'. I Ir

standard deviation for these results is 1.6. One major outlier, matrix 196 memplus, was

ordered much better by AMDBAR than COLAMD. For this matrix, using the AMDBAR










ordering resulted in 30 million flops as compared to 5.6 billion for COLAMD. Overall,

COLAMD is very similar to AMDBAR in terms of quality.

Looking at Table 5.3, one can see that COLAMD produced orderings yielding LU

factors which had non-zero counts within 1I of the best ordering for all but 29 matrices.

COLMMD had 97 matrices for which the non-zero counts of LU were not within 1;' of

the best. AMDBAR only had 18 matrices that were not within 211' of the best. 1 Inl-,

COLAMD is clearly more robust than COLMMD, but AMDBAR is perhaps slightly more

robust than COLAMD.

An additional observation which needs to be made is the advantage of COLAMD

versus AMDBAR in terms of ordering storage. I Ij i were no matrices for which COLAMD

required more storage to order than was required to factorize the matrix. However, from

Table 5.5, one can see that several matrices required more ordering storage for the AMD-

BAR ordering than was required for factorization. Two matrices, numbers 165, ,./.' and

number 196, orani678, required more than three times the storage to order with AMDBAR

than was require to factorize the matrices. I I!-i demonstrates a clear advantage of using

COLAMD versus AMDBAR.

5.2 I Ir Rectangular (RRA) : i i.i

Rectangular matrices are another class of matrices to which COLAMD is well suited.

I Ir, I are two major groups of matrices which define this class.

I hr first is the case where m > n where A is m-by-n. I I!-' situation arises in

least-squares problems. It is often solved by performing a C('! .. -I:; factorization on A A

or by performing QR factorization on A.

'I -l other set of matrices arise in linear programming problems. In these, m < n

and solving them using the interior-point method involves the factorization of AD2A'

where D is a diagonal matrix. 'I hl matrix AD2A has the same nonzero pattern as AA.

For these problems, a column ordering of the transpose of A was found. Once this was










done, performance analysis of these matrices could be done in the same manner as the

least-squares matrices.

As discussed in Section 2.8, a column minimum degree ordering of A often performs

well in minimizing fill for both a ('C!r -1:, factorization of A'A or a QR factorization on A.

Performance tests of COLAMD were done with COLMMD and AMDBAR for comparison.

A group of 157 test matrices were used, most of which came from linear programming

problems. Each matrix was ordered directly by COLAMD and COLMMD. For AMDBAR,

A A (or AA as appropriate) was first formed and then ordered. Once the ordering

was performed, the symbolic ('C!.l! -I:, of A'A (or AA') was done for each method. As

Section 2.8 shows, the ('I! ,!, -I:, factor of A'A, L, is an structural upper bound on R from

QR factorization. I Ih i I the predicted size of the ('C!. -I:, factor L and the predicted

floating point operation count for this operation determined the ordering quality. 1 r.

number of nonzeros in L and the flop count required to compute the ('!. 1!, -I:. factorization

can be found without having to perform the numerical factorization. I I!i- is done using

a symbolic factorization [1iI, I I -, numerical factorization was not performed for these

experiments.
5.2.1 Results Tables

Fi ii, 5.8 shows the matrices in order of increasing best flop count. I I. vertical

line separates big matrices from the smaller ones. Big matrices are those which had a

minimum of 107 flops or more to factor. Table 5.6 lists the matrix number, dimension (m

by n), number of nonzeros in A, number of nonzeros in the symbolic pattern of A A (or

AA' as appropriate) after dropping dense rows and columns, the number of dense rows

and columns, and the best flop count. Best flop count refers to the best predicted flop

count in millions of the symbolic ('! -II -I:; factorization for the three ordering methods. As

with the RUA matrices, rows and columns are dense if they contain -.iI I' or more nonzero

entries.





































0
O



g


0 20 40 60 80 100 120 140 160
Matrices in increasing flop count order


Fi _,ii 5.8: Plotting order of the RRA matrices: by best flop count.














Table 5.6: General information for the RRA matrices.


n n |A| JIATA or dense dense best flop
IAATI rows cols count


Ipigalenet
lpiitest6
lpiitest2
Ipibgprtr
Ipiwoodinfe
Ip_afiro
lpiforest6
lp_sc50b
lp_sc50a
1pinmondou2
lp_scl05
lpiex73a
Ip_adlittle
lpiex72a
Ip_fitld
lp_scagr7
lpfit2d
Ipiboxl
lpsc205
lp_kb2
lp_gfrdpnc
Iprecipe
Ip_stocforl
lp_share2b
Ip_sharelb
Ip_blend
Ip_scorpion
lp_scagr25
Ipicplex2
illc1033
we111033
Ip_scsdl
Ip_sctapl
Ip_lotfi
lpi_kleinl
Ipibgdbgl
lp_ship04s
lp_scsd6
Ipvtpbase
Ip_standata
Ip_standgub
lp_grow7
Ip_shell
I, 1, 1ll
lp_bore3d
lp_beaconfd
lp_ship08s
lp_shipl2s
lp_e226
lp_scsd8


8 14 22
11 17 29
9 13 26
20 40 70
35 89 140
27 51 102
66 131 246
50 78 148
50 78 160
312 604 1208
105 163 340
193 211 457
56 138 424
197 215 467
24 1049 13427
129 185 465
25 10524 129042
231 261 651
205 317 665
43 68 313
616 1160 2445
91 204 687
117 165 501
96 162 777
117 253 1179
74 114 522
388 466 1534
471 671 1725
224 378 1215
1033 320 4732
1033 320 4732
77 760 2388
300 660 1872
153 366 1136
54 108 750
348 629 1662
400 1506 4400
147 1350 4316
198 346 1051
359 1274 3230
361 1383 3339
140 301 2612
536 1777 3558
400 2166 6380
233 334 1448
173 295 3408
776 2467 7194
1149 2869 8284
223 472 2768
397 2750 8584


24
51
27
96
137
153
286
236
252
1140
557
755
712
775
168
1129
175
1401
1107
847
2286
1087
1125
1646
1885
1560
3814
4315
3046
3974
3974
2189
3072
2239
2222
3956
6224
4051
3828
2833
2835
3040
3946
8856
4617
4757
10232
11839
5423
8163


no. I name


0.000
0.000
2 0.000
0.000
0.000
0.000
0.001
0.001
0.001
0.003
0.003
0.003
0.004
0.004
11 658 0.005
-0.005
12 5388 0.005
-0.006
-0.007
-0.007
-0.008
-0.009
-0.009
-0.012
-0.016
-0.016
-0.022
-0.022
-0.022
-0.025
-0.025
-0.029
-0.030
-0.032
-0.034
-0.039
-0.044
-0.047
-0.052
1 0.053
1 0.053
-0.056
-0.063
-0.068
-0.069
3 0.070
-0.073
-0.076
-0.086
-0.091













Table 5.6: continued.


m n |A| |ATA or dense dense best flop
|AAT rows cols count


51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100


no.


Ip_bandm
Ip-scfxml
illcl850

Ipbrandy
Ip-czprob
Ipirefinery
Ip-standmps
Ipiqual
Ipivoll
Ipi_-hemcom
lp_growl15
Ipfinnis
lp-ship081
Ipi-reactor
lp-scrs8
Ipcapri
lpjshipl21
wml
lpgrow22
lp-scfxm2
Ipipang
lp-ken_07
wm2
wm3
Ip1sierra
Ip-modszkl
Ip_tuff
lp-scfxm3
lp-sctap2
Ip_bnll
lp-sctap3
lp-stocfor2
Ip-agg
Ip-crec
Ip-crea
ll, l,..l. !



lpdegen2
]1-IM -11 11
Ip-stair
Ip_etamacro
Ipibgetam
11, -. 2
lp-agg3
Ipisrael
Ip-maros
Ip_ganges


name


305 472 2494
330 600 2732
1850 712 8758
1850 712 8758
219 303 2202
929 3562 10708
323 464 1626
467 1274 3878
323 464 1646
323 464 1646
288 744 1590
300 645 5620
497 1064 2760
776 4363 12882
318 808 2591
490 1275 3288
271 482 1896
1149 5533 16276
207 277 2909
440 946 8252
660 1200 5469
361 741 2933
2426 3602 8404
207 260 2942
207 260 2948
1227 2735 8001
687 1620 3168
333 628 4561
990 1800 8206
1090 2500 7334
643 1586 5532
1480 3340 9734
2157 3045 9357
488 615 2862
3048 6411 15977
3516 7248 18168
210 225 1050
410 1123 5264
410 1123 5264
722 2928 9265
444 757 4201
524 1028 6401
356 614 4003
400 816 2537
400 816 2537
516 758 4740
516 758 4756
174 316 2443
846 1966 10137
1309 1706 6937


7143
6136
9126
9126
5301
15073
5391
5101
5473
5473
2340
6560
6847
17800
3932
3906
5953
22495
8805
9640
12312
9011
14382
10023
10459
11099
12003
10487
18488
12100
9433
16252
27633
22854
40858
45012
4060
13232
13232
10670
14180
20706
12786
5918
5918
26282
26302
8900
23670
16621


0.091
0.096
0.100
0.100
0.101
0.106
0.110
0.111
0.113
0.113
0.125
0.127
0.137
0.138
0.141
0.153
0.171
0.174
0.181
0.189
0.199
0.208
0.210
0.242
0.262
0.267
0.292
0.297
0.302
0.363
0.432
0.472
0.478
0.612
- 0.681
- 0.697
0.830
0.843
0.843
- 0.931
- 0.931
- 0.937
- 0.962
- 0.970
- 0.970
- 1.039
- 1.039
3 1.051
- 1.217
- 1.376













Table 5.6: continued.


in n |A| lA A or dense dense best flop
|AAT rows cols count


lpwoodlp
lp_perold
lp-pds_02
lp_80bau3b
genmatl
Ip-voodw
lp_25fv47
Ip_truss
lposa_07
lp-stocfor3
lpken_11
lpgreenbea
lp_greenbeb
Ipigreenbea
lp_nug06
-i, ,,-i .I
ll ,_l ,,i l ,
lp_osa_14
lpcycle
lp-d6cube
Ipibgindy
1p_bnl2
lp_osa_30
lp_degen3
1pi_klein2
lp_ken_13
lp_nug07
lpd2q06c
Ipigran
lp_osa_60
because
beaflw
ll ,_l il. ,,
beacxc
Ipigosh
lpnug08
lpqap8
pfitlp
1pi_klein3
,_ ,il.. 7
lppds_06
Ipcreb
lpcre-d
lpmnaros-r7
lp_ken_18
lppds_10
Ipicplexl
lpdflO01
lpiceria3d
lpqapl2


244 2595 70216
625 1506 6148
2953 7716 16571
2262 12061 23264
4929 10595 47369
1098 8418 37487
821 1876 10705
1000 8806 27836
1118 25067 144812
16675 23541 76473
14694 21349 49058
2392 5598 31070
2392 5598 31070
2393 5596 31074
372 486 2232
940 2267 14977
975 2446 13331
2337 54797 317097
1903 3371 21234
415 6184 37704
2671 10880 66266
2324 4486 14996
4350 104374 604488
1503 2604 25432
477 531 5062
28632 42659 97246
602 931 4214
2171 5831 33081
2629 2525 20111
10280 243246 1408073
497 507 44551
497 507 53403
1441 4860 44375
497 496 50409
3792 13455 99953
912 1632 7296
912 1632 7296
627 1677 9868
994 1082 13101
2030 6680 74949
9881 29351 63220
9647 77137 260785
8894 73948 246614
3136 9408 144848
105127 154699 358171
16558 49932 107605
3005 5224 10947
6071 12230 35632
3576 4400 21178
3192 8856 38304


34888
13491
23281
22410
95017
41940
22969
26122
101104
234819
82454
70074
70074
70087
8724
29696
25195
219745
57331
24347
126747
29238
422412
101859
139985
161804
16576
56153
60511
969184
107069
121489
124517
218740
206010
28816
28816
274963
562640
238624
88003
398805
372234
664080
609271
149658
2265521
82267
1963306
152376


no. I name


3 1.700
-1.843
-2.252
-2.270
-2.356
-2.577
-2.602
-3.048
3 3.074
-3.200
-4.081
-4.559
-4.559
-4.617
-4.649
-5.871
-5.886
3 6.563
-8.404
3 9.589
-10.110
-11.292
2 12.433
-16.169
-16.802
-16.822
-18.477
-26.986
-27.020
3 28.097
56 6 35.304
66 7 35.454
-37.737
65 1 39.096
-46.128
-69.899
-74.090
11 82.361
143.649
164.446
179.098
220.520
221.294
240.686
248.619
918.737
1135.196
1195.922
1509.781
3087.902










Table 5.6: continued.

no. name m n I|A| ATAI or dense dense best flop
|AAT rows cols count
151 1p_nugl2 3192 8856 38304 152376 3127.031
152 lp_pds_20 33874 108175 232647 320196 8597.795
153 lp-fit2p 3000 13525 50284 6286616 12 9004.501
154 1p_qapl5 6330 22275 94950 378480 24019.623
155 1p_nugl5 6330 22275 94950 378480 31052.356
156 lp_nug20 15240 72600 304800 1216840 419631.155
157 lp_nug30 52260 379350 1567800 6265860 15690052.005



Table 5.7, gives the ordering time in seconds for each method. Ir. ordering time

for AMDBAR includes the time to form ATA (or AAT if m < n). I Ir last column lists


the time to form ATA prior to calling AMDBAR.














Table 5.7: Ordering times in seconds for the RRA matrices.

no. name COLAMD COLMMD A IlI !:.\I, /w A A ATA or
or AAT time AAT time
1 Ipigalenet 0.01 0.00 0.00 0.00
2 Ipiitest6 0.01 0.00 0.00 0.00
3 Ipiitest2 0.00 0.00 0.01 0.01
4 Ipibgprtr 0.01 0.00 0.00 0.00
5 Ipiwoodinfe 0.00 0.00 0.00 0.00
6 Ip_afiro 0.00 0.00 0.01 0.00
7 lpiforest6 0.01 0.00 0.00 0.00
8 lp-sc50b 0.00 0.00 0.00 0.00
9 lp-sc50a 0.00 0.00 0.01 0.01
10 lpi_mondou2 0.00 0.00 0.00 0.00
11 lpjscl05 0.01 0.00 0.00 0.00
12 lpiex73a 0.01 0.00 0.01 0.00
13 Ip-adlittle 0.01 0.00 0.00 0.00
14 lpiex72a 0.00 0.00 0.00 0.00
15 lpfitld 0.01 0.01 0.01 0.01
16 lp-scagr7 0.01 0.00 0.00 0.00
17 lpfit2d 0.07 1.68 0.09 0.09
18 Ipiboxl 0.01 0.00 0.01 0.00
19 lp-sc205 0.00 0.00 0.00 0.00
20 lpkb2 0.00 0.00 0.01 0.00
21 lpgfrdpnc 0.01 0.01 0.01 0.00
22 lprecipe 0.01 0.00 0.00 0.00
23 Ip-stocforl 0.01 0.00 0.00 0.00
24 lplshare2b 0.00 0.00 0.00 0.00
25 Iplsharelb 0.00 0.00 0.00 0.00
26 Ipblend 0.00 0.00 0.00 0.00
27 Ip-scorpion 0.01 0.01 0.01 0.01
28 lp-scagr25 0.01 0.01 0.01 0.00
29 Ipicplex2 0.01 0.01 0.01 0.00
30 illc1033 0.01 0.02 0.01 0.01
31 we111033 0.02 0.02 0.01 0.00
32 1Ipscsdl 0.00 0.00 0.01 0.01
33 Ipjsctapl 0.00 0.00 0.00 0.00
34 IpJotfi 0.01 0.01 0.00 0.00
35 IpikleinI 0.01 0.00 0.01 0.00
36 lpibgdbgl 0.01 0.01 0.00 0.00
37 lp-ship04s 0.02 0.03 0.01 0.00
38 lp-scsd6 0.00 0.01 0.01 0.01
39 Ipvtp_base 0.00 0.00 0.01 0.00
40 lp-standata 0.01 0.03 0.00 0.00
41 Ipjstandgub 0.01 0.03 0.01 0.00
42 lp_grow7 0.00 0.01 0.01 0.00
43 Ip-shell 0.02 0.02 0.01 0.00
44 li,_- ,iiii ii 0.02 0.05 0.02 0.01
45 lpbore3d 0.01 0.01 0.01 0.00
46 Ipbeaconfd 0.01 0.01 0.01 0.00
47 lp-ship08s 0.02 0.04 0.01 0.00
48 lp-shipl2s 0.02 0.03 0.01 0.00
49 lpe226 0.01 0.02 0.01 0.00
50 lp-scsd8 0.01 0.15 0.01 0.01








80




Table 5.7: continued.

no. name COLAMD COLMMD A.: 11 I:. /w ATA ATA or
or AAT time AAT time
51 Ip_bandm 0.00 0.01 0.01 0.00
52 lp_scfxml 0.01 0.02 0.00 0.00
53 illc1850 0.02 0.04 0.01 0.00
54 we111850 0.02 0.03 0.01 0.00
55 Ip_brandy 0.01 0.01 0.00 0.00
56 Ip_czprob 0.12 0.16 0.10 0.00
57 lpi-refinery 0.00 0.00 0.00 0.00
58 Ip_standmps 0.01 0.05 0.01 0.00
59 Ipi-qual 0.01 0.01 0.01 0.00
60 Ipivoll 0.00 0.01 0.01 0.00
61 lpi-chemcom 0.01 0.01 0.01 0.01
62 ip_growl5 0.01 0.02 0.02 0.01
63 lpfinnis 0.02 0.03 0.02 0.01
64 ip_ship081 0.04 0.06 0.03 0.01
65 Ipi-reactor 0.01 0.02 0.01 0.00
66 ip_scrs8 0.02 0.02 0.01 0.00
67 Ip_capri 0.01 0.02 0.01 0.00
68 ip_ship121 0.04 0.08 0.03 0.01
69 wml 0.02 0.03 0.02 0.01
70 lpgrow22 0.03 0.04 0.02 0.01
71 lp_scfxm2 0.01 0.04 0.01 0.00
72 Ipipang 0.01 0.02 0.01 0.00
73 1p_ken_07 0.04 0.04 0.03 0.00
74 wm2 0.01 0.04 0.01 0.00
75 wm3 0.01 0.05 0.02 0.01
76 Ip_sierra 0.02 0.04 0.02 0.01
77 Ip_modszkl 0.02 0.05 0.02 0.01
78 Ip_tuff 0.01 0.03 0.01 0.01
79 lp_scfxm3 0.02 0.06 0.02 0.00
80 1p_sctap2 0.01 0.02 0.02 0.01
81 Ip_bnll 0.02 0.08 0.02 0.00
82 1p_sctap3 0.02 0.03 0.02 0.01
83 ip_stocfor2 0.03 0.07 0.03 0.01
84 1p_agg 0.01 0.02 0.02 0.00
85 Ip_crec 0.18 0.29 0.22 0.02
86 Ip_crea 0.19 0.31 0.24 0.01
87 ip_nug05 0.00 0.00 0.00 0.00
88 ip_pilot4 0.02 0.05 0.02 0.00
89 Ip, pl..i l 0.02 0.05 0.02 0.00
90 ip_pilotwe 0.02 0.06 0.02 0.01
91 lp_degen2 0.02 0.04 0.02 0.00
92 Ip ll "i-ii 0.02 0.25 0.03 0.01
93 Ip_stair 0.02 0.03 0.01 0.00
94 Ip_etamacro 0.01 0.02 0.01 0.00
95 Ipibgetam 0.01 0.03 0.01 0.01
96 i,- .,..--' 0.01 0.02 0.01 0.00
97 1p_agg3 0.02 0.03 0.02 0.01
98 Ip_israel 0.01 0.01 0.00 0.00
99 Ip_maros 0.02 0.11 0.03 0.01
100 Ip_ganges 0.01 0.07 0.02 0.01








81




Table 5.7: continued.

no. name COLAMD COLMMD AMDBAR /w ATA ATA or
or AAT time AAT time
101 lpwoodlp 0.15 1.39 0.13 0.13
102 lp_perold 0.02 0.06 0.02 0.00
103 lp_pds_02 0.06 0.13 0.09 0.01
104 lp_80bau3b 0.07 0.34 0.07 0.02
105 gematl 0.69 0.87 0.23 0.04
106 lpwoodw 0.28 0.67 0.12 0.03
107 lp_25fv47 0.03 0.15 0.03 0.01
108 Ip_truss 0.05 0.14 0.04 0.02
109 lp_osa_07 3.11 1.77 0.42 0.10
110 lpjstocfor3 0.21 0.56 0.27 0.07
111 lpken_ll 0.38 0.32 0.45 0.04
112 lp_greenbea 0.13 0.45 0.14 0.03
113 p_-greenbeb 0.13 0.44 0.15 0.03
114 Ipigreenbea 0.13 0.45 0.14 0.02
115 lpnug06 0.01 0.01 0.01 0.00
116 l,--,l.,, -i., 0.07 0.19 0.06 0.02
117 lp ,p..--i i,. 0.05 0.13 0.04 0.01
118 lp_osa_14 13.91 8.19 1.87 0.20
119 p_-cycle 0.04 0.27 0.06 0.02
120 lpd6cube 0.09 2.82 0.04 0.03
121 Ipibgindy 1.40 0.75 1.05 0.07
122 lp_bnl2 0.06 0.37 0.05 0.01
123 lp_osa_30 76.10 28.20 10.44 0.46
124 lpdegen3 0.28 0.44 0.40 0.05
125 lpiklein2 0.01 0.02 0.07 0.03
126 lpken_13 0.97 0.62 1.15 0.07
127 lpnug07 0.02 0.03 0.02 0.01
128 lp-d2q06c 0.07 0.44 0.08 0.03
129 Ipigran 0.04 0.07 0.06 0.03
130 lp_osa_60 441.23 153.19 62.84 1.27
131 because 0.10 0.90 0.20 0.15
132 beaflw 0.11 0.89 0.27 0.23
133 Ip,'-,l. 0.27 1.46 0.21 0.09
134 beacxc 0.07 0.23 0.27 0.25
135 Ipigosh 0.52 3.14 0.49 0.11
136 lpnug08 0.05 0.06 0.04 0.00
137 lp_qap8 0.05 0.04 0.05 0.00
138 pfitlp 0.01 0.01 0.08 0.04
139 lpiklein3 0.04 0.06 0.32 0.15
140 Ip, l.,. 7 0.56 2.60 0.52 0.20
141 lp_pds_06 0.51 1.28 0.57 0.04
142 lpcreb 3.48 72.00 1.69 0.18
143 lp-cred 3.38 68.19 1.54 0.18
144 lpnmaros-r7 3.60 114.22 0.62 0.43
145 lpken_18 9.52 4.35 12.50 0.35
146 lp_pds_10 1.07 2.61 1.19 0.09
147 Ipi-cplexl 0.19 0.06 0.87 0.21
148 lp_dflOO1 0.39 1.13 0.44 0.03
149 lpi-ceria3d 1.03 1.18 2.69 0.29
150 lp_qapl2 0.41 0.34 0.46 0.03










Table 5.7: continued.

no. name COLAMD COLMMD A: Il ,i:.\I /w ATA ATA or
or AAT time AAT time
151 lp_nugl2 0.36 0.49 0.31 0.04
152 ip_pds_20 3.86 10.53 3.92 0.21
153 lpfit2p 0.04 0.04 2.29 1.21
154 ip_qapl5 1.45 1.19 1.63 0.10
155 lp_nug15 1.27 1.52 1.02 0.10
156 lpnug20 7.19 8.22 5.19 0.35
157 lpnug30 80.02 125.76 51.56 2.68



Table 5.8 lists the size of the ('l!, II -1:. factor, L, of A A for each ordering method.

SIr column labled |A Al list the number of nonzeros in the upper and lower diagonal

halves of A'A (or AA' as appropriate) after dropping dense rows and columns. I In,! is a

good measure of the memory requirements for a symmetric code such as AMDBAR.








83





Table 5.8: IL| for the RRA matrices.

no. name COLAMD COLMMD AMDBAR |ATAI or
IAATI
1 Ipigalenet 17 17 17 24
2 Ipiitest6 32 33 32 51
3 lpiitest2 32 32 34 27
4 Ipibgprtr 80 78 77 96
5 Ipiwoodinfe 102 102 102 137
6 Ipafiro 110 119 113 153
7 lpiforest6 240 244 235 286
8 lpsc50b 242 244 231 236
9 lp-sc50a 238 267 242 252
10 lpimondou2 923 945 902 1140
11 lpjscl05 573 602 558 557
12 lpi_ex73a 785 806 784 755
13 lp-adlittle 434 411 411 712
14 lpi_ex72a 801 828 804 775
15 Ipfitid 300 300 298 168
16 lpjscagr7 766 815 762 1129
17 lpfit2d 325 325 324 175
18 Ipiboxl 1164 1163 1244 1401
19 lp-sc205 1182 1387 1124 1107
20 lp-kb2 503 518 503 847
21 lp gfrdpnc 2525 2338 2153 2286
22 lprecipe 928 694 678 1087
23 Ipjstocforl 979 1084 948 1125
24 lpsshare2b 1052 1292 1004 1646
25 lpssharelb 1429 1296 1251 1885
26 Ip-blend 1177 1051 1006 1560
27 Ipjscorpion 3332 2676 2528 3814
28 lpjscagr25 2980 3381 2958 4315
29 lpicplex2 2167 2148 2146 3046
30 illc1033 2997 2601 2569 3974
31 well1033 2997 2601 2569 3974
32 Ipscsdl 1481 1557 1398 2189
33 Ipsctapl 2593 2731 2564 3072
34 IpJotfi 1906 2190 1917 2239
35 lpikleinl 1250 1217 1208 2222
36 lpibgdbgl 3226 3241 3090 3956
37 lp-ship04s 3520 3516 3660 6224
38 lp-scsd6 2735 2772 2545 4051
39 ip_vtpbase 2912 3169 2892 3828
40 Ipjstandata 3384 3410 3346 2833
41 Ipjstandgub 3386 3412 3348 2835
42 lpgrow7 2745 3062 2775 3040
43 Iplshell 4463 4956 4351 3946
44 li,--_ihlpn 4832 4828 4832 8856
45 lp_bore3d 3168 3529 3098 4617
46 Ip beaconfd 3270 2957 2902 4757
47 lp-ship08s 6020 6076 6220 10232
48 lpsshipl2s 7090 7558 7330 11839
49 lp-e226 3887 3894 3701 5423
50 lp-scsd8 5915 6191 5879 8163








84




Table 5.8: continued.

no. name COLAMD COLMMD AMDBAR iATAi or
IAATI
51 lpbandm 5119 6376 4673 7143
52 Ipnscfxml 6237 5965 4748 6136
53 illc1850 8890 7680 7406 9126
54 well1850 8890 7680 7406 9126
55 Ipbrandy 3611 3706 3444 5301
56 Ip-czprob 8414 8391 8395 15073
57 lpi-refinery 4987 4843 4412 5391
58 lpstandmps 5419 5500 5511 5101
59 Ipiqual 5067 5892 4506 5473
60 1pivoll 5067 5892 4506 5473
61 Ipichemcom 4311 4712 4384 2340
62 lp_growl5 6105 7048 6135 6560
63 lpfinnis 6890 6925 6788 6847
64 lpjship081 9796 9680 9764 17800
65 lpi-reactor 5326 5740 5001 3932
66 lp-scrs8 7494 7060 6235 3906
67 Ipcapri 5582 7182 5518 5953
68 lpshipl21 12418 12414 12468 22495
69 wml 4771 4741 4678 8805
70 lp-grow22 9045 10682 9058 9640
71 lpnscfxm2 11786 12200 9681 12312
72 Ipipang 7864 11721 7399 9011
73 lpken_07 15529 16250 15529 14382
74 wm2 6002 5901 5700 10023
75 wm3 6133 6245 5987 10459
76 Ip-sierra 12744 13426 13078 11099
77 lpmodszkl 12434 13970 12901 12003
78 lptuff 8469 8189 8262 10487
79 lpjscfxm3 17807 18287 14614 18488
80 lp1sctap2 14937 13338 13147 12100
81 Ip bnll 12073 13519 12100 9433
82 lp-sctap3 17816 18327 17673 16252
83 lp-stocfor2 30736 68654 27754 27633
84 lpagg 17092 21579 16016 22854
85 Ipcrec 34778 36025 34255 40858
86 Ip-cre-a 37116 39239 36275 45012
87 lpnug05 11942 11091 12991 4060
88 Ip,_p. I 14742 14444 13956 13232
89 lpipilot4i 14742 14444 13956 13232
90 lIp p,- _-re 20168 23785 18184 10670
91 lp-degen2 16368 16506 16249 14180
92 Ipll .l-!ii 19148 20709 18675 20706
93 Ipstair 15465 20007 16419 12786
94 lp_etamacro 15611 16406 15400 5918
95 lpibgetam 15611 16406 15400 5918
96 11,- .,.- .- 24033 35832 21047 26282
97 lpagg3 24033 28285 21047 26302
98 Ipisrael 11773 11810 11792 8900
99 lpmaaros 29630 54709 24677 23670
100 Ip_ganges 30344 33407 28400 16621













Table 5.8: continued.


no. name COLAMD COLMMD AMDBAR |ATA| or
IAATI
101 Ip-woodlp 24266 18876 20660 34888
102 lp_perold 26425 29364 26340 13491
103 lp-pds_02 44486 50190 44668 23281
104 lp_80bau3b 44263 47739 42280 22410
105 gematl 98078 101182 86971 95017
106 Ip-woodw 45438 47534 49573 41940
107 lp_25fv47 39498 52210 34488 22969
108 Iptruss 51088 63804 54210 26122
109 lp_osa_07 59306 55901 55911 101104
110 lp_stocfor3 232278 543463 202360 234819
111 lp_ken11 133221 136225 133461 82454
112 lp_greenbea 79425 141291 80281 70074
113 Ip_greenbeb 79425 141291 80281 70074
114 lpiLgreenbea 79738 129390 81473 70087
115 lp_nug06 35901 34244 39339 8724
116 lp_pilotja 54544 58973 57064 29696
117 lp_pilotnov 55648 58296 55182 25195
118 lp_osa_14 121898 118497 118506 219745
119 Ip_cycle 92074 221104 100304 57331
120 lp-d6cube 53285 55556 54939 24347
121 Ipibgindy 129655 118971 128117 126747
122 lpbnl2 81141 95503 87978 29238
123 lp_osa_30 226104 222861 222876 422412
124 lp_degen3 123387 134198 122983 101859
125 lpiklein2 81073 81784 80031 139985
126 lp_ken_13 353474 358639 352967 161804
127 lp_nug07 92624 85545 102475 16576
128 lpd2q06c 175041 271307 146994 56153
129 Ipi-gran 190918 157500 143298 60511
130 lp_osa_60 519037 515636 515733 969184
131 because 114355 114499 115127 107069
132 beaflw 115129 117151 116640 121489
133 Ip-pilot 180661 197114 193053 124517
134 beacxc 119812 119812 119812 218740
135 Ipi-gosh 275933 412741 259461 206010
136 lp_nug08 210091 201831 230252 28816
137 lp_qap8 204290 226529 214808 28816
138 lpfitlp 196878 196878 196878 274963
139 lpiklein3 339304 346577 336520 562640
140 lppilot87 422518 475016 406261 238624
141 lppds_06 593816 561709 578744 88003
142 lp_creb 775334 1143846 947714 398805
143 lp_cre_d 763499 1007022 835928 372234
144 lpmaros-r7 853785 854616 1188479 664080
145 lp_ken_18 2229341 2314792 2327294 609271
146 lp_pds_10 1633555 1652109 1603825 149658
147 Ipi-cplexl 1139265 1139265 1139265 2265521
148 lp_dfl001 1544706 1862425 1526195 82267
149 lpi-ceria3d 1590872 1573595 1590872 1963306
150 lp_qapl2 2407592 2760492 2475876 152376











Table 5.8: continued.


name


lpnugl2
lppds_20
lpfit2p
lpqapl5
lpnugl5
lp_nug20
lp_nug30


COLAMD COLMMD A: 11 ':.\ ,


2713968
6827078
4501500
9342598
10732660
63208943
750645329


2447356
8159007
4501500
10747312
11071813
61977353
688096967


2777024
6928795
4501500
9341202
10853272
63252812
736511765


|ATA| or
IAAI1
152376
320196
6286616
378480
378480
1216840
6265860


Table 5.9 lists the predicted flop counts in millions for each method. Flop counts


are a good measure of the work required to factorize the matrix.








87





Table 5.9: Flop counts in millions for the ('lC!. -!.;, factorization of A A .

no. name COLAMD COLMMD A: 11 :.,
1 Ipigalenet 0.000 0.000 0.000
2 lpi_itest6 0.000 0.000 0.000
3 lpiitest2 0.000 0.000 0.000
4 Ipibgprtr 0.000 0.000 0.000
5 Ipiwoodinfe 0.000 0.000 0.000
6 Ipafiro 0.000 0.001 0.001
7 lpiforest6 0.001 0.001 0.001
8 lp-sc50b 0.001 0.001 0.001
9 lp-sc50a 0.001 0.002 0.001
10 lpimondou2 0.003 0.003 0.003
11 lpjscl05 0.003 0.004 0.003
12 lpiex73a 0.003 0.004 0.003
13 lp-adlittle 0.004 0.004 0.004
14 lpiex72a 0.004 0.004 0.004
15 lpfitld 0.005 0.005 0.005
16 lp-scagr7 0.005 0.006 0.005
17 lpfit2d 0.006 0.006 0.005
18 Ipiboxl 0.006 0.006 0.007
19 lp-sc205 0.007 0.011 0.007
20 lpkb2 0.007 0.007 0.007
21 lpgfrdpnc 0.013 0.010 0.008
22 lprecipe 0.019 0.009 0.009
23 lp-stocforl 0.010 0.014 0.009
24 lplshare2b 0.013 0.020 0.012
25 lp-sharelb 0.020 0.016 0.016
26 Ipblend 0.023 0.018 0.016
27 lp-scorpion 0.048 0.026 0.022
28 lp-scagr25 0.022 0.031 0.022
29 lpicplex2 0.023 0.022 0.022
30 illc1033 0.039 0.026 0.025
31 we111033 0.039 0.026 0.025
32 1Ipscsdl 0.033 0.036 0.029
33 Ip-sctapl 0.031 0.037 0.030
34 IpJotfi 0.032 0.043 0.034
35 Ipikleinl 0.036 0.034 0.034
36 lpibgdbgl 0.044 0.045 0.039
37 lplship04s 0.044 0.044 0.050
38 lp-scsd6 0.056 0.057 0.047
39 Ipvtp_base 0.053 0.066 0.052
40 lp-standata 0.054 0.056 0.053
41 lp-standgub 0.054 0.056 0.053
42 lp_grow7 0.056 0.073 0.058
43 Ip-shell 0.071 0.096 0.063
44 11,_--lpI II 0.068 0.068 0.068
45 lpbore3d 0.073 0.095 0.069
46 Ipbeaconfd 0.092 0.074 0.070
47 lp-ship08s 0.073 0.076 0.082
48 lp-shipl2s 0.076 0.095 0.087
49 lpe226 0.094 0.099 0.086
50 lpjscsd8 0.092 0.100 0.091








88





Table 5.9: continued.

no. name COLAMD COLMMD A: Il i:.\
51 Ip_bandm 0.112 0.202 0.091
52 lp_scfxml 0.188 0.172 0.096
53 illc1850 0.146 0.108 0.100
54 well1850 0.146 0.108 0.100
55 Ip_brandy 0.112 0.120 0.101
56 Ip_czprob 0.107 0.106 0.106
57 lpiLrefinery 0.152 0.143 0.110
58 Ip_standmps 0.111 0.118 0.119
59 Ipiqual 0.154 0.233 0.113
60 lpivoll 0.154 0.233 0.113
61 Ipichemcom 0.125 0.162 0.130
62 lpgrowl5 0.127 0.173 0.129
63 lpfinnis 0.141 0.143 0.137
64 ip_ship081 0.142 0.138 0.141
65 lpiLreactor 0.154 0.201 0.141
66 ip_scrs8 0.210 0.226 0.153
67 Ip_capri 0.171 0.333 0.171
68 ip_ship121 0.174 0.174 0.175
69 wml 0.191 0.188 0.181
70 lpgrow22 0.189 0.268 0.189
71 lp_scfxm2 0.312 0.366 0.199
72 lpipang 0.233 0.681 0.208
73 1p_ken_07 0.210 0.228 0.210
74 wm2 0.278 0.264 0.242
75 wm3 0.277 0.291 0.262
76 Ip_sierra 0.267 0.303 0.299
77 Ip_modszkl 0.292 0.413 0.333
78 Ip_tuff 0.339 0.297 0.319
79 lpscfxm3 0.475 0.539 0.302
80 ip_sctap2 0.575 0.376 0.363
81 1p_bnll 0.432 0.615 0.457
82 ip_sctap3 0.476 0.524 0.472
83 Ip_stocfor2 0.586 5.293 0.478
84 Ip_agg 0.696 1.137 0.612
85 Ip_crec 0.684 0.771 0.681
86 Ip_cre_a 0.737 0.930 0.697
87 1p_nug05 0.996 0.830 1.172
88 lp_pilot4 0.944 0.889 0.843
89 Ip ,_ l.. I 0.944 0.889 0.843
90 lppilotwe 1.047 1.694 0.931
91 1p_degen2 0.957 0.983 0.931
92 Ip11.1111 0.964 1.262 0.937
93 Ip_stair 0.962 1.811 1.150
94 Ip_etamacro 0.974 1.106 0.970
95 lpibgetam 0.974 1.106 0.970
96 11, -- -' 1.391 3.683 1.039
97 1p_agg3 1.391 2.005 1.039
98 Ip_israel 1.051 1.060 1.059
99 Ip_maros 1.681 8.062 1.217
100 Ip_ganges 1.498 1.952 1.376








89





Table 5.9: continued.

no. name COLAMD COLMMD AIl: 1! :.
101 lpwoodlp 3.186 1.700 2.177
102 lp_perold 1.843 2.346 1.880
103 lp_pds_02 2.252 3.441 2.297
104 lp_80bau3b 2.565 3.283 2.270
105 gematl 2.983 3.902 2.356
106 lpwoodw 2.577 2.905 3.365
107 lp_25fv47 3.044 6.344 2.602
108 Ip_truss 3.048 5.227 3.493
109 lp_osa_07 3.758 3.074 3.075
110 lpjstocfor3 4.358 46.990 3.200
111 lpken_ll 4.081 4.190 4.096
112 lp_greenbea 4.559 24.672 4.647
113 lpgreenbeb 4.559 24.672 4.647
114 Ipigreenbea 4.617 19.476 4.918
115 lp-iug06 5.299 4.649 6.280
116 I I,-, 1.. 1 5.871 6.731 6.674
117 Ipp-,.l ,.., 5.936 6.220 5.886
118 lp_osa_14 7.246 6.563 6.564
119 Ipcycle 8.404 66.143 12.189
120 lpd6cube 9.589 10.761 10.558
121 Ipibgindy 14.631 10.110 13.700
122 lp_bnl2 11.292 16.998 14.318
123 lp_osa_30 13.080 12.433 12.435
124 lp_degen3 16.404 19.844 16.169
125 lpiklein2 17.204 17.540 16.802
126 lpken_13 16.828 17.117 16.822
127 lp-iug07 22.474 18.477 27.125
128 lp-d2q06c 34.972 97.438 26.986
129 Ipigran 47.820 32.612 27.020
130 lp_osa_60 28.781 28.097 28.109
131 because 35.304 35.319 35.838
132 beaflw 35.454 36.994 36.676
133 I, il .., 37.737 44.829 46.140
134 beacxc 39.096 39.096 39.096
135 Ipigosh 46.439 118.714 46.128
136 lpnug08 78.427 69.899 91.592
137 lpqap8 74.090 90.222 82.833
138 p-fitlp 82.361 82.361 82.361
139 lpiklein3 145.453 152.254 143.649
140 p, -pil.. i 7 168.779 194.888 164.446
141 lp_pds_06 208.034 179.098 183.784
142 lpcreb 220.520 566.221 373.112
143 lpcred 221.294 421.049 266.031
144 lp_marosr7 240.686 240.997 491.296
145 lpken_18 248.619 272.767 273.916
146 lp_pds_10 948.648 1036.736 918.737
147 Ipi-cplexl 1135.196 1135.196 1135.196
148 lp_dfl001 1254.635 1892.358 1195.922
149 lpiceria3d 1561.353 1509.781 1561.353
150 lpqapl2 3087.902 3989.004 3306.011










Table 5.9: continued.

no. name COLAMD COLMMD AMDBAR
151 lp_nug12 3897.217 3127.031 3982.961
152 lp_pds_20 8597.795 12640.426 8812.459
153 lpfit2p 9004.501 9004.501 9004.501
154 lp_qapl5 24019.623 30414.559 24090.640
155 lpnugl5 31052.356 32272.100 31133.672
156 lpnug20 449954.265 419631.155 447771.334
157 lp_nug30 18615322.190 15690052.005 17864222.786



5.2.2 Comparison Plots

In addition to the tables, plots comparing COLAMD with COLMMD and AMD-

BAR are shown below. Following each plot is a table of ratio statistics and any outlier

matrices. Outlier matrices are those which did not fit onto the plot. : i.i I -- which had

a minimum predicted ('C!, --:;, flop count greater than 107 are considered big matrices.

Another row of statistics is shown for big matrices as well.

Finii, -' 5.9 and 5.10 show plots indicating the ratios of ordering times of COLAMD

with respect to COLMMD and AMDBAR. Since the Log2 is taken of the ratio, points

below the zero line indicate COLAMD ran faster than the other method. As in the RUA

experiment, if the ordering time from Table 5.7 was equal to zero, then it was replaced with

0.01 so that a valid ratio could be computed.













































0 00 0




1o0 cco cacooo oc o o0


2 00 0
or0 OD 0 0 0 0 0 0

S0 00 0
G 0 00 00 0 0
0 0
20 00 0

S0 0 0


0
4- O

'0


20 40 60 80
Matrix number


100 120


Ratio Statistics.
Group Min Mean Max MIedian -'.I Dev. Covariance
ALL -4.99 -0.85 1.66 -0.58 1.22 1.48
BIG -4.99 -1.00 1.66 -0.65 1.67 2.79



Filli 5.9: RRA matrices: Plot of Log2(TinmecoLAMID I *"" COLMAID)




Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EQLZN0KFI_6YMYZG INGEST_TIME 2017-07-20T21:14:40Z PACKAGE UF00101370_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES



PAGE 2

!"#$%" &% '#()* +#, ') "& &-*'*.!'

PAGE 3

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

PAGE 4

/ 0 ... 2. 2.. .7 .!# !*##2.#% -""# '.)' 9 9 &4 &-) "#' 9 : ) &-) "#' ; 9 2 #2.#% : + (.< !.("#'.' 9 0 / 9 ''&..( !.( = 99 *-#(,#' = 9: *!.#( ! -",#& 9> ()-&#!##+##-, !# 9 9? &.@*#4'-!.( 9 9; *&!.-&#&..( !.( 9 9A !#( &#+## : 9 ,#.(+' : 9 -7. !##+##' ? 9= -7. !#.(. *#8).#() ; : : # !*#' :9 "###,#*.'!.) :: "# &&+.!"#').-!.( 9= > 9A > ! !*)!*#' 9A > %!*)!*# 9 >9 &*(!*)!*# (, #( ! ## 9 .2

PAGE 5

>: (,#7 := >> "##+## .'! (, '" 4&# := >9 "#&+.!"' := >9 .(*!.(# : >99 (.! %' &' : >9: (.! ).(+ : >9> .(, ,#.(+ :9 >9? 4 +# &&#)!.( :? >9; #!#)! *-# &' :? >9A ,# ".&,#( :; ? :A ? "#@* #('#!.)B C !.)#' : ? #'*&! 4&#' : ?9 .'(&!' ;: ?: .')*''.( A= ?9 "##)! (+*& BC !.)#' A9 ?9 #'*&!' 4&#' A: ?99 .'(&!' = ?9: .')*''.( ; ?: "##!.)BC !.)#' A ?: #'*&!' 4&#' ?:9 .'(&!' A ?:: .')*''.( 9: ; 9A 9 ? / ;9 2

PAGE 6

? #(# &.( !.(!"# !.)#' >= ?9 ,#.(+!.#'.('#)(,'!"# !.)#' >; ?: D !"# !.)#' ?9 ?> &-)*( !'.(.&&.('!"# )!.< !.( ? ?? !.)#' %".) " ,#.(+' #@*.#, # '! +# !" ( !"# )!.< !.( ;> ?; #(# &.( !.(!"# !.)#' A? ?A ,#.(+!.#'.('#)(,'!"# !.)#' A ? !"# !.)#' : ? &-)*( !'.(.&&.('!"#"&#'1 )!.< !.( A ?= #(# &.( !.(!"# !.)#' == ? ,#.(+!.#'.('#)(,'!"# !.)#' => ?9 !"# !.)#' = ?: &-)*( !'.(.&&.('!"#"&#'1 )!.< !.( : ?> #)!,#('# %E)&*(# 2 &(,#.(+!.# (,F-)*( !'!"# *-! !.)#' 9? 2.

PAGE 7

&) 1!.G,. +( & : 9 *4&G4,##,4&) 1 : : *4&G4,##,4&) 1 (,.!'"&#'1 )! > 9 "#4 '.).(. *,#+## &+.!" 99 #& !.(4#! % ##( (, ? : (.!. &,#+##').'(H 9B C 9 ? &!!.(+,#!"# !.)#'H 4 4#'!F-)*( ! : ?9 !.)#'H &! B C ;? ?: !.)#'H &! B C ;; ?> !.)#'H &! B D D C ;A ?? !.)#'H &! B D D C ; ?; !.)#'H &! B C ; ?A !.)#'H &! B C A= ? &!!.(+,#!"# !.)#'H 4 4#'!F-)*( ! A> ? !.)#'H &! B C ?= !.)#'H &! B C 9 ? !.)#'H &! B C : ?9 !.)#'H &! B C > ?: !.)#'H &! B C ? ?> !.)#'H &! B C ; 2..

PAGE 8

?? &!!.(+,#!"# !.)#'H 4 4#'!F-)*( ! ?; !.)#'H &! B C ?A !.)#'H &! B C ? !.)#'H &! B C 9= ? !.)#'H &! B C 9 ?9= !.)#'H &! B C 99 ?9 !.)#'H &! B C 9: ?99 !.7 $('" *&*-! 9> 2...

PAGE 9

4'! )!"#'.' #'#( !#,!!"# ,* !#) "& !"#(.2 #'.! &., .( !. & *&8&&#( !!"# #@*.##( !'!"##+## '!#(+.(##.(+ !# ( .# #)# 4# " . (H .!" 2.' 5#!#( !H -*!# (,( !.().#()# (,(+.(##.(+ ( -7. !# .(. *,#+## )&*(,#.(+ &+.!"BC -#G ,#.(+ ( *('#!.) ''# !.7 -. ! ( *#.) & )!.< !.( .' -#'#( !#, '.(+!"#@*!.#( !+ -"#-#'#( ! !.( (, !.+" ! (, '! -7. !#,#+##$ !".' &+.!".' +, &!#( !.2 #! *'.(+'#!.) ),#',.#)!&( !"##7-&.).!#-G #'#( ! !.( ,,.!.( && $ .' '" %( ! 4# '*-#. ! $ !"# )&*(.(. *,#+##,#.(+),#.( .7

PAGE 10

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

PAGE 11

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

PAGE 12

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

PAGE 13

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

PAGE 14

? .()#$ (, # 4 (,#,$ % # ) ( 4# ''*#, !" ! 8&&G.("## %.&& 4# ' && % #2 #$ %.!" % # " 2 # ' '*) " +* ( !##' !*( !#& $ .! '*Æ)#'! " 2 # (& $ $ (, '&2.(+!"#''!#J?K .''#)!.(+ !.7) (4#*!"#,(#( !!"#,. +( &'*4G !.)#'-,*)#,.( (#G% ,.''#)!.($ ' 4 2 # #+# J>K'" % #,!".',.''#)!.() (4#,(##)*'.2 #& *( !.& '*Æ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Æ).#( !& )&& -'#,$ !"# !.!.(.(+ 4#+.(' "# !.!.(.(+ ) ( 4# .( !

PAGE 15

; ( 4.! ( * 4# '#!' ,#-#(,.(+ ( !"# &+.!" (, !"# -4&# '!& $ !"# + -" .' *() '#(#, (, !"# .(.!. &!.!.( .' #8(#,$ ( && *'.(+ 2 . !.(' !"# /#(.+" (G.("#*.'!.)J:9K (!"# #)#( ! !#) "(.@*# 4 '") ! (, .* .' ) &&#, *&!.'#)!.( ,#.(+ J9K ".' &+.!"8'!8(,' +,'#!%".) ",.2.,#'!"#+ -".( !! % #'*4'#!' "#'#!.'(!!"#1 #!!".' &+.!" "###$ ( + -"!.!.(.(+ &+.!" 4 2 # ) ( 2.,# !"# '#! ()# '#! .' *(,$ # ) " !.!.( !"#( .' ,#.(+ *'.(+ %" ! .' ###, ! ' &+.!" "# )(G '! .(#,.(. *,#+## &+.!".()&*,#'!"#'#!(,#'.(!"#,#+##) &)*& !.(' !"#!.!.( (,#' "# *&!.'#)! .' &' ,##, *'.(+ '! (, , .(. *,#+## "# !.!.( (, '#! ,#.(+' # %" ! .2 # !"# @* &.! -4&#' %".) " + -" !.!.(.(+ ,#.(+' ! -.) && , -& "###$ *'.(+ *&!.'#)! ,#.(+' " '4##('" %(!-,*)#+,#'*&!' %.,# (+#-4&#! -#' : ) &-) "#' ()( ! '! !+&4 & -) "#'$&) &#!",'-.) 1(#-.2 ! ! !.#$*-, !.(+ !"# !.7 .!' + -" !# # ) " ) ".)# .()# !"#'# # &) & .(..< !.( '! !#+.#'$ % # #(!+* ( !##,!" !!"#8&&G.(.'+&4 &&4*(, % #2 #$ #)#( ! ,2 ()##( !' .( !.# (, ')# )-.! " 2 # ,# !"# 2 # --*& "# && %.(+ ,#').4#' '# &) &"#*.'!.)',#.(+ !.7 (#!"#8'!"#*.'!.)',#.(+ !.7% ' ,*#! 1 %.!< J:?K ".'"#*.'!.)$*'#,(*('#!.) !.)#'$ )!* &&2.,#, )&*( (, %,#.(+ (,% '4 '#,*-(4!"!"#!!#( (,( *#.) &2 &*#'!"# !.7 .2 #( !" ! !"# '!#)!.< !.($ B C .'!"#,#+## % (, B C .' !"# ,#+## )&*( $ !"# -.2 ! #&##(! .' ) "'#( '*) " !" ! B B C CB B C C .' .(..<#, (,%"##!"#( *#.) &2 &*#!"#-.2 !.' ))#-! 4&#( *#.) &'! 4.&.! 0.!"!".') ".)#-.2 !$!"#8&&G.(-!#( !. &!".''!#-.'.(..<#, "# 5-4&# %.!"!"# 1 %.!<'! !#+.'.!'#&. ()#(( *#.) &2 &*#' (,!"#"# 2'# ) ")'!' ''). !#,%.!"8(,.(+ ( ))#-! 4&#-.2 ! ,,.!.( && $! % !.)#'.,#( !.) &((G<#

PAGE 16

A !!#(4*!%.!",.3##( !( *#.) &2 &*#'% *&," 2 #! % ,.'!.()!,#' !.''#!.#' ,#'. 4&#! +#! ( ,#.(+4 '#,'&#&( !"# ((G<# !!#( ' !" ! -.2 ! ,#.(+ ) ( 4# *'#, ( *&!.-&# !.)#' && %.!" !"# ' # ((G<# !!#($ 4*! %.!" ,.3##( ! ( *#.) &2 &*#' ,#.2 !.2 # !"# 1 %.!< &+.!".'!"# .((# (,0 &1 # &+.!"J:K$ & !#( #,!"# ".' &+.!".'!"# '#!.) )*( !#! ! 1 %.!< '.! .'*'#,'#!.) -'.!.2 #,#8(.!# !.)#'$ -.2 ! ) ".)#' , (! (##,!4#) "#) 1 #,( *#.) &'! 4.&.! "###$(&!"#((G<#!!#(.'(##,#, ".' "#*.'!.) ) "'#' !"# -.2 ! #&##( ! !"# % B )&*(C .(. * ,#+## .()#,. +( &-.2 !.(+.'*'#,$!"# !.7# .(''#!.)$%".) " &'.-&.#'!" !!"# -.2 !#&##( !.'.(!"#)&*(B %C.(. *,#+## .1 # 1 %.!<$!".' &+.!" &) &&4*(,'!"#8&&G.( !# ) "'!#'!#'# ) ".(&) & -) "-#,#.(+ &+.!"'" '4##((#.!"#,.8) G !.('!!"# 1 %.!K%"#,*)#,!"#'# ) "')#!"# 1 %.!<"#*.'!.) 4 &..!.(+!"# -.2 ! '# ) " ! %' & % ,#+## #)#( ! % 1 " ' 4##( ,(# ! #,*)# !"# )'! ) &)*& !.(+ ,#+##' .( !"# .(. * ,#+## &+.!" 4 *'.(+ -7. !# ,#+##' J?$ ;$ 9?K ( !"# ) '#'$ !!#-!' " 2 # 4##( ,#! .2 # !"# @* &.! "#*.'!.)' .((# (,0 &1 #J:K--'#, .(. *,#8).#()2 #'.(!"#.(. * ,#+## &+.!" ".' )!* &&.(..<#'# &8&&G.( !# ) "'! +# % #2 #$,#'-.!#'&.+" ! @* &.! .2 ##( !'$ !"# !.# )'!' # -".4.!.2 #& ".+" #)#( ! % 1 ( .(. * ,#8).#() 4 !" 4#+ (, .'#('! ,! J:;K *'# -7. !# ,#8).#().#' ! , !.) && .2 #*( !.#' "# &' '" %.2 #,@* &.! ')#, ! !"# .(. *,#+## &+.!"

PAGE 17

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

PAGE 18

9 0 / "## " 2 # 4##( ( .2 ##( !' ! !"# .(. *,#+## &+.!" '.()# !"# 8'!.-&##( ! !.(4 .((# (,0 &1 #J:K #+# (,.*JAK-#'#( !!"##2 &*!.( !"#.(. *,#+## &+.!" "#.+.( & &+.!"% '4 '#,*-(!"##&..( !.( + -",#& ".',#&.' (#7-&.).!+ -"#-#'#( ! !.(!"# !.7 !# ) " '!#!"# #&..( !.( #! !".' + -" 4# ! # ) " '!##&..( !.( !"# (,# .(. * ,#+##.') "'#( (,,##, #!!".'(,#4# ,+#' # ,,#,! '*) "!" ! &&(,#' ,5 )#( !! #*&&)((#)!#, #7!$ &&#,+#' ,5 )#( !! #!"#(# 2 #,!"# + -" (, .' &' # 2 #, "# #'*&!.(++ -" .'#-#'#( !#, 4 ".' '#@*#()# .' #-# !#,*( !.& &&(,#'" 2 #4##(#&..( !#,##.+*#9 -'#*,G),#,#').-!.( !"# &+.!" "## # !"## 5 -4&#'%.!"!".'.-&##( ! !.( !"# .(. *,#+## &+.!" .()# ! # ) " '!#% # ,, (,,#&#!# #,+#' !"# + -"$ % #) ((! -#,.)! "# , !.#!"# 7. *'! +# #@*.##( !'!"# &+.!" "#'#)(,-4&# ''). !#, %.!" !"# .(. * ,#+## &+.!" .' ) &)*& !.(+ !"# ,#+##' (# ) ( '## !" ! ,#+## *-, !#' # #@*.#, # ) " (,# ,5 )#( ! ! !"# #&..( !#, (,# ! " ' 4##( '" %(!" ! !"#'# ,#+## *-, !#' # !"# '! )-*! !.( && .( !#('.2 #! !"# I+.2 #('#!.)+ -" I '#&#)! (,,#(,# .(. *,#+##.( I .+*#9H "#4 '.).(.*,#+## &+.!"

PAGE 19

= &+.!"JK "#8( &-4&#.'*'.(+!"#.(. *,#+## &+.!"! 4! .( +, )&*(,#.(+*('#!.) !.)#' "# # .(,# !".' ) " -!# .' ,#2 !#, ! ,.')*''.(+ 2 .*' '&*!.(' ! !"#'# ! % 5-4&#' 9 ''&..( !.( (# '&*!.( ! !"# ,#+## *-, !# -4&# .' ,*# ! #+# (, )( ! # JK "# (!#, !" ! !#( !.#' '#! (,#' ) ( 4# ,##, !+#!"#$ !" *' #,*).(+ !"# ( * 4#!.#'!"#,#+##*-, !#-" '# *'!4#,(# ".''#!) (4#,#!#.(#,4 !"# && %.(+!"## #! B C!"#( B CI B C . (,(&. B C I B C ,#' !" ! ) ( 4# +*-#, !+#!"# 2. !".' !"## # &' 1( %( ' .(,.'!.(G +*.'" 4&# (,#' ".' &# ,' *' ! !"# (#7! 5 .2 ##( ! .( !"# .(. * ,#+## &+.!" 99 *-#(,#' '"##'! !#'$! % (,#' (, #.(,.'!.(+*.'" 4&#. (,(&. B C I B C !" ' &'4##('" %(4 #+# (,.*J;K!" !.! % (,#' # .(,.'!.(+*.'" 4&#.( !"#( !"# # .( .(,.'!.(+*.'" 4&#.( /( %.(+ !".'$ .(,.'!.(G +*.'" 4&#(,#') (4##-#'#( !#, ' (#(,#.(!"#, ! '!*)!*#' ".'.'###,! ' 0"#( !".' '*-#(,# .' ,##,$ && )('!.!*#( ! (,#' # ,##, "# *'# '*-#(,#') ( '.+(.8) ( !&#,*)# !"# *( ! .( !#( ()# % 1 #@*.#,,*G .(+#&..( !.( (&!"# ,#+## (# #-#'#( ! !.2 # (,# !"# '*-#(,#(##,'! 4# .( ! .(#,

PAGE 20

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

PAGE 21

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

PAGE 22

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

PAGE 23

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

PAGE 24

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

PAGE 25

; ".') (4#*(,2 #@*.) 1& "#, %4 ) 1!".' -) ".''#%" !& % #@* &.! -# *! !.('%".) "#'*&!.(#8&&G.(!" (!"#*'##7 )!,#+##' #'! $ 2.'$ (, *3 *'# # '-".'!.) !#, -7. !.(.( %".) " " '4##('" %(!.#&,".+"@* &.! ,#.(+' ".' -7. !.(-#''#!G,.3##()#' 4#! % ##( !"# (#%&G#,)&.@*# %.!" &&!"# )&.@*#'! %".) " &&(,#'.( 4#&(+ "# ,#+## # ) " (,# .( .' ,#!#.(#, ' !"# '* (, !"# '.<# !"# '#! ,.3##()#'# ) " ,,.!.( &)&.@*#!%".) "!"#(,#4#&(+' ".'.'!"# -7. !.( &'*'#,4 9= -7. !#.(. *#8).#() #)#( ! % 1 4 !" 4#+ (, .'#('! ,! J:;K '" % !" ! ,.8) !.(' ! ( -G 7. !# .(. *,#8).#() ,#.(+ &+.!" # &'! ' '! (, -,*)#'4#!!# ,#.(+' !" ( )4&# .(. * ,#+## &+.!" "# .(. * ,#8).#() &+G .!"!.#'! .(..<# )!* & 8&&G.( !# ) " '!##&..( !.(.('!# ,.(..<.(+!"# 4*(,(8&&G.( '!"#.(. *,#+## &+.!",#' "#--'# ) "# -7. G !.( ! .(. *8&& (, ! % (#% "#*.'!.)'4 '#, *-( !"# .(. * 8&&%".) " -# % #&& !" 4#+ (, .'#('! ,! --'# 4*(, ( !"# .(. * 8&& 4 '#, *-( !"# @*!.#( !+ -"$ #! B C4#!"##7 )!8&&#'*&!.(+.(!"#'#&#)!.((,# '-.2 ! "#(!#!" !+.2 #( )&.@*#$ .( $ &&.'(,#'.(!"#)&.@*# # &# , ,5 )#( ! "###$!"##7-#''.( -7. !#8&&(,# %".) ".' # 4#)&.@*# $.' +.2 #(4 H B CIB C 9 B C 9%"## I & $%"## .'!"#,#+## (,# "# *(, !" ! &#!!.(+ 4# !"# '! #)#( !& )# !#, )&.@*# )( ! .(.(+ .' '*Æ).#( !!4! .( +,4*(,(!"#8&& "#8'!"#*.'!.)!"#) && & ".'"#*.'!.)(!#' !" !!"##&..( !.( '*-#(,# # ('!"##&..( !.( & .+.( &(,#' "###$ . !"# !! & 8&& ''). !#, 4 !".' '#&#)!.( .' B C$ !"#( !"# 2 # +# 8&& .' B C & "#*(,4#!!##'*&!'4 *'.(+ B C &

PAGE 26

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

PAGE 27

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

PAGE 28

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

PAGE 29

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

PAGE 30

9 50 100 150 200 250 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Matrix numberLog2 of ratio .+*#:H (.!. &,#+##').'(H 9B C ##! !"#.+.( &*('#!.) !.7 "###$'*-#(,#'4#)# "# !# %.&& 4# *'#, .('!# , &'$ !"# !# %.&& 4# *'#, .('!# , % #2 #$ !"## ,#'"*&,1 ##-.(.(,!&&#&'4#! % ##(!"#'#!.) (, *('#!.)!#' "#@*!.#( !+ -" ) (( %4#,#').4#, ' '#! %' (,)&*(' ".'.'#-#'#( !#,.(4 !"#'#!' # (, # I H I= # I H I= # .' ) &&#, !"# % '!*)! (,$ )*'#$ # )" # .' )&.@*# .( .' 1( %( '!"#)&*('!*)! #-#'#( !'(,# .(!"#+ -"

PAGE 31

99 &*(' # ,.2.,#, .( ! ! % +*-' "# 8'! +*# !"# )&*(' -.().&)&*( ( #' !"# '*-#)&*(! %".) " .! .' # 4# "# '#)(, ! -# .' % )&*( ((G-.().&)&*(.' # 4# '*-#)&*( % #2 #$ .!,#'(!#-#'#( !!"#'*-#)&*(.(!"#, ! '!*)!*# #7 -&#$&#!)&*( 4# -.().& (, 4# ((G-.().&)&*(4#&(+.(+!!"#' #'*-#)&*( ' "#($ !"# '*-#)&*( .' ,#(!#, ' I #! ,#(!# !"# '#! '.<# '*-#)&*( ,,.!.( && $&#! # I ( *-#)&*(,#!#)!.( .' -##, !# !"#,#+## -7. !.( -" '# .')G -&#!# *.(+ !"# ,#+## -7. !.($ " '" *()!.( # ) " )&*( .' ) &)*& !#, 4 '#, *-(.!' )('!.!*#( ! % .(,.)#' *.(+ '*-#)&*(,#!#)!.($ )&*(' # )G #,,.#)!&%.!"!"#)&*('%.!"!"#' #" '"2 &*# ! % )&*(' #*(,!4# .(,.'!.(+*.'" 4&#$(#)&*(.') "'#( 4.! .& '!"#((G-.().&)&*( ''*.(+ (, #*(,!4#.(,.'!.(+*.'" 4&#B!" !.'$ I C&#! ( I (, I .' ) "'#( ' !"# ((G-.().&)&*($ !"#( '*-#)&*( I (, ( .' # 2 #, !"#, ! '!*)!*#' !#$ I; && %.(+!"#'*-#)&*(#+.(+ ".' '*-#)&*(,#!#)!.( .'''# .(,.'!.(+*.'" 4&#)&*(' ".'.'4#) *'# .(,.'!.(+*.'" 4.& .! .( !"##&..( !.( + -" B # I # C ,#' (! .-& .(,.'G !.(+*.'" 4.&.! .(!"#@*!.#( !+ -"B I C % #2 #$!"##2 #'#.'!*# (,'.()#!".' #!",.' '!#!" ()-&#!#'*-#)&*(,#!#)!.($ .!.'*'#,.( #! B C 4# !"# '#! )&*(' ,5 )#( ! ! )&*( '#, ( !"# % (, )&*(#-#'#( ! !.($!".') (4#%.!!#( '&& %' B CI # #! P I B C 4#!"##7 )!#7!#( &,#+##)&*( "##7 )!,#+##.'! )'!&!)-*!# !"#.(.!. &(,#,#+##'*'#'!"#,#+## -7. !.( .&4#!$&#$ (,) "#.4#J9?K ".'" '4##(,#('! !#,.(#)!.( :9 ! -,*)# 4#!!#@* &.! ,#.(+!" (*'.(+!"#!.+" !#,#+###'! $ 2.' (,*3

PAGE 32

9: #! I B # C P 4# !"# .(.!. & -7. !# )&*( ,#+##' 4 '#, *-(!"# -7. !.(.(J9?K #! 4#!"#-.2 !)&*(B!"#)&*(' &&#'!,#+## C .()# .'-.().&$ &&.!')('!.!*#( !)&*(' *'!4#,##,( % ".'.'!"# '' #&..( !.('!#! !".'-.( !$ (#% % .' #, '*) " !" ! # I B C #! !".' % 4# 1( %( ' !"# -.2 ! % !# .(+ # $ && %' # ,#&#!#, !"# , ! '!*)!*#' (,!"#.(,.)#'! !"#'# %' # #2 #, # ) " H # !#$ ' '! !#, .( "##9$ # ! 1 #' *-'!.)!&&#''')# !" ( # "###$ !".' &+.!" ) ( -# !# .( ( *( ! # !" ! .' 4*(,#, 4 !"# '.<# !"# .(.!. & @*!.#( !+ -" ! !".' -.( !$ && )&*(' # " 2 # !"#. ,#+##' *-, !#, *'.(+ !"# #'! $ 2.'$ (, *3 -7. !.( "# -7. !# ,#+## )-*! !.( ! 1 #' -& )# .( ! % -" '#'"#&& %.(+ *& ,#').4#'!"# -7. !#,#+##H I # D # # ( "#8'!-" '#.'!"#) &)*& !.(!"# %'#!,.3##()#'$ ! B CI # # (.!. && $ ! B CI # # ) " "#'#! ,.3##()# # ) " %%".) ")( ! .(' )&*( &' .( !"# -.2 ! % .' -##, 4 ! 2 #'.(+ !"# '#! # ) " )&*( # ' # ) "% .'#()*( !##,$!"#'*-#)&*('#!'.<# ( .''*4! )!#, ! B C !#$ ( ! B C4#)#'#@* &!=$!"# % .' '*4'#!!"#-.2 ! % (,!"### 4## 2 #,!"#, ! '!*)!*#' ".'.'###,! ' ++#''.2 # 4'-!.( "# '#)(, -" '# !"# ,#+## -7. !.( .' !"# '*.(+ !"# % '#! ,.G ##()#'$ ! B CN' # ) " )&*( # $ &#! !"# ,#+##$ I ! B C !#$ !"# N' # (! !"# 8( & ,#+##' '.()# !"# ,#+## !"# -.2 ! % # % ' (! .()&*,#, .( !"# ! !".' -.( !$ . ( !#,#+##' #@* & <#$ *!"# '' #&..( !.( ) ( 4# ,(# ".' '.!* !.( ))* ,#'-.!# '*-#)&*( ,#!#)!.( 4#) *'# ' #( !.(#, 4 2 #$ '#.(,.'!.(+*.'" 4&#)&*(' 4#.''#, ''*.(+)&*( .'*(,! 4# .(,.'!.(+*.'" 4&#%.!"!"#-.2 !)&*( (,#&..( !#, !!".'-.( !$!"#( # I #

PAGE 33

9> '#( !.(#,4##$'*-#)&*(,#!#)!.(.'-##,( % !#$'*-#)&*( ,#!#)!.( #,*)#!"#( * 4#-.().&)&*('.( # 4*!%.&&(! 3#)! # "#& '!-" '#!"# &+.!".'-#.(+!"#8( &,#+##)-*! !.( ".'.' '.-&# !!# ,,.(+!"#,#+##!"#-.2 ! %$ # ! &&# .(.(+-.().& )&*(' (,'*4! )!.(+!"#'*-#)&*('#!'.<# I D # ( "#'*-#)&*( '#! '.<# ( .' '*4! )!#, ! 4! .( !"# #7!#( & ,#+## '!& $ !"# -.2 ! % *'! 4# ,,#,!# ) " H # &+.!"'" %'!"#+#(# &*!&.(# ''*.(+ !.7 %.!" .!' )&*(' &# , ,##, 4 .' +.2 #( ,,.!.( && $. !"# "&#'1 )!$ $ .' &' 2.,#,$!"# !.# )-.! ) (4#4*(,#,4 !"#&& %.(+%"## .'!"# %.( (, .'!"# )&*(.( $ ".' ) ( &' 4# #7-#''#, ' !"# !.# (##,#, ! )-*!# )( ! .(' *&& %$ %.&& 4# ,#('# (, !"# !.# )-.! 4#)#' $ B C )( ! .(' *&& )&*($ %.&&)( ! .( *&& % (,!"#!.#)-.! .' $ B C % #2 #$#2 #(.(!"#'# % '!) '#')#( .'$'*-#)&*(,#!#)!.($ ''#&..( !.($ %#+.(+$ (, ++#''.2 # 4'-!.(%.&& &&#,*)#!"# )!* &*( !.#.(!"#&& %.(+ ((#H *-#)&*(,#!#)!.( (, ''#&..( !.(#,*)# %#+.(+#,*)#' ++#''.2 # 4'-!.(#,*)#' "###$!"#!*#*( !.#) (4##7-#)!#,!4#% #&&4#& %!"#!"##!.) &*--# 4*(,

PAGE 34

9? % ')*+ , -./# ) , I % # I H I= , , I % I H I= ( I I B # C , ) I I '#&#)! (,,# + '*) "!" ! .'.(..<#, I D + I D ) # IB # C , # ) " , # I , %* ) , # ) " # , ''*# ! B CI # && I , # ) " , , ! B C = % ! B CI # , ! B CI ! B C ( ) , ! B CI= I # I , , ,

PAGE 35

9; % % ) , # ) " # I ! B C ) , I= % ,# ( I D ( # I # I , , ) , # ) ". (, # , (, #.(,.'!.(+*.'" 4&# % I ( # I # I , , & ) , # ) " # I D # ( I , I

PAGE 36

> ".') " -!# -#'#( !' !"# .-&##( ! !.(,#! .&' .()&*,.(+4!"!"# &+.!"' (,, ! '!*)!*#'*'#, "## ,#.'###,!--#(,.7$%".) ")( ! .(' !"#)-&#!#),# "# ),# .' '#!#, .( ! * 5 *!.(#'$ && ) &&#, !"# .( *!.(# "#'# # $ $ & $ (, "## # &' ! % '-#). & *!.(#' (, ( ,,.!.($ !"## #'.7 5, ! '!*)!*#' .( ! .(#,,*.(+!"##7#)*!.(!"# &+.!" "# && %.(+,#').4#'.(,#! .&!"#, ! '!*)!*#' (, &+.!"'*'#,4 > ! !*)!*#' "## #'.7, ! '!*)!*#'*'#,4 "#'# #!"# $ $ $ $ $ (, ( ,,.!.($ !"## # 82 # -. +*#( !'H $ (, ,,.!.( && $ !"# *'# '' '.7!" +*#( !$ "# +*#( !' #,#').4#,4#& %H H ( * 4# %'.(!"# !.7 H ( * 4#)&*('.(!"# !.7 H %.(,.)#'!"# !.7-&*' ,,.!.( &'! +# H &#(+!" .(% ,' H )&*('! ! -.( !#'.( ! "# % .(,.)#')&*( # &) !#, .( J B C B DC K "#( * 4##( !.#'.(!"# !.7 .' I B C H -!.( &*'#'#!!.(+'!)( !&,#('# % (,)&*(-#)#( ! +#' 9A

PAGE 37

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

PAGE 38

9 '" #,2 . 4&#.(!"# %'!*)!.'#.!"#!"# % -.( !#2 &*# "#,#+## .(,.) !#' !"# ( * 4# .+.( & *(G#&..( !#, )&*(' !" ! # # .(.(+ .( !"# % "#-.( !# .'*'#,,*.(+.(.!. &.< !.(%"#(4*.&,.(+!"# %!"# !.7 .( && $ % # " 2 # (!"# '" #, 2 . 4&# 1( %( ' #.!"# .' *'#, ,*.(+ 2 .*' -" '#' ! .(,.) !# !" ! +.2 #( % " ' 4##( '##( ! .' &' *'#, ! '!# !"# % '#! ,.3##()#' ,*.(+ !"# -7. !# ,#+## )-*! !.( -" '#$ ! B C .( &+.!" .' *'#, !"# *!.(# B'## #)!.( >9?C (,.'*'#,!!#.&'!#!"#8'!)&*(.(,#7!"# % >9 &*(!*)!*# (, #( ! ## "#)&*('!*)!.'2 #'..& !!"# %'!*)! .1 #!"# %'!*)!$!"## # (, %".) " # .,#( !.) & ! !"# % '!*)!N' *' +# ,,.!.( && $ !"## # #% '" #, 2 . 4&#' "# 8'! .' -.( !# %".) " .' *'#, 4!" !"# " '" ! 4&# (,,#+##&.'!, ! '!*)!*#' "#'#)(,'" #,2 . 4&#.'#.!"# ## )&*(.',##, 4'4#,4 #( !'*-#)&*($.!.'.!'#& '*-#)&*( %.!" !".) 1(#'' +.2 #( 4 )&*( .' 4'4#, .( ! '*-#)&*($ .(,.) !#' !"# #( ! !"# )&*( !"# #( ! !## , ! '!*)!*# "# !"., '" #, 2 . 4&#.'#.!"# ## )&*(.'#&..( !#,$% # .( ! .(!"#)&*(')# "## !# #&..( !.($ !"# ,# !"# )&*( .' .( ! .(#, %.!" "# & '! '" #, 2 . 4&# !"# )&*( '!*)! .' *'#, ' !"# ,*.(+'*-#)&*( ,#!#)!.($!"# -.( !#.(!"#,#+##&.'!$ (, '-#). &-*-'#) &&#, &&)&*('.(!"#,#+##&.'! .( ! .( -#2.*'-.( !#!.(,.) !#!"#)&*( "# , .!.(!"#&.'! ( )&*('*(,#+.(+#G').(+ #(!.(!"#,#+##&.'! (,!"###) ( *'#!".'')#!#),!"#" '"2 &*#'*-#)&*(,#!#)!.( "#'-#). &-*-'# .' #'*&!!"# )!!" !!"#,#+##&.'! (,!"#" '"! 4&# #4!")('!*)!#, (!"#' # (!"##2 #( ! !" ! )&*(4#.(+-*!.( !!"# " '"! 4&#)&&.,#'%.!" )&*( !!"#"# , ,#+## &.'!$!"# )&*( ! !"#"# , ,#+## &.'!" '((##,

PAGE 39

:= .!'-#2.*'-.( !# (,) ( !"### *'#!".'')# ! -.( !! !"# " '"#, )&*( '..& ,#+##&.'!.'*'#,.(9AJ=K >: (,#7 "#.(,#7 .''.-& ( '.<#9 D "##7! $ .'+.2 #( '% 1.(+')# (,.'*'#,4 !"#,#.(+-" '#!"# &+.!"!)# !#(#% #+#, %' "# "&,' !"# )&*( $ B && C$ && % #, 4 !"# % $B # && C !"#')#&& %.(+!"# %.'.('*Æ).#( !!)# !# (#% #+#, %$+ 4 +# )&&#)!.(.'-##,!#)& .*( *'#,')# >> "##+##.'! (, '" 4&# "# &+.!" *'#' ( &#(+!" D !"# "# , 4!" !"# ,#+##&.'! (,!"#" '"! 4&# "#'#, ! '!*)!*#''" #!"# "#'" #,2 . 4&# .(!"#)&*('!*)!*#.'*'#,! " (,&#)&&.'.('4#! % ##( !"#" '"! 4&# (, !"#,#+##&.'! "##!"#!".' .'+.2 #(4 !"#*'#.(!"#!"#)&*( -.( !# "#,#+##&.'!' #,*4&&.(1 #,!#( 4&# 4.! ,,.(+ (,# 2 & )&*(' "#" '"&.'!' #'.(+&&.(1 #, >9 "#&+.!"' "## # * .( *!.(#' *'#, 4 "#'# # && ) &&#, !"# .(*!.(#$ %".) ".'*'#) && 4&# "# &+.!" 8'! .(.!. &.<#' && !"# , ! '!*)!*#' (, -#' .(-*! # ) "#) 1.(+ .( !"# *!.(# "# *!.(# -#'-#G#&..( !.((4!"( *&& (,,#('# %' (,)&*(' ".'.'!"#(&& % #,4 .(.!. &)&*(').(+$ &'.(!"# *!.(# "# & *!.(#-#' '! !"# % 1 ! .' "## !" ! && !"# '*-#)&*(' # ,##, "# *!.(#'.-&,#' &&!"#((G-.().&)&*(' 4'4#,.( !'*-#)&*(' ( ,,.!.($! % #*!.(#' #) &&#,4 !"# & *!.(#!-#G + 4 +# )&&#)!.( (, '*-#)&*(,#!#)!.( "#'# # !"# (, *!.(#'#'-#)!.2 #&

PAGE 40

: >9 .(*!.(# "#*'#8'!) &&' (,''#'!"# +*#( !' $ (, !# '# -#&..( .(-*! # ) "#) 1.(+$ !"# .( *!.(# !"#( ) &&' !"# * .( *!.(#' .( '#@*#()# *.(+ !"# #7#)*!.( !"#'# ),#'$ !"# .(-*! +*#( !' (, #,#'! #, ,,.!.( && $!"# +*#( ! .' 2 #%.!!#(%.!"*!-*!'! !.'!.)' !"#*'# "#'# #H !"#( * 4#,#('#E( *&& %'*(,Q( * 4# ,#('#E( *&&)&*(' *(,Q!"#( * 4#+ 4 +#)&&#)!.('-##,Q (,!"#'! !#!"#.(-*! +*#( ! "#.(-*! +*#( ! ) (" 2 #(#!"#&& %.(+'! !#' (,'!.&&4#2 &.,H && %.(,.)#' .( )&*( # .( '!#, ,# (, ((G#-# !.(+Q !"## # *('!#, (,E #-# !#, % .(,.)#' .( (# # !"# )&*(' "#& !!# '! !# #@*.#' # .( !#('.2 # .(.!. &.< !.(!)#)! >99 (.! %' &' "# .(.!. &.< !.( '#)!.( '#!' *!"# )&*( (, % , ! '!*)!*#' (, &' ) "#) 1' )&*( -.( !#' (, % .(,.)#' *! (+# (, ,#)# '.(+ )&*( -.( !#' .( (#!"#'# .'*(,$!"#.(.!. &.< !.(#!*(' ( #),# && %.(+!".''!#-$ !"# % !"# !.7.'+#(# !#, *.(+!".''#)!.($ .! .',#!#.(#,%"#!"# (! !"# )&*(' # 5* 4&#, &*(' # 5* 4&#, . ( !"# % .(,.)#' %.!".( ( !"# )&*(' # #-# !#, ((G.()# '.(+ !"# )&*(' # 5* 4&#,$ ( ,,.!.( & -" '# *'! 4# #7#)*!# ! #G4*.&, !"# )&*( ".' '#)!.( " ' !.# )-.! SB D D C >9: (.! ).(+ "#.(.!. &').(+-" '#.',.2.,#,.( !'#2 # &'!#-' "#'# #,#('#E( *&&)&*( (, %# 2 &$.(.!. &')#)-*! !.($ (,)&*(.('#!.(.( !!"#,#+##&.'!' "#8'!'!#-,#' ( #-! )&*('& '! ".'.',(#'!" !.(!"##2 #( !!" ! '.(+*& !.7 .' +.2 #( ! $ )!.< !.( ) ( -)##, ' ' -''.4&#

PAGE 41

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

PAGE 42

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

PAGE 43

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

PAGE 44

:? >9? 4 +# &&#)!.( 4 +# )&&#)!.( .' ) &&#, 4 !"# & *!.(# %"#( !"## .' .('*Æ).#( ! !)# !# (#%#+#, % 4 +#)&&#)!.(&1' ! &&&.2 #)&*('.(!"#, ! '!*)!*# (, 2 #' !"# )( !.+**'& ! !"# ( ! ".') ( 4# ,(# 4 &1.(+ ! )&*('!"*+" '.()#!"#-" '.) &,#!"#)&*('# .('*() " (+#, '!"# -)##,' % #2 #$ 4#) *'# !"# -" '.) & &) !.( %' ,#' (! (#)#'' .& )#'-(,!!"# %N'.(,#7$!"#(#')-&.#( !!"# %( * 4#.(# ) " &.2 # %.' '!#,.(!"#-& )#!"#8'!)&*(.(,#7 "#8'!)&*(.(,#7.''!#,!#.& .( !"# % '!*)!*#N' 2 . 4&# .' !"#( '# ) "#, ( (#+ !.2 # 2 &*#' 0"#(*(,$!".' 1'!"#'! ! % (,))!.() ('! !!"## .()#!"#*'# +.2 #' !"# &+.!" ! &# '! #7! #&4 % ! )# !# !"# 8'! -.2 ! %$ !# + 4 +#)&&#)!.($!"##.'+* ( !##,!4##(*+"')# ( '*4'#@*#( !-.2 ! %4 "##9 "#!.#)-.! + 4 +#)&&#)!.(.' $ B C "#+ 4 +#)&&#)!.( '! !#+*'#,.(.''..& !!" !*'#,4 B9ACJK (,B>ACJK >9; #!#)! *-# &' *-#)&*(,#!#)!.(.') &&#,4 !"# & *!.(#&& %.(+!"#'*.(+ %'#!,.3##()#'# ) ")&*(.(!"#-.2 ! % !.'-##,4 &1.(+ !# ) " )&*(.(!"#-.2 ! % .2 #( )&*($8'!.!.' ,#'*#!" !.!" '(!4##( 4'4#, .( ! (!"# '*-#)&*( .! .' *( 4'4#,$!"#( % # &1 ! !"# " '" &.'!+.2 #( 4 !"# )&*(N'" '"( * 4# &&)&*('.(!".'" '"&.'! #!"#(.G%.'#)#,%.!"# ) " !"# ( ! % .(,.'!.(+*.'" 4&#)&*(' ##+#,.( !(#'*-#)&*( ".'-)##,' *( !.&!"#" '"&.'!" '4##()-&#!#&'# ) "#, "#(!"## .(.(+)&*('.(!"#-.2 ! % # ) "#) 1 #, !# !" ! !# !"# 8'! )&*( .( !"# -.2 ! % " ' 4##( ) "#) 1 #,$ !"# &+.!" 8(, &# , 4'4#,)&*('%".) ".!%.&&'1.2 # "#% '!) '#!.# )-.! '*-#)&*( ,#!#)!.( .' $ B C *! !".' % *&, (& ))* . & +#( * 4#)&*('% ##.(!"#-.2 ! % (," --#(#,!" 2 #!"#' #" '"2 &*# 4*!% ##(!.(,.'!.(+*.'" 4&#!# ) "!"#

PAGE 45

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

PAGE 46

? "#&& %.(+'#)!.('-#'#( !!"##'*&!'(!"##'#!'!#'! !.G )#'H '@* #*('#!.)$#)! (+*& $ (,'#!.) "#'# # 44#2. !#, '!"# $ $ (,!"# !.)#'#'-#)!.2 #&&& %.(+!"# % #&&G#.(+( .(+)( 2 #( !.( ## !" .( !.(( !"# % #&&*4*G !.(# .4 .' )#, + .('! (, "# 2 #'.( *'#, .(!"#'# #7-#.#( !'.' "# (&,.3##()#4#! % ##( 2 #'.('= (, .' !"# )#)!.( ,#('# % (, )&*( ,--.(+ .( ' #( !.(#, 4#G #$ .'!"# )&*(.(. *,#+## &+.!"*'#,4 ,# *&!$ .! -#' '*-#)&*( ,#!#)!.( (, #&##( ! & 4'-!.( #2 # !"## .!# !.(' ,G ,.!.( && $.! *'#' ( -7. !# ,#+## ,#').4#,.(#)!.( 9 %".)" .'(! ' !.+" ! ' !" !*'#,4 ! &'*'#' *&!.-&##&..( !.( (,#! ) ".#2 #& +#'#!' .(,#-#(,#( !-.2 !'$)# !#' (#&..( !.('#!)( ! .(.(+)&*('%.!",#+##' %.!".( !"#'"&, 9 D$ %"## .' !"# .(. * ,#+## J9?K .' '#!.).(. *,#+## &+.!" !.''..& !>A.(!"# % #&&*4*!.(# .4 >A (, #,#').4#,.(!"#-#4 #'! $ 2.'$ (,*3 JK # !*#' >A .()&*,#!"# *'# @*!.#( ! + -"$ !"# ' # -7. !# ,#G +##*'#,4 $'*-#(,#,#!#)!.($ (, ++#''.2 # 4'-!.( .'!"# ' # '>A #7)#-!!" !.!& ) 1' ++#''.2 # 4'-!.( >A .'(&*'#,.(!"# '#!.)#7-#.#( ! "#!"###7-#.#( !'% ##,(#( *(&! ( !#-.'#>===E?=== %.!"9 .(# (,??2.!* &# ".' ) ".(#" '#.+" !9><0$ &! G ' B(& (# -)#'' % ' *'#,C && !"##7-#.#( !' % ## *( :A

PAGE 47

: !"*+" *'.(+#7*()!.( .( !# )#' (, >A # %.!!#( .( $!"#!"#),#' #.( -&.).! <#' .( !"# % #&&G#.(+ !.7 # !# !#, ' 4&.) && ' ((G<#'.(!"#, ! '!*)!*#' % #2 #$ ,-'#7-&.).!<#' -#2 #( !!".'$ #7-&.).!<#' ##-& )#,%.!"!"#2 &*#= $-.!& ,.(+!"# !.7.( ! ? "#@* #('#!.)B C !.)#' .(,.(+ +, )&*( ,#.(+ )!.< !.( .' !"# -. + & G !"#'# #7-#.#( !'$ .' !#'!#, %.!" 9A? *('#!.) !#'! !.)#' .( (.2 #'.! &., ''# !.7 )&&#)!.( ! T "#'# !.)#' (+# .( '.<# #% " *(,#, ! 2 # .&&.(((G<#' -#G ()# (, @* &.! ).'($ (, B( #7-&.).!&C # &' *'#,(!"#'# !.)#' ) " !.7% ',##,,.#)!&4 (, $!"# ' 4&.)!!#( % '8'!#,B.( #7*()!.($(!.( C (,!"#(,##, "#!.#! .'.()&*,#,.(!"#,#.(+!.# ()# !"# ,#.(+% ' ,(#$ )!.< !.( % ' -##, 4 !"# *-# ) 1 +# *'.(+G .( !.( 4*!*-#'## T '% #&& '###()#JAK ? #'*&! 4&#' .+*# ? '" %' !"# !.)#' .(,# .()# '.(+ + "#4#'!F)*( ! .'!"# & % #'! F !.(+ -.( !-# !.( )*( ! *(,%.!" ( !"# ,#.(+'!#'!#, "# 2 #!.) & &.(# '#!#' !.)#' !"# ' &&# (#' !.7 .' ###, ! ' . .! #@*.#, .(. * = F-' # ! )!.<# 4&# ? &.'!' !"# !.7 ( * 4#$ ( #$ ,.#('.( B C$ ( * 4# ((G<#' .( !"# !.7 B C$ !"# ( * 4# ,#('# %' (,)&*('$ (,!"# + #'!F-)*( !##'!!"#4#'!F !.(+ -.( ! -# !.( )*( ! .( .&&.(' )!.< !.( #'*&!.(+ !"# !"## ,#.(+ #!",' !.)#' #( * 4##,4 .()# '.(+4#'!F-)*( ! (1 ##-.(+%.!"!"#,# *&!

PAGE 48

: 0 50 100 150 200 250 30 0 0 2 4 6 8 10 12 Matrices in increasing flop count orderLog10 of best flop count for all methods .+*#?H &!!.(+,#!"# !.)#'H 4 4#'!F-)*( ! 4#" 2. N'*()!.($ ,#('# %)&*(.')('.,##,L,#('#M ..!)( ! .(' 9#((<##( !.#'

PAGE 49

>= 4&#?H #(# &.( !.(!"# !.)#' ! " " #"$ %" &" # "" ' ' !& ( ' ' #( ' ' ( "" ( ' ' # ' ' " ) ' ' * ) ) ' ' ) * ) ) ' ' &! ' ' !& # ' ' % " ' ' " (( ' ' ( # %! ' ' #+%# ( ' ' % " (( ' ' % " ' ' ( ) ( ( ' ' #+% ( ' ' !& ' ' % " ( ' ' ,( ( ' ' "! ) ) ' ' ( &! ' ' ) !-#( ( ' ' !& ) ' ' # ' ' ) ,( ( ) ' ' % "( ( ' ' &" ' ' &" ' ' , ))( ' ' ( !& ( ' ' ( ( # ) ' ' &! ' ' +" ) ) ( ' ' +" ) ) ( ' ' ) +" ) ) ( ' ' % ( ' ' ( " ) ' ' ( . ( ' ' ( ( " ( ' ' ) ( # %! ' ' (( & ' ' ( &! ' ' ( % " ) ) ' ' ( /-)) )) ( ' ' () ) ( ( +" ) ( ) ' (

PAGE 50

> 4&#?H )( !.( *#, ! " " #"$ %" &" +" ) ) ' ( +" ) ) ' ( # ( ( ' ' ( & () ))) ' ' () ( # "" ) () ' ' ' ' +" ) ) ' % " (( ' ' ) " ' ' " *& ) ) ' ' "-& ) ' # * ) (( ' ' ) " (( ' ' ) # ) ) ' ' ) ( &! ' ' ) % " ' ' ) "-& ' . ) ) ' ' ) % "( ( ' ' & &. ( ' ' . # ) ' ' ! ) ' ' #+% ) ) ) ' ' #+%# ) ) ' ' ( . ) ' ' ) ' ' ( ))( ' ' "-& ( ' ) # ( ) () ' ' ) ( ' ' ) % " )( ' ' ) ! ' ' (( ) +" ( ( () ' ( ) % ' ' )( % () ' ' ) +" ( ( () ' ) +" ( ( ( () ' ) +" ( ( () ' )) !-( ( ) ' ' ) %( ( ) ' ' ( ' ' # ) ) (( ' ' ) # ) ()( ' ' ) !-" ) ) ' ' ) ( # ) ( ' ' (( #+% ) ) ( ' ' ( #+%#) ) ) ' ' ( # ( ) ( ' ' ( ) % ") ) ' ' ( # ) ( ' ' (( . ( ( ( ' ' (

PAGE 51

>9 4&#?H )( !.( *#, ! " " #"$ %" &" # ) ( ' ' ) * ' ' ( &" ' ' ( "-!( ( ) ' ' * ) ' ' *( ' ' &" ' ' ) * ' ' () ' ' * ' ' ( "! ( ' ' ) +" ' ' ) + () ' ' ) ( () ' ' ) () ' ' ) ( () ' ' ) () ' ' ) ) () ' ' ) +" ' ' ) () ' ' ) ((( ' ' # * ) ' ' +" ' ' ( .! ( ) ' ' ) "-! ' ' () .! ( ' ' . ' ' ) ( ' ' ) ' ' ) ' ' ' ' ) (( ( ' ' ( # ) ) ' ' ( &( ) ' ' +",* )) ' ' # * ( ' ' !+ () ' ' ) " ( ' ' )( (( ) ' ' ( ( ()( ' ' ( "! ' ' ( -" () ) ' ' ) ( %( () ( ' ' ( (( ( ( ) ' ' ( ( &( ' ' ( ( " )) ' ' () ( () ' ' () #"() () )) ' ' ( # * ' ' (( ' ' )

PAGE 52

>: 4&#?H )( !.( *#, ! " " #"$ %" &" "% . )( ' ' "% . )( ' ' #"#() () )) ' ' ( " ) ( ' ' ( )() ( ' ' ( "**( ( ( ' ' ( ( ' ' ) " )) ' ' # * ) ) ' ' ) # * ) ) ' ' ) !# ' ' ) #() () ' ' )) # * ( ( ' ' )) ( * ) ( ' ' ) ) ) ' ))( * ) ( ' ' . ( ' ' ) ) ( ' ' (( * ) ( ' ' ( ) ) ' ' ) # * ) ' ' ) " ) ) ' ' 0 %' ' ( ( () ' ' # * ) ) ' ' )) * ) ( ' ' ( " (( (() ' ' ) *) ) ( ' ' ( ) () ' ' ( ) % ) ' ' ) "-! ' ' ( ) ) ) ' ' () ) ( ' ' (( )( &-( ( )) ' ' ( ) % ) ' ' ( ) (( ' ' ) ( ' ' ) )) )( ' ' )) ) # * ( ( ) ' ' ' ' &" ( ' ' ( &" ( ' ' ) ) ' ' ( ( ( ' ' ) ' ' )) !!&" ) ' ' ( &) ' ' ( ) % ' ' ". ( ' ' ( ) ) ' ' (

PAGE 53

>> 4&#?H )( !.( *#, ! " " #"$ %" &" # * ( ' ' ( ) ) ' ' &' ' ( "-! ' ' ) ' ' (( # ' ' ( ! ( ' ' () ) "-! ) ( ' ' ( ) ' ' ()) * ' ' () * ' ' &( ( ' ' * ' ' ( )) ' ' ( *( ' ' )) * ' ' )( * ' ' ) ) " *&( ( ' ' . )) ' ' ) (( ' ' ) ' ' ) ' ' !. ( )( ' ' ) ( ( ' ' )) &-( ( )) ' ' ) ( ' ' ( %( ) (( ' ' ) &) ' ' ( % . ' ' % . ' ' ! ( ( ))( ' ' * ( )) ' ' # (( ' ' ( * ( )) ' ' ( * ( )) ' ' * ( )) ' ' * ( )) ' ' ) *( ( )) ' ' ( * ( )) ' ' ( * ( )) ' ' ( * ( )) ' ' ( * ( )) ' ' ( ( *) ( )) ' ' (( &-( (( ' ' ( ( +",* ( )( ' ' ( ( !.( ) ()( ' ' ( .% ()( ' ' ( ) () &( ) ' ' ( # ) ' ' ( "** () ' '

PAGE 54

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

PAGE 55

>; 4&#?9H ,#.(+!.#'.('#)(,'!"# !.)#' ! 1 1 23% ! 4!". ! 1. # "" !& #( ( "" # " * ) * &! !& # % " " ( # %! #+%# % " % " ) ( #+% !& % " ,( "! ( &! !-#( !& # ) ,( % "( &" &" , !& ( # &! +" ) ( +" ) ( ) +" ) ( % ( " ( . ( " ( ( # %! (( & ( &! ( ( % " ) ( /-)) () ( +" ) ( +" )

PAGE 56

>A 4&#?9H )( !.( *#, ! 1 1 23% ! 4!". ! 1. +" ) # & ( ( # "" ) +" ) % " ) " " *& "-& # * ) " # ( &! % " "-& . ) ) % "( & &. ( . # ! #+% ) #+%# ) ( . ( ) "-& ( ) # ( ( ) % " ) ! ) +" ( ) % )( % ) +" ( ) +" ( ( ) +" ( )) !-( ) %( # ) # ( ( !-" ( # ( #+% ) #+%#) ) # ( ( ) % ") # ( . (

PAGE 57

> 4&#?9H )( !.( *#, ! 1 1 23% ! 4!". ! 1. # * &" ( "-!( ( * ) *( &" ) * ( ( * "! +" + ( ( ( ( ( ( ( ( ( ) ( ( +" ) ( ( ( # * ( +" ) ( .! "-! .! . ) ( ) ) () ( # ) ( &) +",* ( ) ( # * ( ( ) !+ ( ( ) " ) ( ( "! ( ( -" ) ( ( %( (( ( ( &( ) ( " ( ( () #"() ( ( ( # * ) (( ( (

PAGE 58

> 4&#?9H )( !.( *#, ! 1 1 23% ! 4!". ! 1. "% . "% . #"#() ( " ) ) "**( ( ( ) ( ) " ) # * ( # * ) !# ( ) #() # * ( ( ( ( * ) ) ( () () ( * ) . ( ) * ) ( ( # * ( " (( 0 %( ) # * ) ( * ( " ( ) *) ( ( ) ( ( ) % ( ( ( ) "-! ) ) ) ( ) ( ( )( &-( () ) % ( ( ) ) ( ) ( ) ( )) ) ) # * ( ( ( &" ) &" ) ( ) ) ( ) !!&" ( ( &() ) % ". ) ) ( )

PAGE 59

?= 4&#?9H )( !.( *#, ! 1 1 23% ! 4!". ! 1. # * ) ) )( ( &)( ) ( "-! () ) ( ( # ) ! ) ) "-! ( ) ( * ) * &( ) ) ) * ( ( ( ) ( *( * ) * ( ) ( ) " *&( ) ( ( . ) ( ( ( ) !. ( ( (( &-( ( ) ( ( ( %( ( ( ) ) &) ) % . ) % . ) ! ( ( * ( # ) ) ( ( * (( * ( * ( ) * (( ) *( ( ( * ( () ( * ( (( ( ( * ( ( ( * (( (( ( *) ( ( (( &-( ) ( ( +",* ( ( ) ( !.( ) ( .% ( (( () &( ) )( ( # (( ( "** ) (

PAGE 60

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

PAGE 61

?9 4&#?:H D !"# !.)#' ! 1 1 2 # "" ( (( !& ( #( ( "" ) ( # )( " (( ( )) * ) ) ( ( ) * ) ( ( &! !& # ( ( ( % " ( ) ) ) (( (( " ) ( # %! ( )( #+%# ) ( % " ( ( ( % " ) ( ) ( ) ( #+% ) ( !& ( ( % " )) ) )) ,( )) ) "! )( ( ( &! ( !-#( ( (( !& ( ) # )( ) ,( () % "( ) ( &" ) )) &" ( , ( ( () ( !& ) ( ( ( # ) ( &! +" ) ( ) ( ) +" ) ( ) ( ) ) +" ) ( ) ( ) % ) () ( " ( ( ) ) ( . )( (( ) ( " ( ) ) ( # %! ) (( )( (( & ) () ( &! ( ) ( ( % " ) ) ( /-)) ) (( () ) ( ( +" ) ( ( +" )

PAGE 62

?: 4&#?:H )( !.( *#, ! 1 1 2 +" ) # ( )) ) & ) ) ) ( # "" ) ( ) ( ( (( +" ) ( ( % " ( )) ) " ) ( )) " *& ( )) "-& ) ) # * ( " ) ) # (( ))( ( &! ( % " (( ( ) "-& )( ) ( . ) )) )( ( ) % "( )( )( ) ( & &. ( () ) ( ) ) . # ) ( )( ! ( ) ) #+% ) ( ( ( ( ) #+%# ) ( ( . ) ( ))( ) ) (( ( ( "-& ( () ) ) # ( ( (( ) )) () ( ) ) % " ( ( ( ) ! ( ) () ) +" ( ( ) % ( (( )( % )( ( (( () ) +" ( ) ) +" ( ( ( ) +" ( ( )) !-( ( ) () ) %( () ( () ) ) ) # ) # ( ) ) !-" ) (( () ( # ( ) #+% ) (( ( () #+%#) () ( ( # ( ( )) ) ) % ") # ))) )) ( . ( (() ()

PAGE 63

?> 4&#?:H )( !.( *#, ! 1 1 2 # ( ) ( * ) ( ( &" ) ( ( "-!( ( (( ( * )( ) *( )) ( &" ( ) ) * ) ) ( ) ( * ( ( "! )) )) (( +" ) ) + (( (( ( ( ( (( ) ( (( ) ( ( (( ) ( (( ) ) (( () ) +" ) (() (( ( ( ( ) ( ( ) # * ) ) ) +" )() ( ( .! ) ) ) "-! )) .! ) ) ) . ) ) ( )( ( ( ) ) ) () ))( ( ( ( ( )() ( # ) ) ( &(( () () +",* () ()) ( # * ) ) !+ )) ))) )) (( ) " () ) ( ) (( ( ) ( ) ) ( ( ( "! )( ) ( -" )) )) (( ( %( ) ) ( () (( ( )) ( ) ) ( &) (( () ( ( " () ) ) ( ( () #"() ) )( () ( # * ()( ( ) ))

PAGE 64

?? 4&#?:H )( !.( *#, ! 1 1 2 "% . () ( "% . () ( #"#() ( )(() ( () ( " ( ( )) ()( )() "**( ( ) ( () )) ) " ) (( ) # * ) ) # * ) ) ) !# ) ( ) ))) #() ( ) (() # * )( ( ( ) ( ( * )) ( ) ) ) )) )) * ()) ) . ) ( ) ))) )) ( ( * () ) ) ) ))) # * )) ( ( ( " (( ( ( ) 0 %( ) ) ( ( )(( ((( ( # * ) )) ( * ) ) ( ( " (( ( ( ( ) *) ) ( ( ( )( ) % ) )) ) ( ) "-! )() () ) ( ) ) ( )) ))( ) ( )) ( )( &-( ((( ( ( ) % )( ) ) ( ( () ) () ( )) ( ( ( () ) # * ( ) () () (( ) ) ) &" (() ( ) &" ( (( )) ) ) ( ) ( ( () (() ) ()) )))( ( !!&" ( ( )( &( ) ) % ) ( )( (( ". () (( ( ( ) () (() )

PAGE 65

?; 4&#?:H )( !.( *#, ! 1 1 2 # * (() ( ) ) ( ( &) )) )( ( ( "-! ( ) )) )) # ( )( (( )) ! ((( ( (( ) ) "-! ) ( ) () ) ( ) * () ( ) * ) )) &()( )( )) * ) ( ( ( )( ( *( ) ( ( * ) ) )) ( * ) ( (( ( ) " *&( ( (( ( . ) ) ( ) ) ( ( () () ))( (( ) (()( ( !. ( ) ( ( )))( ( &-( ) () ( ( )) ((( %( ( )((( ) () ) &) ( % . ))( ) ( % . ))( ) ( ! ( () ( ) * ( ( ( ((( # ) ) ( ( * ) ((( * )( ((( * ) () ((( * )) ( ((( ) *( () ) * (( ( ) ( * )) ( ( ) ( * ) () ) ) ( * () ) ( ) ( *) ( (()) ) (( &-( ( )( ) ) ( +",* ( ) ) ) ( !.( )( ((( (( ( ( .% )) ))) () &(( ))( (( ( # )(( ( ) )) "** ( ))

PAGE 66

?A 4&#?:H )( !.( *#, ! 1 1 2 . ) ) ((( ( ( ) .-! ( ) ((() ( )( ( +",* )() ))( )(( +",* )() ))( )(( ! ( ) () # () ) ( ) () ( &+ ) ) ) ) )) ! ) (( , ) ) ( () ) ( ) , ( ))()(( )) ( ) , ))( )) ( ) ( + (( ) (() % () +",* (( ) ())() ( +",*( ) ( ) )(( ( ) ) ) ()( "!. )( )()( ( ) )( )) "!. )( ( )) ( ) "!. ) ( ) ) ) % .( ( ( ) ()) % . (( )) ) ) ) ( ##! ( )( ))((( ) ( ( ( ( ( ( 4& " ) 4&# ?>$ '" %' !"# F !.(+ -.( ! -# !.( )*( !' F)*( !' .( .&&.(' &&#!",' "#F-)*( !.' +,# '*#!"#% 1#@*.#,! )!.<# !.7

PAGE 67

? 4&#?>H &-)*( !'.(.&&.('!"# )!.< !.( ! 1 1 2 # "" !& ( #( ( "" # " ( * ) * &! !& # % " ( " ( # %! #+%# ( % " ( % " ( ( ) ( #+% ) !& % " ,( "! ) ( &! ) ) !-#( ( !& # ) ,( ) % "( &" ( ( &" ( ( , ( !& ( ( ( # ) &! +" ) +" ) ) +" ) % ( " ( . ( ( ( " ( ) ( ( # %! (( & ( &! ( ( % " ) () ( /-)) ) ( () ( ( +" ) ( ( ( +" ) (

PAGE 68

? 4&#?>H )( !.( *#, ! 1 1 2 +" ) ( # ( ) & () ( ( ( # "" ) +" ) % " ) ) " ) " *& ) "-& # * ) ) " ) # ) ( &! ) ) % " ) "-& ) . ) ) % "( & &. . # ! #+% ) #+%# ) ( ( . ) ) ( "-& ( ( ) # ( ( () ) % " () ( ) ! (( ) ) +" ( ( ( ) % ) ) )( % ) +" ( ) ) +" ( ( ) ) +" ( ) )) !-( ) ( ) %( ( ( # ) ) # (( ( ) !-" ( ) ) ( # (( #+% ) ( ( ( #+%#) ( (( # ( ( ) % ") ( ()( ( # )( (( . ( ( (

PAGE 69

;= 4&#?>H )( !.( *#, ! 1 1 2 # ( ) * ( ) ) &" ( ( ( "-!( )(( ( * ( ) *( ( &" ) ) * ( ( * ( ) "! ) ) +" ) ) + )( ( ) ( ) ( ) ( ( ) ( ) ( ) ) +" ) ( ) ) )) # * ( +" ( ) ( .! ( ) "-! () ) ) .! ( . ) ( ( ) ( # ) ) ( &+",* ( # * ( ( !+ () ) " )( ( )( (( ( ( "! () ) ( -" ) )() )( ( %( ( ( (( ( (( ( ) ( ( &( () ( " () ( )) () #"() ) ) ( # * ) ) )

PAGE 70

; 4&#?>H )( !.( *#, ! 1 1 2 "% . ) "% . ) #"#() ) ( " ( ) ) ( "**( ) (( ) " # * ) ) ) # * ) )( ) !# ) )) #() )) ) ) # * )) ( ( * ) ) ) ))( ) * ( . ( ( ) (( ( * ( ) # * ) " 0 %) ) ( ( ) ( # * ) )) ) () * () ( " ( ) ) *) )( ( ( ) % ) ) "-! ( ( ) ) () ) ( (( ( ) )( &-( ( )) ) ) % ( )) ) ( ) ( )( ) ) ) ( )) ) )) ) # * ( (() ) &" ( ((( ( &" ( ( ) ( ( ) )) () !!&" ) )( ( &( ) ) % ( () ". ( ( ( ) ( (

PAGE 71

;9 4&#?>H )( !.( *#, ! 1 1 2 # * ( (( ( ) ( () &(( ( ( "-! ( (( )) ) # () ( ! () ( () ) "-! ( ) ( ()) )) ( * () )) ) * ) ) &)( * ( )) ( ( *( )) ) ( * )( ( (( * ) ( ) " *&( () . (( ( ) )( ) (( ) !. ) ) (( ( )) ) &-( ) ( ( ( %( ()) ( ) &( ( ( % . ( ( % . ( ( ! ( )(( ( * )( # ( ( ( * ( ) * ) ) * ) ) * ) ( ) *( ( ) ) ( * ))) ( * ((( ( ( * ( * )) ( ( *) ( (( &-( ( ()) ( +",* ( ( ) ( !.( ( ( ) ( .% )) ( ) () &( ( # ( ))( ( "** )

PAGE 72

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

PAGE 73

;> 4&# ??H !.)#' %".) " ,#.(+' #@*.#, # '! +# !" ( !"# )!.< !.( ! .". +5". ( # # () & &. ) ) # ( )( ( # ) () # ) ( # ( ) ) # ( ( ( # (( ( # ( () ) ) ( ) ) !!&" () ) (( <# &.(#.(,.) !# ( '!# !" ( !"# !"# #!", !"# ,#.(+ !.# 4&# ?9 % ' <#$ !".' % ' #-& )#, 4 !"# ' &&#'! !.# #'&*!.( B== '#)(,'C ! )-*!#!"# !.

PAGE 74

;? 50 100 150 200 250 5 4 3 2 1 0 1 2 3 4 5 Matrix numberLog2 of ratio ) ! 67 # ( ( # %! ( &! ( &! ) )( ( ( ( ( ##! ( )( .+*#?9H !.)#'H &! B C

PAGE 75

;; 50 100 150 200 250 5 4 3 2 1 0 1 2 3 4 5 Matrix numberLog2 of ratio ( ( ) ( ) ) .+*#?:H !.)#'H &! B C

PAGE 76

;A .+*#' ?> (, ?? '" % -&!'.(,.) !.(+ !"# !.' !"# '.<#' !"# )!' %.!"#'-#)!! (, + .($ !"# .'! 1 #( !"# !.$!"###$-.( !'4#& %!"#<#&.(#.(,.) !#-,*)#,' &&# )!' !" (!"#!"##!", "#, ! !"#'#-&!'.' 4&#?: 50 100 150 200 250 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Matrix numberLog2 of ratio ) ) ) ) ( ! 8 8 67 !!&" ( ( ) .+*#?>H !.)#'H &! B D D C

PAGE 77

; 50 100 150 200 250 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Matrix numberLog2 of ratio ( ( ( ! 8 8 67 !!&" ( .% )) ))) ( .+*#??H !.)#'H &! B D D C .+*#' ?; (, ?A$ '" % !"# !. !"# )!.< !.( F)*( !' 2' (,.1 #!"#-#2.*'-&!'$-.( !'4#& %!"#<#&.(#.(,.) !# !" !N',#.(+-,*)#,&#''F-'!" !!"##!",4#.(+)#,%.!" "# , ! !"#'# -&!' .' 4&# ?> !.)#$ (& !"# *!&.#' !"# 4.+ !.)#' # .()&*,#,!"#'#-&!'

PAGE 78

; 50 100 150 200 250 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Matrix numberLog2 of ratio ( ( ( ! 67 ( ( )) ) ) ( (( ( )( ) ( (( ( ) ) )() ()) ) ()) ))()( () !!&" )( )( (( )) ( ( ( )( ( ) )) ( (( (( % . ( % . ( ( .% )) ( ) ( +",*( ))( ) ))() )(( ( % .( ) % . )) ( ##! (() ( ) .+*#?;H !.)#'H &! B C

PAGE 79

A= 50 100 150 200 250 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Matrix numberLog2 of ratio ) ) ( ) ) ) ! 67 ( (() ))) !!&" )( (( ( ( ( ) ( (( ) ( ) ( .% )) ( ( ( ) )(( ) ! ) )(( ( .+*#?AH !.)#'H &! B C ?: .')*''.( !"# 4 2 # #'*&!' !"# *('#!.) '@* # !.)#'$ .! ) ( 4# )()&*,#, !" ! -#' 2 # % #&& ( 4!" !#' ,#.(+ !.# (, ,#.(+ @* &.! $ .' '+,4#!!#!" (!"# &!#( !.2 #' (!#',#.(+!.#$.' &'! &% ' '!#!" ((& !.7 9=$ $, $ ,., ,# '!# !" ( $ (, 4 (& (#

PAGE 80

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
PAGE 81

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

PAGE 82

A: ,(#$ -# ()# ( &'.' !"#'# !.)#' )*&, 4# ,(# .( !"# ' # ((# ' !"# &# '!G'@* #' !.)#' ',.')*''#,.(#)!.(9$ )&*(.(. *,#+##,#.(+ !#(-#' % #&&.(.(..<.(+8&&4!" "&#'1 )!.< !.( )!.< !.(( # ()#!#'!'% ##,(#%.!" (,).'( +*?A !#'! !.)#' % ## *'#,$ '! %".) " ) # &.(# -+ .(+ -4&#' ) " !.7% ',##,,.#)!&4 (, $ B ' ---. !#C % ' 8'! #, (, !"#( ,##, ()# !"# ,#.(+ % ' -##,$ !"# ' 4&.) "&#'1 B C % ' ,(# # ) " #!", ' #)!.(9'" %'$!"#"&#'1 )! $ $.' ('!*)!* &*--#4*(,( )!.< !.( "###$!"#-#,.)!#,'.<#!"#"&#'1 )! (,!"#-#,.)!#, F !.(+ -.( ! -# !.( )*( ! !".' -# !.( ,#!#.(#, !"# ,#.(+ @* &.! "# ( * 4#((<#'.( (,!"#F-)*( !#@*.#,!)-*!#!"#"&#'1 )!.< !.( ) ( 4# *(, %.!"*! " 2.(+ ! -# !"# ( *#.) & )!.< !.( ".' .' ,(# *'.(+ ' 4&.) )!.< !.( J;K " *'$ ( *#.) & )!.< !.( % ' (! -##, !"#'# #7-#.#( !' ?9 #'*&!' 4&#' .+*# ? '" %' !"# !.)#' .( ,# .()# '.(+ 4#'! F)*( ! "# 2 #!.) & &.(# '#!#' !.)#' !"# ' &&# (#' .+ !.)#' # !"'# %".) " " , .(. *= F-'#! )! 4&#?; &.'!'!"# !.7( * 4#$,.#('.(B 4 C$ ( * 4#((<#'.( $ ( * 4#((<#'.(!"# ' 4&.)!!#( B ' ---. !#C !# ,--.(+ ,#('# %' (, )&*('$ !"# ( * 4# ,#('# %' (, )&*('$ (, !"# + #'! F)*( ! ##' ! !"# 4#'! -#,.)!#, F-)*( !.(.&&.('!"#' 4&.)"&#'1 )!.< !.(!"#!"##,#.(+#!",' ' %.!"!"# !.)#'$ %' (,)&*(' # ,#('#.!"#)( ! .(?=R#((<# #( !.#'

PAGE 83

A> 0 20 40 60 80 100 120 140 16 0 0 2 4 6 8 10 12 14 Matrices in increasing flop count orderLog10 of best flop count for all methods .+*#?H &!!.(+,#!"# !.)#'H 4 4#'!F-)*( !

PAGE 84

A? 4&#?;H #(# &.( !.(!"# !.)#' ! ! " " #"$ %" &" & .& ) ( ( ' ' & " ' ' & " ' ( & #. ( ' ' & % + ) ( ' ' & ' ' & +" ( ) ' ' ) & "# ) () ' ' & " ) ' ' & ! ( ) ( ' ' & " ( ' ' & ( ' ' & && ) (( ' ' ( ( & ( ' ' ( & 9 ( ( ( ) ) & ". ) ( ' ' & 9 ( ( )) ) & # ( ' ' & " ' ' & ,# ( ) )( ' ' & .+ (( ) ' ' ) & ( ) ) ' ' & "+ ' ' ( & "-# ( ' ' & "-# )) ' ' & #& ( ( ' ' & " )) ( ( )( ' ' ) & ". ( ( ' ' & & ( ) ( ' ' && ( ( ' ' % && ( ( ' ' & "" )) ) ' ' & " ) ' ' ( & &9 ' ' & ,& ( ) ' ' ( & #.#. () ' ' & "-(" ( (( ( ' ' (( ) & "" ( ( ( ' ' ( & #" ) ( )) ' ' ( & " ( ) ' ( & ".# ) ) ' ( & . % ( ( ' ' ( & "-&& ) ( ' ' (( & "-(& ( ) )) ' ' ) ( & # ( (() ( ' ' ( & #+ () ( ' ( & "-)" ( ( ' ' () & "-" ( ) ))( ) ' ' ( & ( ) ( ' ' ) & "") ))( ) ' '

PAGE 85

A; 4&#?;H )( !.( *#, ! ! " " #"$ %" &" & #! ( ( ( ( ' ' & "+! ' ' &&) ) )) ' ' ( % &&) ) )) ' ' & #* ' ' & 5# ) ' ' & 9* (( ' ' ) & "!" ( ( )) ' & /& (( ( ( ' ' & & (( ( ( ' ' & -!! )) (( ( ' ' & . % ( ' ' & 9" ( ( )( ' ' ( & "-)& ( )) ) ' ' ) & ) )) ' ' ( & "") ( )) ' ' & () ) ' ' ) & "-& ( ( ' ' ( %! )) ' ' ) & . % (( ( ) ( ' ' ) & "+! ( ' ' & . ( ' ' ) & , ( )(( () ' ' ( %! ( ' ' ( %! () ( ' ' & " ) ' ' & !"5, ) ) ' ' ) & : ) ( () ' ' & "+! ) ) )()) ' ' ) & " ( ' ' ) & #& ( ) ( ' ' ( ) & " () ( ( ' ' ( ) & "+ ( ' ' () )( & .. ()) ) )( ' ' ) & () ( ()) ' ' ) ) & () )) ( ' ' ) & . ( ' ' ) )) & &( ( ( ' ' )( ) & &( ( ( ' ' )( & & % ) ' ' & . ((( ( () ' ' & ::+) ( ) ( ' ' & " ( ( ) ' ' ( & ! ( ) ) ' ' & #.! ( ) ) ' ' & .. ) (( ) ' ' & .. ) ( ' ' ) & "& ( (( ) ' & !" )( ' ' & .." ' '

PAGE 86

AA 4&#?;H )( !.( *#, ! ! " " #"$ %" &" & % (( ())) ' & & () ( ' ' )( & " ) ' ' ( & )## ( ( ' ' .! ( ( ' ' & % % ) )() () ( ( ' ' & +( ) ) ' ' ) & "" )) ) ' ' () & " ) (() ( ' ( & "+ ( ( () ' ' & , ( ( ( ( ) )(( ' ' () & . # ) ( ' ' ( & . ## ) ( ' ' ( ( & . # ( ) ' ' ( & . () )( ' ' (( & & 0 ( ( ' ' ) & & (( ' ' )) ) & " ( ( ( ' & *& ( ' ' )(( & # ( )( ( (( ' ) & #.* )) ( ' ' & #& ( (() ( ) ' ' & " ( (( (()) (( ' ( ( & . ( ( ) ' ' & ,& ( ) ' ' ) & , ) ( ( )( ' ' ) & . (( ' ' )( ) & / ) ) ' ' ) & . ' ' & " ) (( () )( ' ) #" ( (( ( #$% ( ( () (( & & (( () (( ( ' ' ( # ( ( ( )( & ."( ' ' () & .) )) ' ' ) & /) )) ' ' ( ) & 9 )) ( ' ) & ,& ( ) ( ' ' (( ( & &) ) ( ( )( ' ' ((( ( & " )) )) ' ' ) ( & # ( ) )) ' ' ( & )) ( () (( ( ' ' ( (( & !" () (()() () ' ' () ( & , ) ( ) ' ' () ( & " ) ( ( ) ' ' ) ( & & ( ( ' ' () & $ ) ' ' ( & (( ) ' ' ) & / )) )( ' ' )

PAGE 87

A 4&#?;H )( !.( *#, ! ! " " #"$ %" &" & . )) )( ' ' & " )( ) ( ' ' ) & 9 )( ) ' ( ( & / ( )() ' ' ( & . ( )() ' ' & . ( () )( ' ' ( & . ) ) ' ' 4&# ?A$ +.2 #' !"# ,#.(+ !.# .( '#)(,' # ) " #!", "# ,#.(+ !.# .()&*,#'!"# !.# ! B . C "# & '! )&*(&.'!' !"#!.#! -.!) &&.(+

PAGE 88

A 4&#?AH ,#.(+!.#'.('#)(,'!"# !.)#' ! 1 1 23% ! ! & .& & " & " ( & #. & % + & & +" ) & "# & " & ! & " & & && ( & & 9 & ". & 9 ) ) & # & " & ,# & .+ & & "+ ( & "-# & "-# & #& & " ) & ". & & && % && & "" & " ( & &9 & ,& & #.#. & "-(" ) & "" & #" ( & " ( & ".# ( & . % ( & "-&& (( & "-(& ( & # ( & #+ ( & "-)" ( () & "-" ( & & "")

PAGE 89

= 4&#?AH )( !.( *#, ! 1 1 23% ! ! & #! & "+! &&) ( ( % &&) & #* & 5# & 9* ) & "!" & /& & & & -!! & . % & 9" ( & "-)& ( & & "") & ) & "-& ( ) %! & . % ( & "+! ( & . & , ( ( ( %! ( %! & " ( & !"5, ) & : & "+! ) & " ) & #& ) ) & " ) & "+ )( & .. ) & ) ) & ( ) & . )) & &( ) & &( & & % & . ( & ::+) & " ( & ! & #.! & .. & .. ) & "& & !" & .."

PAGE 90

4&#?AH )( !.( *#, ! 1 1 23% ! ! & % & & & " ( & )## ( .! ) ( & % % ) & +( ) & "" ( ( & " ( & "+ & , ) ( ( & . # ( ( & . ## (( ( & . # ( ( & . & & 0 & & ( ) & " ( ) ) & *& ( & # ) ( & #.* ( & #& & " ) (( ( ( & . ) (( ( & ,& & , & . ) & / (( ) & . ( & " (( )( #" #$% ) & & ( ( # & ."( ( & .) ( & /) ( ) & 9 ) ( & ,& ( ( & &) ( & " ) ( ( & # () ) ( & ) ) ( ) (( & !" ( ( ( & , ) ( ( & " ( & & ) () & $ (( ( & ) & / ( ( (

PAGE 91

9 4&#?AH )( !.( *#, ! 1 1 23% ! ! & . ( ( & " ) & 9 ( ( ( & / ( & . & . ) & . ) ) 4&#?&.'!'!"#'.<#!"#"&#'1 )!$ $ # ) ",#.(+#!", "# )&*( & 4&#, &.'! !"# ( * 4# ((<#' .( !"# *--# (, & % # ,. +( & " &2 #' B ' ---. !#C !#,--.(+,#('# %' (,)&*(' ".'.' +,# '*#!"###@*.##( !' '#!.)),#'*) " '

PAGE 92

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

PAGE 93

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

PAGE 94

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

PAGE 95

; 4&#?H )( !.( *#, ! 1 1 2 & . ) (( ( & " )) ) ) & 9 ( ( ( ) ( & / ( ) ( ( )() & . ) ) )() & . ) ( ) )( & . ( )) ) 4&# ? &.'!' !"# -#,.)!#, F)*( !' .( .&&.(' # ) " #!", &)*( !' # +,# '*#!"#% 1#@*.#,! )!.<#!"# !.7

PAGE 96

A 4&#?H &-)*( !'.(.&&.('!"#"&#'1 )!.< !.( ! 1 1 2 & .& & " & " ( & #. & % + & & +" ) & "# & " & ! & " ( & ( & && ( ( ( ( & ( ( ( & 9 & ". & 9 ) & # & " & ,# & .+ ) & & "+ ( ( & "-# & "-# & #& ) & " () ) & ". & & && % && & "" & " ( & &9 ( ( & ,& ( ( & #.#. (( ( & "-(" (( (( ) & "" ( & #" ( & " ( ( & ".# ( ( & . % ) ( & "-&& (( & "-(& ) ) ) ( & # ( & #+ ( ( & "-)" ) () & "-" ) ( & ( ) & "")

PAGE 97

4&#?H )( !.( *#, ! 1 1 2 & #! & "+! )) &&) ( ) ( % &&) ( ) & #* & 5# & 9* ( ) & "!" ) & /& ( & & ( & -!! & . % & 9" ( ( ( & "-)& ( ) ( & ( ( & "") & ) & "-& ( ( %! )) ) & . % ) ) ) & "+! & . ) ) & , ) ( %! ) ( ( %! & " & !"5, ( ) & : & "+! ( ) & " ) & #& ( ( ) & " ( ( ( ) & "+ ) () )( & .. ) & )( ) ) & ) & . ) )) & &( (( )) )( ) & &( (( )) )( & & % ( ( & . ) & ::+) ( & " ) ( & ! ( & #.! ( & .. ) & .. ) & "& & !" ) ) & .." ( )

PAGE 98

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

PAGE 99

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

PAGE 100

20 40 60 80 100 120 140 5 4 3 2 1 0 1 2 3 4 5 Matrix numberLog2 of ratio ( ) ) () ( .+*#?H !.)#'H &! B C

PAGE 101

9 20 40 60 80 100 120 140 5 4 3 2 1 0 1 2 3 4 5 Matrix numberLog2 of ratio )( )( ) ( ! 67 & 9 ( )( .+*#?=H !.)#'H &! B C

PAGE 102

: .+*#' ? (, ?9 '" % -&!' .(,.) !.(+ !"# !.' !"# "&#'1 )! '.<# %.!"#'-#)!! (, + .($ !"# .'! 1 #( !"# !.$!"###$-.( !'4#& %!"#<#&.(#.(,.) !#-,*)#, ' &&#"&#'1 )!!" (!"#!"##!", "#, ! !"#'#-&!'.' 4&#? 20 40 60 80 100 120 140 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Matrix numberLog2 of ratio ( ( ) ) ( ! 67 ) & "+ )( & "+ ) (( & *& ( ( .+*#?H !.)#'H &! B C

PAGE 103

> 20 40 60 80 100 120 140 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Matrix numberLog2 of ratio () ( ( () ( .+*#?9H !.)#'H &! B C .+*#' ?: (, ?> '" % !"# !. !"# -#,.)!#, F)*( !' 2 #'*'!"#!"#! % #!",' .1 #!"#-#2.*'-&!'$-.( !'4#& %!"#<# &.(#.(,.) !# !" !N',#.(+-,*)#,&#''F-'!" !!"##!",4#.(+)#,%.!" "# , ! !"#'#-&!'.' 4&#?

PAGE 104

? 20 40 60 80 100 120 140 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Matrix numberLog2 of ratio ( ( () ) ( ! 67 & )) ( & . )( ) & "+ )) & .. ) ( & !" )( ) & +( ((( ()) & "+ () ( ) ) ( & . # () ( (( & . ## () ( (( ( & . # ( () ) & *& )(( (( ) ) & / ( ) () () & ."(( )( ( & # ( ) .+*#?:H !.)#'H &! B C

PAGE 105

; 20 40 60 80 100 120 140 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Matrix numberLog2 of ratio ) ) ) ) ! 67 & )) ) & " ()( (( & !" ()) ( .+*#?>H !.)#'H &! B C ?9: .')*''.( !) (4#'##(!" !,#.(+!"##)! (+*& !.)#'$.' +,) ".)# !.' *) "4#!!#!" (.(!#' ,#.(+!.# (,@* &.! 0"#()#, !$.''#!.#' '!# (," ''..& @* &.! .(+!"#,#.(+!.#%.!"$.' 4*!? ' '! ! -.) && "## # (& #% !.)#' %".) " ,##, '!# !"# !.)#'$.' 4*!; !.#' ' '! ' ('.,#.(+ && !.)#'$ .' 4*! ' '! ' !"# 4.+ !.)#'$ .' =R '!# ' &&# !.)#' (, !.)#' %".) " # 2 #

PAGE 106

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

PAGE 107

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

PAGE 108

0 20 40 60 80 100 120 140 160 180 20 0 0 2 4 6 8 10 12 Matrices in increasing flop count orderLog10 of best flop count for all methods .+*#??H &!!.(+,#!"# !.)#'H 4 4#'!F-)*( !

PAGE 109

== 4&#?=H #(# &.( !.(!"# !.)#' ! #"$ #""! () () #""! #""! ( #""!( #""! ) ) #""! #" % ) #" % ( #""! ( ( #""! () () % % ) ( ( ( #""! ) ) #""!) ( ( #""! ) ) ) % #" % ) ( #"", ( #""! ( ( #""! % ) ) ( ( " % ( ( )( ) #""! ( ( "-) ) #""!( ( #""! ( #"", ) ( "( ( ( ( ( ( #" ( ( #" % (( #"", () ( (( (( ) #" %( ( ) % ) ) ( " ( ( % ( ( ( ( ) #" ) (( ( ) (( % ) ( #" (( ( #"", () ( ( #""! ( ( () % ( % ( ( #" % ((

PAGE 110

= 4&#?=H )( !.( *#, ! #"$ #" % )( % ( #""!) ) () ( #" %) ( ( ) #" ) ( ( ") ) ) ( ) ( ) #""! ) #" % ( % (() #"", ( ( & ( (( ( & # ( (( ( #""! ( ( % (( ) &"( #""! ( ( ( ( ) ) ) % ( ( ( ( #"", ) ) % ) ) ( % " ) #"",( () ) % ) % ( ( ) ) &"( ( ) ) #"", ( ) ) % ( ( )( % ( ( ) % ) ) ) ( ) 0.!") ) 0.!") )) &")) ) 5" ) & ) 0.!") ( #" % )( ( % ( 0.!"-) ( ( % ) ) )( )( % )) )) (() ) % " ) &") )

PAGE 111

=9 4&#?=H )( !.( *#, ! #"$ 0.!"( (( (( ) #""! ( () ( 0.!"-( (( ( % ) ) #"", ( ) ) #"", ( ) ) ) . (( ( 0.!"( ( 0.!") ( 0.!") ( &") ( #""! ) ()( ( ( ( % ( ( ( #"", ) ((( ) % ) &")) ( ( ( ) )) )) #""! ( ""!& ( ( ) ( &") ) " () ( " ( )) % (( ) ) " )(( &"-)) )) ( % ) ) ( #"", (( !" ) #"",) ( ( &"( #"", ( (( #"", ( (( &"-( ( ) ) ) #& ,-& () ( #"", ( ( #""! ( ( &") ) ( & ( &"-( ( ) (( #"",( )) () ( #"", ) )( ) ( #""!( )) ( ( #""! ( ( () #"", () ( ")( )( ) ( #"",( ) (( )

PAGE 112

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

PAGE 113

=> 4&#?'" %!"#,#.(+!.#'.('#)(,'!"#*#!",' ,#.(+!.#' (,.()&*,#!"# !.(!"# / )&.@*# !.7 4&#?H ,#.(+!.#'.('#)(,'!"# !.)#' ! ; ; 2 ( #""! #""! #""! ( #""!( #""! #""! #" % ) #" % #""! #""! % % ( ( #""! #""!) #""! ) % #" % #"", #""! #""! % ) ( " % ( ) #""! "-) #""!( #""! #"", "( ( ( ( #" #" % #"", (( ) #" %( % ) ( "

PAGE 114

=? 4&#?H )( !.( *#, ! ; ; 2 ( ( " ( % ( ( ) #" ( (( % ( #" ( #"", ( ( #""! () % ( % ( #" % #" % % #""!) ( #" %) ) #" ") ) #""! ( #" % % #"", ( & & # #""! ( ( % ( ) &"#""! ) % ( ( #"", % ) % " ) #"",( % ) % ( ) &"( ) #"", ) % )( % ) % ) ) 0.!"( ) 0.!")) &") 5" () & (

PAGE 115

=; 4&#?H )( !.( *#, ! ; ; 2 ( 0.!"#" % ) ( ( % ( 0.!"-) % ) % )) ) % " ) &") 0.!"(( #""! ( 0.!"-( ( % #"", #"", ) . 0.!"0.!"0.!"&"#""! ( ( ( % ( #"", ) % &"( ( )) ( #""! ( ""!& ( &"" " % ) " &"-)) ( % ) ( #"", !" #"",) ( ( ( &"#"", ( #"", &"-( ) #& ,-& #"", ( ( #""! (

PAGE 116

=A 4&#?H )( !.( *#, ! ; ; 2 ( ( &"( ( & ( &"-( ( ) (( #"",( ( ( #"", ) ( #""!( ( #""! () #"", ( ")( ( #"",( ) ) !"(( "( ) ( !"( ( ( "& * !" ( " *"! ) ) ) #"",( ) ) ( ( ", ( (( #"",) ) ( ( "(( ( ( #"", ( ( #""! ( #"",) ( #"", ) #"",) ) % ) ( (( ) #"", ) ) ) #"", ( ( ( #"", (( 9 *", ) ) #"", ) ) ( . ( ) ) ( #"", ( #"", !")() ) ( ) #"", ) *"! ) ) ) !" ( ( ) ) #"", ()) ) ) #"", ( ) ## ) (( ( )( #"", ) ) " ( ) ) *"! ) ) #"", ) ) )) . ( ) )( ) #"", ( ( "( ( )

PAGE 117

= 4&#?H )( !.( *#, ! ; ; 2 ( . ) *", ( ""# (( ( ( "+ ( ( ( *", ) ( () # %, () ) ) + ( ( ) .# ) ((( ( ) + ()) 4&#?9$'" %'!"#-#,.)!#,'.<#!"#"&#'1 )!' # ) ",#.(+ #!", !#$ '.()# .' & % # !. (+*& $ .! .' -''.4&# !" ! % #2 #$ B C 9D

PAGE 118

= 4&#?9H !"# !.)#' ! ; ; 2 ( #""! () () () () #""! #""! ( #""!( #""! ) ) ) ) #""! #" % ( ( ( ) #" % #""! ( ( ( ( #""! () () () () % )( )( )( % )( ( )( )( ( ( #""! ) ) ) ) #""!) ( ( ( ( #""! ) ) ) ) ) % ( (( (( #" % ) ) ) #"", )( (( )( )( #""! ( ( ( ( #""! % ) () ( (( (( ( ) " ( ( ( % ( ) ( ) ) )) (( ) ) #""! ( ( ( ( "-) ) #""!( #""! #"", ) () )) ) "( ( ( ( #" ( ( (( (( #" % ( ( ( (( #"", ( ( () () (( ( ) #" %( ( (( ( % ) () ( " )( )( )( ( % ( ( ( (( ( ( ) #" ( ) ( ( )( )( )( (( % ) ) ( #" ( ( #"", ( ( #""! ( ( ( ( () % ) ) ( % ( ) ) ) #" % ( ()) ())

PAGE 119

= 4&#?9H )( !.( *#, ! ; ; 2 ( #" % ) % ( ( #""!) ( #" %) ) ) #" ")) ( ( ) ( ) ) ) ( #""! #" % ( (( (( % ( ( ( #"", ( ( & & # ( ( #""! ( ( ( ( % )) ) ) &"(( ) ) #""! ( ( ( ( )( )( ) ) ( ( ) ) % ( ) ( #"", ((( ) ( ) % ) ) % () ) () () " ) ( ( ) #"",( ) ( % ) ()) ()) ) % ( () ( ) ) &"( ) () () ) #"", ) % )) )( % ((( () () ) % ) ((( () ) 0.!") ) ) 0.!"()) ()( ) ) )) &")) ) ) ) ) 5" ( ) )) )) & ) ) ) 0.!"() ) ( ( #" % ) )() ) % ) () ) ) ( 0.!"-) ( ( ( ( () (( (( % ) )) % )) ( ) (( (( ) % ) " ( () &") (

PAGE 120

4&#?9H )( !.( *#, ! ; ; 2 ( 0.!"((( ( () ( () (( ( ( ) ) #""! () )) ( ( 0.!"-( ) () % #"", ) ( ( #"", ) ( ( ) . ) () () 0.!"( ( 0.!") ( ) ) 0.!")) &")) #""! ) ) ( ( ) (( ( % ( () #"", ) ) ) ) () () ) % ( ) &"() ) ( ) )) ( ( ) ( ) ( #""! ) ( ""!& ( (( ) ( &"( " ) ( )() )( )( " ) ) ( ) ( % ) ) ) ) " ) ( &"-)) )) () )) )) % ) ( ) ( ) #"", (( ( ( ((( !" ) () ) #"",) ( ( ( &"( ( () () #"", ( ) #"", ( ) &"-( ) ) ) #& ,-& ( #"", )) ( ( #""! )) ( ( &"( )) ( ( ( & ) )) ( &"-( )( ( )( )( (( #"",( )) ( ( ( #"", )( ( )( )( ( #""!( ( ) ((( ((( ( #""! (( )) () (()(( () #"", )( ( ")( )) #"",( ) ( ) ) )

PAGE 121

9 4&#?9H )( !.( *#, ! ; ; 2 ( !"(( () ( ( "( ( ( !"( ( ( "& * !" " ( )) )) *"! ) ( ( ) #"",( ) ) ) ) ", ( ( () () #"",) ( )) ( ( "(( ) ( (( )(( ) #"", ) ) ( ( ( #""! )( )))) )() )() ( #"",) ( ) #"", (() ( ((( ((( #"",) ()) ))( ( % ( ) (( ) ) ) #"", ( ) ( #"", ( ( #"", ) )) )) 9 )) )( ) )( *", ( () ) ) ) #"", () ) ( () ( . ( #"", ( ))) )) )) #"", ( !")() ) (( (( ) #"", )( )() ) *"! ( )) )) ) !" )( ) #"", (( ) #"", () )) ) )( ) ## ( (( (( )( #"", ) ( ) ( )) ) " ) ( ()) ) *"! ( ( ( ( ) #"", )( )) . ))( ( ( )) ) #"", () ( ( )( "( )) ) . ))) ) ( *", ( (( )) ) ) ) ) ) ""# ( ( ) ) ( "+ )( ) *", () ) ) # ) %, ))() ) )) () )() ) + )) ( .# ( ) () ()(( + () ((( ( ) )

PAGE 122

: 4&# ?:$ &.'!' !"# -#,.)!#, F)*( !' .( .&&.(' # ) " #!", ".' .' +,# '*##( !!"#% 1#@*.#,! )!.<# !"# !.7 4&#?:H &-)*( !'.(.&&.('!"#"&#'1 )!.< !.( ! ; ; 2 ( #""! #""! #""! ( #""!( #""! #""! #" % ) #" % #""! #""! % % ( ( #""! #""!) #""! ) % #" % #"", #""! #""! % ) ( " ( % ( ( ) #""! "-) #""!( ( ( ( ( #""! ( ( ( ( #"", ( ( ( "( ( ( ( ( ( #" #" % #"", (( ) #" %( ) ) ) % ) ( "

PAGE 123

> 4&#?:H )( !.( *#, ! ; ; 2 ( ( % ( ( ( ) #" ( (( % ( #" ( #"", ( ( ( #""! () % ( % ( #" % #" % % ) #""!) ( ( #" %) ( ) ) #" ( "( ) ( ) ) ) ) #""! ) ) ) ) #" % ( ( ) ( ( % ( #"", ( ) ( & ( ) ( ( & # ( ( ( #""! ( ( ( ( % (( ( ( ( ) &"( ( ( #""! ( ( ( ( ) ( ( ) ( % ( ) ( ( ( #"", % ) ) ) % ) ) " ) ) ) #"",( ) % ) % ( ) ) &"( ) ) ) #"", ) ) ) ) ) % ( )( % ) ( ( ) % ) ( ) 0.!") 0.!")) &") ) 5" () & )

PAGE 124

? 4&#?:H )( !.( *#, ! ; ; 2 ( 0.!"( ) #" % )( ( % ( ( 0.!"-) ( % ) ( )( )( % )) (( ) % ( " ( ) &") ) 0.!"(( ()) #""! () ( ( ( 0.!"-( ) ( % ) ) #"", ( ( ) ( #"", ( ( ) ( ) . ( ( ( 0.!"() ) ( ( 0.!"( ( ( 0.!"( ( &"( ( #""! ()( ( ( ( ( % ( ) #"", ) ) ) ) ) % ) &") ) ( ( ( )( )) #""! )) ""!& ) ) ) ) ( &"( ) ) " ( ) " ) )) )) % ) ) ) " ( ( ) &"-)) % ) ( )) () ( #"", (( (( !" ) ) #"",) ) ( &") ( ( #"", #"", &"-( ) ) ) ) #& ,-& ( )) ( ( #"", ) ( ( #""! ) (

PAGE 125

; 4&#?:H )( !.( *#, ! ; ; 2 ( ( &") ) ) ( & ((( ( &"-( ) ) (( #"",( ) ) ( ( #"", () ) ( ( ( #""!( ((( ((( ( #""! ( ( () (( () #"", (( ( () ( ")( ( ( ( #"",( ) ) !"(( "( ((( ) !"( ( ( ( "& * ) ( !" ( ( " )( *"! )(( ) #"",( )) ", ( () #"",) ) ( ( ) ) "(( (( #"", #""! ) ( #"",) ) ()( )( ) #"", (( () #"",) ( % ) ( ( ) #"", )) ( (( #"", ( ( #"", )( ) )() )() 9 ( ) () *", ) ) (( (( #"", )() ( ) ) ( . ( ) #"", ( ((( #"", )( ()( )) )( !")() ) )( ) ) ) #"", ) ) ( *"! ( ( )) ) !" ( ( ( ( ) #"", ( ( ) #"", ) ( (( ) ## )( () ) ) )( #"", )( ))( ) ) ) " ( ))) ) *"! ( (( ) #"", ( ((( ( ( )) . )( )) ) #"", )) () ( "( ) (

PAGE 126

A 4&#?:H )( !.( *#, ! ; ; 2 ( . ) )) ) )) *", ( ) (( (( ""# ()(( )( ( ()()) ()()) ( "+ ) ( ()) *", ) # ) ()( %, (( ( )) ( (() ) + (( ))) (()) (()) .# ( )( (( )) ()( ( + ( )(( (( (( ?:9 .'(&!' "# && %.(+ -&!' )# ,.#)!& %.!" (, >A ).'(' # #7)&*,#, ,*# ! .!' '..& .! %.!" ,,.!.( && $ % ') "'#(!#-#'#( !!"#'#!.)),#' && %.(+ # ) " -&! .' ! 4&# !. '! !.'!.)' (, ( *!&.# !.)#' *!&.# !.)#' #!"'#%".) ",.,(!8!( !!"#-&! !.)#'%".) "" , .(. *-#,.)!#, "&#'1F-)*( !+# !#!" (= #)('.,##, !.)#' (!"# %'! !.'!.)' .''" %(4.+ !.)#' '% #&& .+*#'?; (,?A'" %-&!'.(,.) !.(+!"# !.',#.(+!.#' %.!"#'-#)!! (,.()#!"# .'! 1 #(!"# !.$-.( !'4#& % !"# <# &.(# .(,.) !# ( '!# !" ( !"# !"# #!", !"# ,#.(+ !.# 4&#?% '#@* &!<#$.!% '#-& )#,%.!" !.#=='#)(,'!)-*!# !"# !.

PAGE 127

20 40 60 80 100 120 140 160 180 20 0 5 4 3 2 1 0 1 2 3 4 5 Matrix numberLog2 of ratio ( ) ( ) ( ( ! 67 ) 5" () " ) ( #"",) ( ( #"", ) #"", ) ) #"", ) )( #"", ) ) #"", ( ( "+ ( ( ( %, () ) .+*#?;H !.)#'H &! B C

PAGE 128

20 40 60 80 100 120 140 160 180 20 0 5 4 3 2 1 0 1 2 3 4 5 Matrix numberLog2 of ratio ) ) ) ) ( .+*#?AH !.)#'H &! B C

PAGE 129

9= .+*#' ? (, ? '" % -&!' .(,.) !.(+ !"# !.' !"# '.<# !"# "&#'1 )! ,##,4 %.!"#'-#)!! ,##,4 (, .()#!"# .'! 1 #(!"# !.$-.( !'4#& %!"#<#&.(#.(,.) !#,#.(+ -,*)#, ' &&#"&#'1 )! !" (!"#!"##!", "#, ! !"#'#-&!' .' 4&#?9 20 40 60 80 100 120 140 160 180 20 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Matrix numberLog2 of ratio ) ( ! 67 9 )) )( ) ## ( .+*#?H !.)#'H &! B C

PAGE 130

9 20 40 60 80 100 120 140 160 180 20 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Matrix numberLog2 of ratio ( ( .+*#?H !.)#'H &! B C .+*#'?9= (,?9 '" %!"# !. !"#-#,.)!#,F-)*( !'4! .(.(+!"# "&#'1 )! ,##,4 2' !"#!"#! % #!",' .1 #!"#-#2.*' -&!'$-.( !'4#& %!"#<#&.(#.(,.) !#!" !N',#.(+-,*)#,&#''F-'!" ! !"##!",4#.(+)#,%.!" "#, ! !"#'#-&!'.' 4&#?:

PAGE 131

99 20 40 60 80 100 120 140 160 180 20 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Matrix numberLog2 of ratio ( ( ) ! 67 % % ( ) *"! )((( ( ) #"",) 9 ( ) ( *", ) ( ) *"! ( ))( ) ) ## )( ()( ) *"! ( ) (() *", ( )(( () ( "+ ) ( ) + (((( ))) .+*#?9=H !.)#'H &! B C

PAGE 132

9: 20 40 60 80 100 120 140 160 180 20 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Matrix numberLog2 of ratio ) ) ! 67 9 ( ) () .+*#?9H !.)#'H &! B C ?:: .')*''.( !"# '#!.) #'*&!' 4 2 #$ .! ) ( 4# ' ., !" ! .' '*-#. &!#( !.2 #! % #2 #$) ((!)-#!#%.!"!"#( !.2 #'#!.) ,#.(+),#'.(!#'*(G!.# 1.(+ ! ,#.(+!.#'$ .'! -.) &&! %.)# ' '! ' !"# !.)#' !".' )! .2 #' ! 2 # ! % #&2 # !.#' ' '! ( )!$ .' (#2 # '!# !" ( !"# 4.+ !.)#' (, .' (& )) '.( && '!# !"# ' &&(#' % #2 #$ .' (#2 # '!# !" ( -.) && $ .' ? !.#' '!# !" ( !"# 4.+ !.)#'$ .' * !.#' ' '! "#

PAGE 133

9> % '! -# ()# 4 % ' ( !.7 3 !".' !.7 #@*.#,$:=A '#)(,'!,# ')#,!(&=:'#)(,' % #2 #$ #@*.#,=:'#)(,'!".' * &.! ).'( 4#! % ##( (, '" %' !" ! .' *) " 4#!!# ,#.(+' ! -.) && #@*.# =R &#'' )! '! +# (, 99R &#'' F-' ! )!.<# !" ( ,#.(+' + .($ !"# 4.+ !.)#'$ .' #2 #( 4#!!# !"# 4.+ !.)#'$ ,#.(+' ) ( 4# #7-#)!#, ! #@*.# ?R &#'' '! +# (, " 2 # :=R &#'' F)*( ! !" ( ,#.(+' (#2# #@*.#'# !" ( AR #F-'!" ( % #2 #$ ) ( #@*.#*!>!.#'!"#F-)*( ! ' %.!" !"# #)! (+*& #'*&!'$ (, -,*)# ,#.(+' &'!.,#( !.) &@* &.! "#'..& .! .''" %(#2 #(#4 !"#' &&F-)*( !'! (, , ,#2. !.(=A 0 0.5 1 1.5 2 2.5 3 x 104 0 0.5 1 1.5 2 2.5 3 x 104 nz = 2164210 .+*#?99H !.7 $('" *&*-! "# !.)#''!,*!.(!"##'*&!' '4#.(+2 #!.#)('*.(+!,# 4 && #!",' ".'.' 4#) *'# !"# *-! !.)#' " 2 # ( ,#('# %' (, )&*(' .()#((#!"##!",'# 2 #,#('# %' (,)&*('.(!".'#7-#.#( !$!"#,#.(+

PAGE 134

9? !.#' # #7!##& ".+" (# ) ( '## !"# ,#('# %'E)&*(' !"# !.7 .( .+*#?99 '## " % # 2.(+ ,#('# %' (, )&*(' 3#)!' !"# *( !.#' !"# ,#G .(+'$ ' && #7-#.#( ! ( 5*'! !"# *-! !.)#' % ' ,(# %.!" ,#('# %E)&*( !"#'"&, == 4&# ?> 4#& % .' *( !.# ( &'.' '" %.(+ !"# , !.) '-##,*-' 4! .(#,)#,%.!"(,#('# %E)&*(# 2 & !# &'" %!"#,#.(+@* &.! .'.2 #, &&!"##!",'4 !"## 2 &,#('# %' (,)&*('-.!,#.(+ 4&# ?>H #)! ,#('# %E)&*( # 2 & ( ,#.(+ !.# (, F)*( !' !"# *-! !.)#' " %" " %3&+ " %3&+ ! &!" ! $ ! $ ( . ( ( )) . ) ) ( )( . ) ) ) " %" " %3&+ " %3&+ ! &!" ! $ ! $ ( . ) ) ) )) . ) . ) ) )) " %" " %3&+ " %3&+ ! &!" ! $ ! $ ( . )) ) )) . ((( ) . ( ) " %" " %3&+ " %3&+ ! &!" ! $ ! $ ( . () ( )) . ) )( )) . )) ( )()&*'.($ %"#( ,#.(+'#!.) !.)#'$ .' *) " 4#!!# !" ( ( !.2 # '#!.) ,#.(+ &+.!" '*) " ' .' 2 .& 4&#$ .! '"*&, 4# *'#, 2 # % #2 #$ . '*) " ),# .' (! 2 .& 4&#$ %.&&

PAGE 135

9; -,*)# ".+"@* &.! ,#.(+4*!%.&&#@*.# *) "#!.#!" ( ( !.2 #'#!.) ),#

PAGE 136

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

PAGE 137

0 .' 2 .& 4&# ' *'#)( !.4*!#,) 1 +# ?9 !H # ! .' &' 2 .& 4&# ! (, . 2.'N% #4'.!# T "# && %.(+ ! % 8&#' # !"# (& G) && 4&# 8&#' (##,#, "# ,.3#!"#,.'!.4*!.(),#.(!" !!"#,#4*++.(+),#.'(!.()&*,#, !""# $ !% & ' () * + , () *%!+ .,/0&123,4 5% ) , *%5+ .,/0&1 5% () *%+ *%+ () .,/0&1&63617,8 % () *!+ *!+ () .,/0&1&6361.,/ ! *5+ .,/0&1&697:1.,;3 5 9

PAGE 138

9 * + <) .,/0&1=;04/6&1.,/ > ?) 1 @ ' ' 1 ) 1 ) A B 11 @ () ) () *.,/0&123,4+ A B 7;6 9/6 @ 1 ) 1 ) ' *+ ' *+ ' 1C! ) () *.,/0&123,4+ A B 9 ? > . D .( ' @DAE@DA ? F E >@DA /; D ' > ' . /; />.2 )G ) ?) 0) H5 ) 08( G @(A ) .,/00& 0) 4 I 0) @ .,/00&A /; ' )? 0) /;@A G / & @JA ; 9 ) = :) K >7. 6 3 ,( 7 3 /) & !""# $ !%

PAGE 139

:= ( ( ) 3 9 &0?"H%L"ML &0?"#% H"" 3 . @A !""# ) ; 9 7 7 NG 067G/ G >7,$G&6& G 8GN 4,/;6/O 3, 8773O 6K>766& ,7 G0>/G6& 3O ;6 G O,;7 ,83 7G2 > ) ) ; . P; ) P G ) 0) P P PP @ A > ) ) ) O ) ) ) F 0) 0) ) 3G . @ ;? A & ?) 6 ?) @ >;4/G.A ) ) ) 7 ???????????????????????????????????????????????????????????????????????????? 1 ???????????????????????????????????????????????????????????????????????????? ; 1 @' 1 1A B > 7 ) 7 ?! ' B 3) ' ) *1+ ? 1 B 3)

PAGE 140

: 1 B 3) ???????????????????????????????????????????????????????????????????????????? 11 ???????????????????????????????????????????????????????????????????????????? ; 11 @()A B > ) () *.,/0&123,4+ B , 7 @() *.,/0&1&63617,8+ 1A . @() *.,/0&1&6361.,/+ 1A & () ) %H . () *%+ () *!+ ) () G ) ) 11 11 @) A 3;// () ???????????????????????????????????????????????????????????????????????????? ???????????????????????????????????????????????????????????????????????????? ; @1 1 ()A B > . @DA >@DA/; @DAED//E ? F ' E 1 B 3) 7 1 Q % . 9/6 1 1 B 3) 7 1 Q % . 9/6 1 B 7 @ A Q 5' C R@1C!A C L@1C!A C 1 C .,/0&1 . 9/6 3 ( ' E ) 8

PAGE 141

:9 Q 1 @' 1 1A ) *+ B G ' ) ) ) ) 9 ) ) F 1 @' 1 1A 'H ) ) , *@ *+A @ *C!+?!A+ ) ) ) N ( ) @. A %?) % 1?! % 1?! . 9/6 ) @ *%.,/0&1?!+A *%+ ) *!+ ) @ A *5+ ) ) * + % ! @ (A 3 ) PP PP PP ) PP PP PP 9 ) L *1C!+ B 4 ' 1C! , PP . *@ *+A @ *C!+?!A+ *%+ ) ' *+ S *C!+ % 1?! *1+ ) . 9/6 , 7;6 @D >@DA/; @DAE@DA//EA *%+ *1?!+ *(+ < < ( D ( % 1?! @ *%+ < < DA G 9/6 ) () *.,/0&123,4+ B G 11 G () @ @) A 3;// A

PAGE 142

:: G @G310KA SQ () ' ' ) PP 7 . .G1 T B &6& B ) T ( B ) ) B ? U ! B T B B U 5 B T B )( B B @) A U B T 1 B 1 B U L B U .G B 7G1 T B B ) T B ) V ? B 11 @A U ! B T ( B ( 1 B @ ) A U 5 B U 7G B &

PAGE 143

:> 0K@)A @@@A Q @)AA W @A @)AA 0G3@)A @@@A S @)AA W @A @)AA ,361.,0>/6063@A @?@A?!A 7;6 @!A 9/6 @%A 60>O @?!A 7 /G$6 @%A &6& @?!A . &6&1>7G3.G>/ @?!A &6&13,31>7G3.G>/ @?5A 0 ( 7,81G1&6&@A 7,81G10726&1&6& @7*+5(A 7,81G10726&1&6&@1(A @1( S /G$6A 7,81G1/G$6@A @7 *+5( Q /G$6A .,/1G1&6&@A @. *+ S /G$6A .,/1G1/G$6@A @. *+ Q /G$6A .,/1G1&6&1>7G3.G>/@A @. *+ &6&1>7G3.G>/A -2G//17,8@A T 7 *+5( &6& B U 2G//1>7G3.G>/1.,/@A T . *+ &6&1>7G3.G>/ B U 2G//13,31>7G3.G>/1.,/@A T . *+ &6&13,31>7G3.G>/ B U 7 >;4/G. @?)A >7G$6 @ ?)A >;4/G. >7G$6 > >7G$6 >7G$6 11 @ 1 1 7G 7 *+ .G . *+ *+ *+ A B >7G$6 1 @ 1 1 7G 7 *+ .G . *+ *+ *+ ) () *.,/0&123,4+ 15 15 1 A B >7G$6 1 @ 1 1 7G 7 *+

PAGE 144

:? .G . *+ *+ *+ 15 1 A B >7G$6 1 @ 1 .G . *+ *+ A B >7G$6 11 @ .G . *+ *+ *+ 1 1 A B >7G$6 )1 @ 1 1 7G 7 *+ .G . *+ *+ A B >7G$6 1( @ 1 7G 7 *+ A B ;67?.//4/6 7,;G36 1 1 ' ) ) ) ) ) F >;4/G. 1 @ > ' ) ' 1 ) 1 ) A T / ) B ) F B

PAGE 145

:; @' S % XX 1 S % XX 1 S %A T ?! @?!A B U 5 @'A C @@@1A C !A ' @.GA ' @AA . C @@@1A C !A ' @7GA ' @AA 7 C 1 ) C .,/0&1 B F ( ) ) C 'H B @A B U 11 11 ? ) () *%+ ()*%+1 () *!+ ()*!+1 () *5!"+ ) >;4/G. 11 @ > ) () *.,/0&123,4+ () A T / ) B @Y()A T B () ' U @ % B S .,/0&123,4 B CCA T () *+ % B U () *.,/0&1&63617,8+ %H B H%Z () *.,/0&1&6361.,/+ %H B H%Z U

PAGE 146

:A D /; ' >@DA /; > ) D .( ' @DAE@DA //E , ' % 1?! %?) @ % 1?! % 1?!A 7 *@ *+A @ *C!+?!A+ *%+ % *1+ ) ) @ A ) N ( 0) H5 ' ) ' @ ' ) A ' C ' C R@1C!A .G . *%1+ @ ' @.GA R EA C L@1C!A 7G 7 *%1+ @ ' @7GA L EA C )1 ) 1 8 'H G ) C .,/0&1 . 9/6 7;6 , 1 ) 1 ) ' *% '?!+ ' *1+ *' ?!+ @ ) ' ) A *% 1+ *%+ % *1+ ) . * *+ *C!+?!+ () *% !"+ ) G() 3;// ?) 11 ?) G . 7;6 *% 1?!+ *%+ *1?!+ *(+ < < ( D @ A *%+ ) @ A *!+ ) @ A *5+ ) ) G

PAGE 147

: * + % ! O * + % G . 9/6 >;4/G. 7;6 @ > 1 ) 1 ) *+ *+ ) () *.,/0&123,4+ @ 3;//A A T / ) B ' B ' 71' B ' 7 *+ .1' B ' . *+ )1 B 7G 7 B 7 *%1+ .G . B . *%1+ 15 B ) ? ? 15 B ) ? ? ) B ) ) 1 B ) 1() *.,/0&123,4+ B () () 1 B ' .( @1 S % XX 1 S % XX Y XX YA T 1 1 ) ? ) @9/6A BU ' *1+ B @' S % XX *%+ Y %A T ' ) ? *%+ ) ' @9/6A B U G () @Y()A T () 1() B 11 @()A B U 7 . .1' @1 C !A ' @.GA ' @A B 71' @1 C !A ' @7GA ' @A B )1 ? @5' C .1' C 71'A B @)1 S 1 C .,/0&1A T

PAGE 148

: @9/6A B U 5' C )1 B . @.G A V *+ B 7 @7G A V * C .1'+ B . 1 11 @1 1 7 . A B @1 ?!A T @9/6A B U G' ( 1 @1 1 7 . () V15 V15 V1A B , ) 1 @1 1 7 . 15 1 5'A B , ? 1 @1 . A B 7 @ % B S .,/0&1 B CCA T *+ % B U *.,/0&1&63617,8+ 1 ? 15 B *.,/0&1&6361.,/+ 1 ? 15 B *.,/0&1&697:1.,;3+ ) B *.,/0&1=;04/6&1.,/+ 1 B @7;6A B U 3,3?;67?.//4/6 7,;G36 ?) ) 11 ( ) . 7 G ? 7 ?! ! <) % <) 3 ?) >7G$6 11 @ >

PAGE 149

>= 1 ) 1 ) 7G 7 *+ ' 1C! .G . *+ ' 1C! *+ ' *+ ' 1C! A T / ) B B B B B B 1 B B 1 B <)1 B <) G' ( 1 % B @ % B S 1 B CCA T *+ B @ S 1A T ) ? @?!A B U . *+ B . *+ *C!+ ? B . *+!( ! B . *+5 % B . *+ 60>O B . *+L1 60>O B 1 B U ( @ *1+ S 1A T ) ? @?!A B U *%1+ PP )F ( <)1 9/6 B @ % B S 1 B CCA T 7 *+ % B 7 *+5( ?! B U @ % B S 1 B CCA T 1 ?! B V * *++ B V * *C!++ B @ S A T

PAGE 150

> CC B ( @ S % XX Q 1A T @?!A B U @ S 1A T <) <)1 7;6 B U ) @7 *+5( Y A T 7 *+CC B 7 *+5( B 1 B U T ) . *+?? B U U U . 7 *%+ *1+ B 7 *%+! 7 *%+ B 7 *%+5( ?! B @ ! B S 1 B CCA T 7 *+ 7 *?!+ C 7 *?!+ B 7 *+! 7 *+ B 7 *+5( ?! B U . @<)1A T <) @ % B S 1 B CCA T V * *++ B V * *C!++ B @ S A T CC B @7 *+5( Y A T *@7 *+!ACC+ B 7 *+5( B U U U U T <) E ( @ A @ % B S 1 B CCA T

PAGE 151

>9 V * *++ B V * *C!++ B @ S A T *@7 *CC+!ACC+ B U U U . ( @ % B S 1 B CCA T 7 *+5( % B 7 *+! 7 *+ B U ? @<)1A T . *%+ 3 ) ) ) . *%+ % B *%+ . *%+ B @ ! B S 1 B CCA T . *+ . *?!+ C . *?!+ B *+ . *+ B U 7? @ %B S 1 B CCA T V *7 *++ B C 7 *+ B @ S A T *@ *CC+ACC+ B U U @!A B U T <) @ A @%A B U U 1 2 3 ?)

PAGE 152

>: >7G$6 1 @ > 1 ) 1 ) 7G 7 *+ ' 1C! .G . *+ ' 1C! *+ *+ ' 1C! ) () *.,/0&123,4+ 15 ) ? ? 15 ) ? ? 1 A T / ) B B B B @A B 1 B 1 B B 15 B ) ? ? 15 B ) ? ? 11 B 11 B 1 B 1 B 1 B ; 6 () 11 0K @% 0G3 @() *.,/0&1&63617,8+ 1 1AA B 11 0K @% 0G3 @() *.,/0&1&6361.,/+ 1 1AA B 1 % B 15 1 B 15 1 B 2 > /; ' ) @ 1?! B Q % B ??A T . *+ B @ %A T ( . *+5 ??15 B 2G//1>7G3.G>/1.,/ @A B U U 2 > @ 1?! B Q % B ??A T ( @.,/1G1&6& @AA T B U . *+ B

PAGE 153

>> @ Q 11A T ( . *+5 ??15 B V *. *++ B C . *+ B @ S A T 7 *CC+!?? B U 2G//1>7G3.G>/1.,/ @A B U U 2 @ % B S 1 B CCA T 7 *+! B @ Q 11 XX %A T ( 2G//17,8 @A B??15 B U T ( ( 1 0K @1 A B U U . ) PP @?A 3 PP ) ) @ 1?! B Q % B ??A T ( @.,/1G1&6& @AA T B U % B V *. *++ B 1 B C . *+ B @ S A T CC B ( @7,81G1&6& @AA T B U 1CC B E C 7 *+! ? ! B 0G3 @ 1A B U

PAGE 154

>? 1 @A @1 ? V *. *++A B @1 %A T ? @ PP ) (A . *+5 ??15 B 2G//1>7G3.G>/1.,/ @A B U T . *+ 1 B . *+5 B U U PP @ A @ A7 ) G' )( @ % B S 1 B CCA T *+ 60>O B U 1 1 B ?)( @ 1?! B Q % B ??A T @.,/1G1/G$6 @AA T &/ . *+5 B / 1 *+ B . *+ 60>O B . *+L1 1 B @1 Y 60>OA T . *1+ B U *+ B 1 0G3 @1 A B U U 7) 15 15 B 15 15 B 1 1 B U

PAGE 155

>; 1 , @ A ; 3 ?) >7G$6 1 ) ) @ > 1 ) 1 ) ' 5' C )1 7G 7 *+ ' 1C! .G . *+ ' 1C! *+ *+ ' 1C! 15 7 1 0 @5' A A T / ) ( B 1 B B B 1 B 1 B 1 B 11 B 11 B 11 B 11 B 1 B B B B B 1 B ) 1 B B 1 B )( 1 B )( 1( B ( ( 1( B 7 *+5( 1 B ' 1 B 1( B P(P @A 1(B 1( 11( B ) ) 1 B ; ) & 1 B ; ) & ) B ) ) G' ( 1( G310K ? 1 B G310K SQ 1( 1( @1 7A B 1 % B ) % B ,

PAGE 156

>A @( % B ( S 15 B E(E ) A T @ *1+ 60>O VV 1 S 1A T 1CC B U 1 *1+ B 1 . *1+L1 B *1+ 1 B @1 Y 60>OA T . *1+ 60>O B U ) ( 11 . *1+5 B ( . *1+5 ( B ) ( 11( . *1+!( B ( C 11( B :)1 1 0G3 @11 1 ? (A B @ C 1 Q A T )1 @1 1 7 . V *+A B )CC B ) ) 7*+5( 1( 1( @1 7A B U . 11 B ' ' 11 % B ) E . *1+!( ?11( B V *. *1++ B C . *1+ B @ S A T CC B ( @7,81G1&6& @AA T B U V *7 *++B C 7 *+ B @ S A T

PAGE 157

> CC B 1( . *+!( B @1( Q % VV .,/1G1/G$6 @AA T . *+!( ?1( B *CC+ B 11 C 1( B U U U . *1+!( 11( B 1 0K @1 11A B 2 ( V *. *1++ B C . *1+ B @ S A T ) ( CC B 2G//17,8 @A B U 11 ? 11 B @11 Q %A T ( PP ) @ A 1 *. *1++ B U T ' 1 60>O B U N ) PP ' @P(PA @ A ( ) @ A ' ) ? ' ' ' @ ' PP ) A ) ? . ? )

PAGE 158

> V *11+ B C 11 B @ S A T CC B 1( ?. *+!( B . *+!( 1( B 7 1 . *+5 B 1 . *+ B 1 . *+L1 B @1 60>OA T *1+ 1 B U T . *1+L1 1 B U @1 Y 60>OA T . *1+ 1 BU V *. *++ B C . *+ B @ S A T CC B 1( 7 *+5( B ( @7,81G10726&1&6& @1(AA T B U 1 1( ? 1( B ( ) @1 S %A T 1 7 *+! B U ) ( E 1 ? 1( B )) ) ' @1 %A T 2G//17,8 @A B U T ( 7 *+5( 1 C 1( B U U U

PAGE 159

?= V *11+ B C 11 B @ S A T CC B % B 1 % B V *. *++ B 1 B C . *+ B @ S A T CC B 1( 7 *+5( B ( @7,81G10726&1&6& @1(AA T B U 1CC B C B 1 C 1( ? 1( B 1 0G3 @1 1A B U E . *+ @A @1 ? V *. *++A B 9 @. *+ %A T ) 2G//1>7G3.G>/1.,/ @A B 11 ? . *+!( B . *+5 ( B ) ( ( C . *+!( B U T > . *+5 1 B ) Z 1 C ! B 1 *+ B @1 Q 60>OA T PP ? @ A )( 1 . *1+ B . *1+ B U T

PAGE 160

? PP )( 1 ? @1 C 5A B *+ ? @ C 5A B U . *+L1 1 B . *+ . *+ @A B U U 11 @. 11 11A B 2 2G//1>7G3.G>/1.,/ @1A B . ( 1( C @1 C !A B @1( Q 1(A T1( 1( @1 7A B U 9' V *11+ B 1 B C 11 B @ S A T CC B ( @.,/1G1&6& @AA T B U 1CC B *. *+ C @. *+CCA+ 1 B E @ E A 1 . *+5 C 11 B ) ) E(E ( 1 1 ? ( ? . *+!( B ( ?? 1 ? . *+!( B ( F 1 0G3 @1 1A B . *+5 1 B > )(

PAGE 161

?9 1 *1+ B . *+L1 1 B . *+ 60>O B @1 Y 60>OA T . *1+ B U *1+ B 1 0G3 @1 1A B U 7 @11 Q %A T ( ? 7 *1+ 11 B 7 *1+ @A @1 ? V*11+A B 7 *1+! 11 B 7 *1+5( % B U U ) @)A B U 1 1 @ A ? ) ) ( @ )) A *% 1?!+ *%+ ) *1?!+ ) G E ( ( ) ) ( ) ) ( ) , @1A ) 3 ?) >7G$6 1 @ > 1 ) .G . *+ ' 1C! *+ *% 1?!+ A T /) B B B E B E

PAGE 162

?: , ? @ % B S 1 B CCA T ? ? @Y.,/1G1&6&1>7G3.G>/ @A VV . *+5 60>OA T B T . *+! B U @Y.,/1G1&6&1>7G3.G>/ @AA B ? ? B . *+5 B T . *+5 CC B . *+! B . *+! B ) ) U @. *+5 60>OA B ? 1 . *+5 B U U : @ % B S 1 B CCA T *.*+5+ B U U 11 & ) ) )( .( *1 1 C 1?!+ ) )( )( F *+ Q % *+ *+ )( . * *++ )(

PAGE 163

?> ?@ *+ C 5A )( 9 )( . *+ 3, P P . *+ ) . *+L1 )( PP ( ) ' @A < ) 3 ?) >7G$6 11 @ > .G . *+ ' 1C! *+ *+ )( 1 ( 1 ) ( A T / ) B B B 1 B )) ! B 1 B B 1 1 B )( B B B ( 1 B )( 1 B )( . V *1+ B C 1 B @ S A T CC B @.,/1G1&6& @AA T B U ) . *+ B : )( 1 *+ B @1 Q 60>OA T 1 . *1+ B U T 1 ? @1 C 5A B U

PAGE 164

?? . )( @1 1 B 1 Y 60>O B 1 . *1+L1A T . *1+ B 1 )( 1 1 B . 1 @ . *1+L1 B Y 60>O B . *+L1A T @. *+ Y XX . *+5 Y . *1+5A T 1 B B U ! V *. *1++ B V *. *++ B @ % B S B CCA T PP @ A ) @!CC Y CCA T )( B U U ? P)(P @ Y A T 1 B B U : Y . *1+!( C. *+!( B . *+! 1 B 2G//13,31>7G3.G>/1.,/ @A B 1@A . *+5 60>O B )( . *1+L1 . *+L1 B U U 6 )( @1 Q 60>OA T PP . *1+ 60>O B U T PP

PAGE 165

?; *+ 60>O B U U U )1 & ( ; ) ) 7 ) ( ) ' ) ' 3 ?) >7G$6 )1 @ > 1 ) 1 ) 7G 7 *+ .G . *+ *+ *% ?!+ V *%+ A T / ) B B < B B B B & V*%+ B @ % B S 1 B CCA T @.,/1G1/G$6 @AA T V *. *++ B . *+ @A @ ? V *%+A B . *+ B @< % B < S B @ % B S 1 B CCA T

PAGE 166

?A @7,81G1/G$6 @AA T @7 *+ %A T ' ( 2G//17,8 @A B U T 7 *+51 V *7 *++ B 7 *+51 B E ,361.,0>/6063 @A B U U U & B @ S A T ) @CC S %A T ?? B ,361.,0>/6063 @A B 7 *+51 B 7 *+ @A @ ? V *%+A B 7 *+ B @< % B < S B 7G$6 1( 1( @ > 1 ) 7G 7 *+ 7 *% 1?!+5( '

PAGE 167

? A T / ) B @ % B S 1 B CCA T @7,81G1/G$6 @AA T 7 *+5( % B U U @!A B U

PAGE 168

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

PAGE 169

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Æ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Æ).#( ! "#*.'!.) -)#,*# !.!.(.(+ + -"'$"##&&'!# #) "(.) &6*( &$9BA=C$ --9U:=A

PAGE 170

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

PAGE 171

/ !# ( .# % ' 4( .( ,#( !($ &., $ .( A: # + ,* !#, ( !##.+") "&.( (,#)#.2 #,".' ) "#&).#()#.()-*!##(+.(##.(+ !"# (.2 #'.! &., .( *+*'! ; *#( !& "# .' -*'*.(+ '!# (+.(##.(+,#+##.()-*!##(+.(##.(+ !!"#(.2 #'.! &., .'.( !##'!' (, "44.#'.()&*,#0 &,0 ".'! $)& ''.)) '$! 2 #&$ (,)-*!#!#) "(&+ ;9