A column pre-ordering strategy for the
unsymmetric-pattern multifrontal method *
Timothy A. Davis t
January 2, 2002
Technical report TR-02-001. Department of Computer and Information
Science and Engineering, University of Florida.
Abstract
A new method for sparse LU factorization is presented that com-
bines a left-looking column pre-ordering strategy with a right-looking
unsymmetric-pattern multifrontal numerical factorization. I Il col-
umn ordering is selected to give a good a priori upper bound on fill-in
and then refined during numerical factorization (while preserving the
bound). Pivot rows are selected to maintain numerical stability and
to preserve sparsity. Left-looking methods cannot select pivot rows to
preserve sparsity. As a result, the new method nearly always obtains
better orderings than both left-looking methods (such as that used in
MATLAB), and the prior unsymmetric-pattern multifrontal method
on which it is based (: i\.;f or UMFPACK Version 2).
Categories and Subject Descriptors: G.1.3 L' -iti,, i, .! Analysis]: 1-itii, i-
ical Linear Algebra linear systems (direct methods), sparse and very large
systems G.4 [A.l 11 ii,, i, s of Coil iill in ] '.11i I,11 ini, i, .,! Software algo-
rithm analysis, fI. ', M I
General terms: Algorithms, Experimentation, Performance.
Keywords: sparse nonsymmetric matrices, linear equations, multifrontal
method, ordering methods.
*This work was supported by the National Science Foundation, under grants D _IS-
9504974 and D_, 1S-9803599.
tDept. of Computer and Information Science and E, 1i_;.. i;i_ Univ. of Florida,
Gainesville, FL, i S.\ email: t .- !- -, til .- ,11 http://www.cise.ufl.edu/~davis.
1 Introduction
This paper considers the direct solution of systems of linear equations, Ax =
b, where A is sparse and unsymmetric. Tli, matrix PAQ is factorized into
the product LU. Tli, column ordering Q is selected to give a good a pri-
ori upper bound on fill-in and then refined during numerical factorization
(while preserving the upper bound on fill-in). Tli, row ordering P is se-
lected to maintain numerical stability and to preserve sparsity. TIl, method
is a combination of a column pre-ordering and symbolic analysis phase based
on COLA .l) [16, 17, 39], and a numerical factorization phase based on a
modification of the unsymmetric-pattern multifrontal method (U'. iPACK
Version 2, or 1.A38 [14, 15], which does not have a pre-ordering and symbolic
analysis phase).
Ti, pre-ordering and symbolic analysis of the method presented here is
similar to that used by left-looking methods such as SuperLU [18] or '. AT-
LAB's LU [33, 34]. In these methods, the column pre-ordering Q is selected
to provide a good upper bound on fill-in, no matter how the row ordering
P is chosen during numerical factorization. However, left-looking methods
cannot select the row ordering to preserve sparsity. '.1.A38 can select both
the row and column ordering to preserve sparsity, but it lacks an analysis
phase that gives good a priori bounds on fill-in. It can thus experience un-
acceptable fill-in for some matrices. In contrast to both of these strategies,
the numerical factorization in the method described here has the same a
prior upper bound on fill-in as left-looking methods (something that ': 1 A38
lacks), and the new method can select the row ordering P based on sparsity
preserving criteria (something that left-looking methods cannot do).
Section 2 gives an overview of the prior methods that the new method
is based on or related to: column pre-orderings, a priori upper bounds on
fill-in, left-looking methods, and right-looking multifrontal methods. Sec-
tion 3 describes the new algorithm. Performance results and comparisons
with other codes are given in Section 4, before a few concluding remarks and
information on the availability of the code are given in Section 5.
2 Background
Tli, new method is related to left-looking methods, since it uses a column
pre-ordering that gives the same a priori bounds on fill-in. Tli, numerical
factorization phase is based on the right-looking multifrontal method, guided
by the supernodal column elimination tree. T111 related methods are de-
scribed below.
2.1 Column pre-orderings
Fill-in is the introduction of new nonzero entries in L and U whose corre-
sponding entries in A are zero. Ti, row and column orderings, P and Q,
determine the amount of fill-in that occurs. Fin'ling the best ordering is an
NP-complete problem [45], and thus heuristics are used.
Suppose the column ordering Q is fixed, and let C = AQ. Sparse Gaus-
sian elimination with partial pivoting selects P via standard partial pivoting
with row interchanges, and factories PC into LU. If C has a zero-free di-
agonal the nonzero pattern of U is a subset of the nonzero pattern of the
C11i -I.y factor Lc of CTC [30]. Tli, entries in each column of L can be
rearranged so that their nonzero pattern is a subset of the nonzero pattern
of Lc. This subset relationship holds no matter how P is chosen during
Gaussian elimination on C.
This observation leads to a useful method for finding an ordering Q that
gives a good upper bound on the fill-in in the LU factors of C = AQ. Simply
use for Q an ordering that reduces fill-in in the C11.i1 -I y factorization of
(AQ)TAQ [28, 30, 31]. Tli, COi .1'.i) [33] and COLA:.i) [16, 17, 39]
routines in :iATLAB find an ordering Q without constructing the nonzero
pattern of ATA.
2.2 Left-looking methods
Left-looking methods such as LU organize their computation with the col-
umn elimination tree (the elimination tree of CTC [40]). SuperLU uses the
supernodal column elimination tree to reduce execution time by exploiting
dense matrix kernels (the BLAS [19]) in the computation of each supercol-
umn (a group of columns of L with the same upper bound on their nonzero
pattern). ',iA48 in the Harwell Subroutine Library is another example of a
left-looking method [26]. It differs from LU and SuperLU by using a partial
right-looking numerical factorization as its pre-ordering strategy.
At the kth step of factorization of an n-by-n matrix A, the kth column
of U is computed. Tli, pivot entry is chosen in the kth column, permuted to
the diagonal, and the kth column of L is computed. Columns k + 1 to n of
A are neither accessed nor modified in the kth step. Tli, advantage of this
approach is that it can be implemented in time proportional to the number of
floating-point operations [34]. This is not known to be true of right-looking
methods such as the multifrontal method. However, the disadvantage is that
the kth pivot row cannot be selected on the basis of sparsity, since the nonzero
patterns of the candidate pivot rows are unknown. Tli, pre-ordering Q is
found by assuming that all candidate pivot rows at the kth step have the
same upper bound nonzero pattern. Tli, pivot row is selected solely on the
basis of maintaining numerical accuracy. Only a right-looking method (one
that modifies the columns k + 1 through n at the kth step of factorization)
has access to the true nonzero patterns of candidate pivot rows at the kth
step of factorization.
2.3 Right-looking multifrontal methods
Tli, multifrontal method is one example of a right-looking method. Once
the kth pivot row and column are found, the elimination is performed and
the outer-product is applied to the remaining (n k)-by-(n k) submatrix
that has yet to be factorized.
Ti, factorization is performed in a sequence of frontal matrices. Each
frontal matrix is a small dense submatrix that holds one or more pivot rows
and their corresponding pivot columns. Consider the first frontal matrix. TIl,
original entries in the corresponding rows and columns of A are assembled
into the frontal matrix. Tli, corresponding eliminations are performed, and
the contribution block (a Schur complement) is computed. This contribution
block is placed on a stack for use in a later frontal matrix. Tli, factorization
of subsequent frontal matrices is the same, except that it is preceded by an
assembly step in which contribution blocks (or portions of them) are assem-
bled (added) into the current frontal matrix. After the assembly step, the
current frontal matrix has a complete representation of a set of pivot rows and
columns. In all multifrontal methods, more than one pivot row and column
can be held in a frontal matrix. Computing the Schur complement can be
done with dense matrix-matrix multiplication (D( il.1 i.1 [19]), an operation
that can obtain near-peak performance on high-performance computers.
i ly approaches have been taken to apply the multifrontal method to
different classes of matrices:
1. symmetric positive definite matrices [8, 41],
2. symmetric indefinite matrices ('.1A27) [23, 24],
3. unsymmetric matrices with actual or implied symmetric nonzero pat-
tern ( .1.A41) [3, 21, 25],
4. unsymmetric matrices where the unsymmetric nonzero pattern is par-
tially preserved ('.iA41u) [5, 6],
5. unsymmetric matrices where the unsymmetric nonzero pattern is fully
preserved ('.1.A38) [14, 15],
6. and QR factorization of rectangular matrices [4, 42].
Tli, ,_ are significant differences among these various approaches. For the
first four approaches, the frontal matrices are related to one another by the
elimination tree of A, or the elimination tree of A + AT if A is unsymmetric
[40, 41]. Tli, elimination tree has n nodes; each node corresponds to one pivot
row and column. TIl, parent of node k is node p, where p is the smallest row
index of nonzero entries below the diagonal in the kth column of L. A frontal
matrix corresponds to a path in the elimination tree whose columns of L have
similar or identical nonzero pattern; the tree with one node per frontal matrix
is called the assembly tree [22] or the supernodal elimination tree. Each
frontal matrix is designed so that it can fully accommodate the contribution
blocks of each of its children in the assembly tree. Tliu-- the assembly step
adds the contribution blocks of each child into the current frontal matrix.
For symmetric positive definite matrices, all of the pivots originally assigned
to a frontal matrix by the symbolic analysis phase are numerically factorized
within that frontal matrix. For other classes of matrices, some pivots might
not be eliminated, and the contribution block can be larger than predicted.
Tli, uneliminated pivot is delayed, and its elimination is attempted in the
parent instead.
In the first three approaches, the frontal matrices are square. In a recent
approach by Amestoy, Duff, and Puglisi [5, 6] (approach I in the list above),
it was noted that rows and columns in the frontal matrix that contain only
zero entries can be detected during numerical factorization and removed from
the frontal matrix. Tli, frontal matrix is rectangular, and the assembly tree
is still used.
Tlii first four approaches precede the numerical factorization with a sym-
metric reordering of A or A + AT, typically with a minimum degree [1, 29]
or nested-dissection ordering [9, 28, 37, 38] as part of a symbolic analysis
phase.
Tli, fifth approach, used in '.1A38, does not use a pre-ordering or sym-
bolic analysis phase. Rectangular frontal matrices are constructed during
numerical factorization, using an approximate '. ,1 i I.>witz ordering. T!lI first
pivot within a frontal matrix defines the pivot row and column pattern and
the size of the frontal matrix. Extra room is added to accommodate subse-
quent pivot rows and columns. Subsequent pivots are then sought that can
be factorized using the same frontal matrix, allowing the use of dense matrix
kernels. Tli, frontal matrices are related to one another via a directed acyclic
graph (DAG) rather than an elimination tree.
Ti, last approach, multifrontal QR factorization [4, 42], is based on the
column elimination tree of A.
3 The algorithm
An overview of the new algorithm (U iFPACK Version 3.2) is given below,
followed with details of its implementation. It uses the fifth approach in the
list above (like 1.A38). Unlike 1.A38, it precedes the numerical factorization
with a column pre-ordering and symbolic analysis phase.
3.1 Overview
U. iFPACK3 first finds a column pre-ordering that reduces fill-in, without
regard to numerical values. 'i \i the analysis phase breaks the factorization
of the matrix A down into a sequence of dense rectangular frontal matrices.
TIi, frontal matrices are related to each other by a supernodal column elimi-
nation tree, in which each node in the tree represents one frontal matrix. This
phase also determines upper bounds on the memory usage, the floating-point
operation count, and the number of nonzeros in the LU factors.
In its numerical phase, U, iFPACK3 factories each chain of frontal ma-
trices in a single work array, similar to how the unifrontal method [27] fac-
torizes the whole matrix. A chain of frontal matrices is a sequence of fronts
where the parent of front i is i + 1 in the supernodal column elimination
tree. Like all multifrontal methods, U: .iFPACK3 is an outer-product based,
right-looking method. At the k-th step of Gaussian elimination, it represents
the updated submatrix Ak as an implicit summation of a set of dense con-
tribution blocks (also referred to here as elements, borrowing a phrase from
finite-element methods) that arise when the frontal matrices are factorized
and their pivot rows and columns eliminated.
Each frontal matrix represents the elimination of one or more columns;
each column of A will be eliminated in a specific frontal matrix, and which
frontal matrix will be used for each column is determined by the analysis
phase. This is in contrast to prior multifrontal methods for unsymmetric or
symmetric indefinite matrices, in which pivots can be delayed to the parent
frontal matrix (and further up the tree as well). It differs from .1iA38, which
has no symbolic analysis at all. Tli, pivot rows are not known ahead of
time as they are for the multifrontal method for symmetric positive definite
matrices, however.
Tli, analysis phase determines the worst-case size of each frontal matrix
so that they can hold any candidate pivot column assigned to them, and
any candidate pivot row. From the perspective of the analysis phase, any
candidate pivot column in the frontal matrix is identical (in terms of nonzero
pattern), and so is any candidate pivot row. A left-looking numerical factor-
ization method does not have any additional information.
However, the right-looking numerical factorization phase of U: 1 iFPACK3
has more information than its analysis phase. It uses this information to re-
order the columns within each frontal matrix to reduce fill-in. Similarly, since
the number of nonzeros in each row and column are maintained (more pre-
cisely, COI -.1. II' )-style approximate degrees [33]), a pivot row can be selected
based on sparsity-preserving criteria as well as numerical considerations (re-
laxed threshold partial pivoting). This information about row and column
degrees is not available to left-looking methods.
Tlhil- the numerical factorization refines the column ordering Q by re-
ordering the pivot columns within each front, and it computes the row order-
ing P, which has the dual role of reducing fill-in and maintaining numerical
accuracy (via relaxed partial pivoting and row interchanges).
3.2 Column pre-ordering and symbolic analysis
Tli, column pre-ordering is a slightly modified version of COLA Ii) [16,
17, 39]. COLA il) finds a symmetric permutation Q of the matrix ATA
(without forming ATA explicitly), and is based on an approximate minimum
degree method [1].
i \ the symbolic analysis phase constructs the supernodal column elim-
ination tree. Tli, tree is post-ordered, with the largest child of each node
being ordered just before its parent. :'I \, each frontal matrix is assigned to
a unifrontal chain. After the post-ordering, two frontal matrices i and i + 1
are in the same chain if i + 1 is the parent of i. Tli, largest frontal matrix
in each chain is found; this determines the size of the work array to be used
to factorize the chain. In the numerical factorization phase, the unifrontal
method will be applied to each chain, with as few as a single contribution
block being stacked per chain (more may be created if this results in too
large of a contribution block with too many explicitly zero entries). Tli,
symbolic phase determines upper bounds on the memory usage, the number
of nonzeros in L and U, and the floating-point operation count. This entire
phase, including the ordering, is computed in space proportional to number
of nonzeros in A.
.1iA38 attempts to find unifrontal chains on the fly. Tli, post-ordering
of U I iPACK3 finds much longer unifrontal chains, which is why it is able
to achieve much higher performance than 1A38. Post-ordering the tree also
reduces memory usage of the contribution block stack. Performance results
are discussed in more detail in Section 4.
3.3 Numerical factorization
Ti,, numerical factorization phase starts by allocating several temporary data
structures, including a work array that can hold the largest frontal matrix
in the supernodal column elimination tree, and a stack to hold the elements.
During numerical factorization, the active submatrix Ak is held as a collection
of rectangular elements, one for each non-pivotal column of A and one for
each contribution block created during numerical factorization. To facilitate
the scanning of rows and columns, element lists [14] for each row and column
hold the list of elements that contribute to that row and column. Tli, are
also used to compute the COi .1..i )-style approximate degrees used during
numerical factorization.
Let Ci denote the set of |Ci| candidate pivot columns in the ith frontal
matrix. Tli, set of non-pivotal columns that can appear in the ith frontal
is Ni. Let Ri denote the set of |Ri| candidate pivot rows for the ith frontal
matrix. Tli, sum of \Cij for all i is equal to n; this is not the case for Ri.
Tli, upper bound on the size of the frontal matrix is IRi,-by-(|C, + INil).
If the matrix is structurally nonsingular, IR|l > ICl for all i will hold. Tli,
parent of node i is the smallest numbered node that contains a column in Ni
as one of its own candidate pivot columns. Ti, Algorithm 1 is an outline of
the method (nB is a parameter, 24 by default).
3.4 Frontal matrix strategy and local pivot search
Figure 1 shows the (n k)-by-(n k) active submatrix Ak being factorized,
and the portion of that matrix that may be held in the work array (the
shaded region, which is the upper-bound of the current frontal matrix). Ti,
current frontal matrix occupies only part of the work array, shown as the two
dark shaded regions. Ti, -, two regions are actually stored as one contiguous
block. During factorization within this frontal matrix, some of the candidate
rows in Ri will appear in the frontal matrix, and some may not (some of
these may never appear). Likewise, some of the columns in Ci will appear
in the front and some will not yet appear (but they are all guaranteed to
appear by the time the frontal matrix is fully factorized). Fifilly, some of
the columns in Ni will currently be in the front, but some will not (and like
Ri, some may never appear). Tli, frontal matrix has been permuted in this
perspective so that the candidate pivot columns Ci are placed first followed
by the non-pivotal columns in the set Ni. Note that the work array is large
enough to hold all candidate pivot rows (Ri), and all candidate pivot columns
(Ci); the part of these rows and columns outside the work array is zero in the
active submatrix Ak. Ti11 (k 1)st and prior pivots are not shown, but some
of these are held in the frontal matrix as well and are removed when their
pending updates are applied to the contribution block. Tli, outer-product
updates are applied via a dense matrix-matrix multiply.
Tli, search for the kth pivot row and column is limited, but it is this step
that allows the method to typically obtain orderings that are better than
left-looking methods. Up to two candidate pivot columns are examined: the
column of least approximate degree in Ci in the current front, and the one of
least approximate degree in Ci but not in the current front. In each of these
two columns, up to two candidate pivot entries are sought: the candidate
row of least approximate degree in the current front, and the row of least
approximate degree not in the current front. Tlii pivot entry must also
be numerically acceptable (by default, an absolute value of 0.1 times the
absolute value of the largest entry in the candidate pivot column, or larger).
Tli, row and column degrees are not exact; COIT :.1'.i)-style approximate
degrees are used, which is simply the sum of the sizes of the contribution
Algorithm 1: UMFPACK3 numerical factorization
initializations
k=0
i=0
for each chain:
current frontal matrix is empty
for each frontal matrix in the chain:
i=i+1
for ICil iterations:
k=k+l
find the kth pivot row and column
if too many zero entries in new contribution block
apply pending updates
create new contribution block and place on stack
start a new frontal matrix
else
if too many zero entries in new LU part of frontal matrix
apply pending updates
end if
extend the frontal matrix
end if
assemble contribution blocks into current frontal matrix
scale pivot row and column
save kth column of L and kth row of U
if # pivots in current frontal matrix > nB
apply pending updates
end if
end for ICil iterations
end for each frontal matrix in the chain
apply pending updates
create new contribution block and place on stack
end for each chain
Figure 1: Ti, active submatrix and current frontal matrix
C, N,
candidate non-candidate
pivot columns
columns
in not in in not in k ...
uppeI ounu UnII
on frontal matrix
zero
1 current frontal matrix
remainder ofAk active submatrix
(not affected by current frontal matrix)
-a C
o .
zero
blocks in each row and column. Tighter approximations were tried (as in
COLA i) and A i1)), but this was not found to improve the ordering quality.
Since the tighter A .i )-style approximation requires a second pass over the
element lists of the rows and columns in the current frontal matrix, the
simpler COi. I-.ii )-style approximation was used instead. Tli, tighter A i)-
style degree approximation is used only by the column pre-ordering in the
symbolic analysis phase.
Ti, candidate pivot entries are shown as four dots in Figure 1. Anywhere
from one to four of these candidates may exist. T111 candidates are eval-
uated, and the exact degrees of up to two candidate pivot columns and up
to four candidate pivot rows are computed. Tli, best one of the candidate
pivot entries is chosen as the kth pivot row and column. Tli, metric used to
evaluate these candidates is a form of approximate minimum fill-in [43, 44];
the pivot entry that causes the least growth in the size of the actual frontal
matrix is chosen.
Increasing the size of the current frontal matrix to include the new pivot
row and column may create new zero entries in the frontal matrix, in either
the pending pivot rows and columns, or the contribution block, or both.
Pending updates are applied if the number of zero entries in the pending
pivot rows and columns (not shown in Figure 1) increase beyond a threshold.
Ti, updates are also applied, and the current contribution block stacked, if
too many zero entries would be included in the old contribution block; in
this case a new frontal matrix is started. Tli, latter step also occurs at the
end of a chain.
After the pivot search and possible update and/or extension of the frontal
matrix, prior contribution blocks are assembled into the current frontal ma-
trix. T111 are found by scanning the element lists, in the same manner as
.1 A38. T!i, assembly DAG used by '.1 A38 is neither used nor computed in
UI. iFPACK3; its role is replaced by the simpler supernodal column elimina-
tion tree computed in the analysis phase. Tli, kth pivot row and column are
computed and a copy is saved in a separate compressed-index data structure
for L and U. FirI, ll-, pending updates are applied if sufficient work has
accumulated.
4 Experimental results
In this section the new method, U:.iFPACK3, is compared with LU, Su-
perLU, '.1.A38, and the latest i71-i iiii, i i" version of .1.A41 [5, 6], referred
to here as .iA41u. Each method (except for LU) was compiled with iden-
tical compiler parameters (the highest level of optimization), and executed
on a Sun Ultra 80 with 4GB of main memory and four processors. Only
one processor was used, although SuperLU and .iA41u both have parallel
versions. Tli, Sun Performance Library BLAS was used. LU was used within
M.iATLAB Version 6.0. It does not make use of the BLAS.
With the exception of '.1A38 and LU, all methods used their default
parameter settings and ordering methods. .1.38 can permute a matrix to
block triangular form [11, 20, 22] and then factorize each irreducible diagonal
block. This can improve performance for some matrices. Tli, other methods
do not include this option, but can be easily adapted to do so via a short
.iATLAB script. This was tested, and the overall relative results presented
here do not change very much. M.iATLAB uses LU with COi-.1i..l) [33] to
solve sparse linear systems (x=A\b), although its built-in COLA .il) routine
is faster and generates better orderings [16, 17, 39].
Tli, UF sparse matrix collection [12] includes 357 real, square, unsym-
metric sparse matrices. All methods were tested on all but three of the very
largest matrices (APPU, PI:l2l and XENON2) which could not be factorized
on this computer. Statistics gathered for each method included:
Ti, CPU time for the pre-ordering and symbolic analysis phase, and
the numerical factorization phase. Tli, total run time is the sum of
these two times.
Tli, number of nonzeros in L + U. This excludes the zero entries that
most methods explicitly store in their data structures for L and U.
It also excludes the unit diagonal of L, which does not need to be
explicitly stored.
Tii i 1i ,,.,ii i!" floating-point operation count. This was computed
based solely on the nonzero pattern of L and U,
S2LU, + Lk
k=l k=l
where Lk is the number of off-diagonal nonzeros in column k of L, and
Uk is the number of off-diagonal nonzeros in row k of U. Both Lk
and Uk exclude explicitly stored entries that are numerically zero. TII
flop count is a function of the quality of the pivot ordering found by
the method (P and Q), and not a function of how the factorization is
actually computed.
Ti, total memory usage, excluding the space required to hold A, x,
and b. LU and iA41u do not report this metric. LU's memory usage
when A is real and square is 12 bytes per nonzero in L + U (excluding
the unit diagonal of L, which is explicitly stored), plus 53n bytes for
P, Q, the rest of the data structures for L and U, and temporary work
space [32]. Til, memory usage of .1 A41u was found via binary search
on the size of its two work arrays (one double precision array and one
integer array).
T11, norm of the residual, ||Ax b~|o.
LU uses a default threshold partial pivoting ratio of 1.0 (true partial piv-
oting), while U:. FPACK3 uses its default of 0.1 (entries in the L computed
by U. IFPACK3 have a magnitude of 10 or less). All methods except LU use
iterative refinement with sparse backward error [7] in their forward/backward
solve step. U. IFPACK3 was found to be just as accurate as LU.
Tli, symmetry of the pattern a sparse matrix is defined as the number
of matched off-diagonal entries over the total number of off-diagonal entries.
An entry aij is matched if i = j and aji is also an entry. A matrix with a
symmetric pattern has a symmetry of one; a completely asymmetric pattern
has a symmetry of zero. Tli, test set is split into two parts, unsymmetric
matrices (symmetry < 0.5) and -; iiii, 1 ni':" (symmetry > 0.5). Only the
larger matrices are considered, defined as those for which LU requires 107 or
more floating-point operations. 'ii ii i .ll singular matrices are discarded.
This leads to a test set of 77 unsymmetric matrices and 94 symmetric ones.
Results for all 77 unsymmetric matrices are shown in Figure 2. Each
of the four plots depicts the relative results of one method as compared to
U. IFPACK3. Each circle on the plot is a single matrix. Tli, x-axis is the
log2 of ratio of the memory required to factorize the matrix using the specific
method over the memory required by U IAFPACK3 to factorize the matrix.
Tli, y-axis is the log2 of the relative total run time. Tli-' a circle in the
upper right quadrant depicts a matrix for which the specific method requires
Figure 2: Results relative to U iFPACK3 (unsymmetric-patterned matrices)
MATLAB LU SuperLU
I .L E
4 I4 I
I I.. . . . . . . . .
" 2 I 2
__ a _____ E _
I 1: I I :
0 I I I I:
-2 : -2 I:
-2 -1 0 1 2 0 -2 -1 0 1 2
log2 (LU memory / UMFPACK3 memory) log2 (SuperLU memory / UMFPACK3 memory)
MA38 MA41 u
w : |
E
-4 4
Io O
*
o I 4 II*
0
80
o< 0
0 <
2 2 2 I o
o o
-2 -1 0 1 2 -2 -1 0 1 2
log2 (MA38 memory / UMFPACK3 memory) log2 (MA41 u memory / UMFPACK3 memory)
more time and more memory than U:. iPACK3. Ti, x-y axes are drawn as
solid lines; these are bracketed by dashed lines representing relative results
of 0.8 and 1.25 respectively. Within these two dashed lines, the results for
the specific method and U :.1PACK3 are comparable. '. 1li.ii relative run
time and memory usage results are shown as dotted lines. Figure 3 reports
the same results for the 94 symmetric-patterned matrices. All eight plots in
the two figures use the same x-y axes for ease of comparison. This means
a few outliers are not shown in some plots of Figure 3. Table 1 reports the
median relative results for these two sets of matrices, as well as two metrics
not shown in Figures 2 and 3: the relative number of canonical floating point
operations, and the relative number of nonzeros in L + U.
A selection of matrices is given Tables 2 through 4. Ti, -, are the
Figure 3: Results relative to U'. iFPACK3 (symmetric-patterned matrices)
MATLAB LU SuperLU
I I E-
-2 -1 0 1 2
log2 (LU memory / UMFPACK3 memory)
-2 -1 0 1 2
log2 (MA38 memory / UMFPACK3 memory)
-2 -1 0 1 2
log2 (SuperLU memory / UMFPACK3 memory)
MA41u
-2 -1 0 1 2
log2 (MA41 u memory/ UMFPACK3 memory)
Table 1: .1 i. iii relative results vs. U. iFPACK3
LU SuperLU MA38 MA41u
I,!!- !!!!,r i, test set
time: 7.47 3.05 1.53 0.81
flop: 1.59 1.58 1.88 1.70
memory: 1.27 1.32 1.58 1.94
nnz in L + U: 1.28 1.27 1.30 1.18
- !i!~i~ 1 ,, test set
time: 4.99 1.83 1.76 0.43
flop: 1.26 1.26 2.06 0.56
memory: 1.37 1.37 1.58 0.90
nnz in L + U: 1.13 1.13 1.27 0.68
Table 2: Selected unsymmetric-patterned matrices
Source Name n nnz in in description
Bai Rv5151 5151 20.2 0.490 Markov chain, random walk
Grund BAYERO1 77 ;, 275.1 0.000 chemical process simulation
Mallya LHR71C II71 I I -2 I 0.002 chemical process simulation
Zi.. RDIST1 4134 94.4 0.059 chemical process simulation
Vavasis AV\1Il 1 _' 41092 1683.9 0.001 irregular finite-element problem
Hollinger G7JAC200 59310 717.6 0.032 economics, G7 social security
Hollinger : 1i ...i i .1 i 64089 376.4 0.074 economics, Intl. Monetary Fund
Table 3: Selected circuit simulation matrices
Source Name n nnz in iin description
Grund MEG1 2904 58.1 0.002 1MB memory circuit
Grund MEG4 5860 25.3 1.000 4MB memory circuit
AT&T TWOTONE 120750 1206.3 0.245 frequency domain, harmonic balance
Bomhof CIRCUIT_1 2624 35.8 1.000 differential algebraic equations
Bomhof CIRCUIT_4 80209 ;. 7.6 0.829 differential algebraic equations
Hamm MEMPLUS 17758 99.1 1.000 memory circuit
Hamm SCIRCUIT 1','1 I' 958.9 1.000 digital circuit with parasitics
Table 4: Selected symmetric-patterned matrices
Source Name n nnz in in description
II II
Li LI 22695 1215.2 1.000 3D,, i I. _. l.1 .h...1,l ( ,1,i -
Wang WANG4 26068 177.2 1.000 3D MOSFET semiconductor
Zhao ZHAO2 33861 166.5 0.922 electromagnetic -1. 1II
Ronis XENON1 48600 1181.1 1.000 complex zeolite/sodalite crystal
Simon BBMAT 38744 1771.7 11 ;I' 2D airfoil
Simon RAEFSKY4 19779 1316.8 1.000 buckling problem for container model
FIDAP EX11 16614 1096.9 1.000 3D, cylinder & flat plate heat exch.
largest matrices within each class (matrix source or problem type). Circuit
simulation matrices occurred in both the unsymmetric and symmetric sets;
a selection of these are listed separately in Table 3.
Results for these matrices are given in Tables 5 through 7, which lists the
run time in seconds (including the symbolic analysis and ordering phase), the
canonical floating-point operation count (in millions), the total amount of
memory used (in megabytes), and the number of nonzeros in L + U (in thou-
sands; this count excludes the unit diagonal of L) for LU, SuperLU, U:. F-
PACK3, '.1 A38, and '. iA41u. Table 6 also shows the results for U: .iFPACK3
with a non-default column ordering from SY:' .1: .1i) applied to A + AT, and
with a non-default option that prefers the diagonal entry as pivot. Results
within 25% of the best result for a particular matrix (excluding the non-
default ordering for U: '. iFPACK3) are shown in bold. Note that a few of the
circuit simulation matrix results do not fit in the axes chosen for Figure 3.
Tli, results must be interpreted with caution; the test set is not a
statistical sample of all sparse matrices encountered in practice, and the
run time results can differ depending on the computer used. SuperLU and
'.1A41u both have parallel versions. LU, U:-.iFPACK3, and '.1.A38 do not.
U:'. iFPACK3 is based on the supernodal column elimination tree, which could
be used to guide a parallel version of U:. iFPACK3. '.1 A38 has no such
tree, although a parallel re-factorize algorithm based on the elimination DAG
found in a prior sequential numerical factorization has been developed [2, 35,
36].
Tli, left-looking methods LU and SuperLU find nearly identical orderings
because they use the same pre-ordering and pivoting strategy. U: .1 FPACK3
behaves most similarly to the left-looking methods. This is to be expected,
since all three methods use the same pre-ordering, are based on the column
elimination tree, and exploit the same worst-case upper bound on fill-in. Be-
cause U:'.iFPACK3 can further refine the row and column ordering based
on the row and column degrees available during factorization, it is typically
able to find a better pivot ordering than LU and SuperLU, leading to fewer
floating-point operations, fewer nonzeros in L and U, and less memory us-
age. U:, iFPACK3 is typically faster than LU and SuperLU in this particular
environment, more so for the largest matrices.
U:'. iFPACK3 makes efficient use of memory, even though it needs a large
work array to hold a contribution block stack, and another large work array
for the current frontal matrix. Table 8 lists the median number of bytes per
nonzero entry in L + U; this is the total memory usage including all work
Table 5: Results for selected unsymmetric-patterned matrices
Matrix LU SuperLU UMF3 MA38 MA41u
Bai time: 1.3 0.6 0.4 0.8 0.5
Rw5515 flop: 28.1 28.3 21.6 48.9 37.7
mem: 4.1 5.0 3.4 5.6 5.7
nnz LU: 333.6 336.0 295.7 392.2 312.4
Grund time: 3.9 2.5 3.1 3.7 2.6
BAYEROI flop: 33.4 33.6 18.3 26.1 37.7
mem: 18.0 27.0 15.8 17.0 34.3
nnz LU: 1320.8 1305.4 1004.9 1044.2 1283.2
Mallya time: 181.2 163.4 123.7 151.4 45.7
LHR71C flop: 492.4 492.1 346.8 932.9 .-" 7
mem: 84.7 89.3 54.3 101.1 128.7
nnz LU: 7086.7 7086.6 5960.4 8783.0 7087.4
tZ. time: 1.0 0.4 0.4 0.3 0.4
RDISTI flop: 14.9 14.9 6.6 7.0 5.6
mem: 4.3 4.7 2.6 3.4 3.8
nnz LU: .,..6 .;-..6 234.1 230.2 210.4
Vavasis time: 3438.9 843.9 197.1 346.7 40.1
AV 11 flop: 74015.6 73965.0 33310.9 63999.3 3543.3
mem: 491.7 436.7 374.9 469.7 168.3
nnz LU: 42782.8 42764.2 39140.1 38139.6 9055.7
Hollinger time: 3164.7 il-,7.9 254.1 331.5
G7JAC200 flop: 61457.9 61821.6 54740.5 44344.0
mem: 532.1 546.4 416.4 388.2
nnz LU: 46235.8 46358.0 40575.3 32484.7
Hollinger time: I II. 2156.2 270.2 2198.1 1485.3
i ,1 ..; ..'1 i flop: 119173.3 110564.0 63216.1 289004.0 391125.9
mem: 641.0 598.7 444.7 1415.0 Il 1i .7
nnz LU: 55730.1 ,2. 7 40215.3 7' 90927.8
Table 6: Results for selected circuit simulation matrices
Matrix LU SuperLU UMF3 MA38 MA41u UMF3 w/
(default) SYMAMD
Grund time: 1.2 1.3 0.5 0.4 0.5 4.1
MEG1 flop: 19.5 42.2 2.5 3.2 30.2 60.3
mem: 3.0 5.9 5.5 2.0 9.2 10.8
nnz LU: 249.0 355.3 122.0 130.9 303.9 410.1
Grund time: 1.2 3.5 0.4 0.1 0.1 0.3
MEG4 flop: 28.1 28.1 14.2 0.2 0.3 0.1
mem: 4.0 10.8 6.0 1.1 1.4 0.2
nnz LU: 322.3 322.3 242.6 45.4 45.0 37.1
AT&T time: 449.8 169.9 60.2 67.5 57.6 3717.5
TWOTONE flop: 8445.3 8480.3 10787.2 10126.3 9360.6 281806.2
mem: 190.7 2 1 154.7 172.2 312.8 1043.0
nnz LU: 16131.4 16201.0 14975.0 14061.6 11544.5 132972.7
Bomhof time: 16.2 4.6 1.3 0.4 0.3 0.3
CIRCUIT_I flop: 442.4 442.4 2.1 0.9 0.8 0.8
mem: 11.8 10.5 6.4 1.2 1.5 3.4
nnz LU: 1019.9 1019.1 54.5 44.0 44.6 42.9
Bomhof time: 5241.7 2090.5 1114.3 12.1 10.8 37.7
CIRCUIT_4 flop: 103079.9 101322.0 139762.6 6.3 5.8 6.3
mem: 435.8 474.7 2279.2 14.4 18.5 55.0
nnz LU: 37728.7 39799.7 43599.1 443.0 487.6 440.6
Hamm time: 152.5 37.4 6.9 0.9 0.4 2.0
MEMPLUS flop: 3422.3 3422.2 10.9 1.6 1.8 1.6
mem: 38.3 35.4 34.2 4.0 4.6 9.4
nnz LU: 3264.8 3264.8 220.9 133.0 140.9 122.4
Hamm time: 30.0 15.4 11.9 7.7 4.2 157.7
SCIRCUIT flop: 582.0 582.0 .;11.6 93.7 57.7 59.3
mem: 80.4 100.7 55.7 49.9 57.4 75.1
nnz LU: 6272.3 6272.3 4819.4 2963.1 2942.3 2552.3
Table 7: Results for selected symmetric-patterned matrices
Matrix LU SuperLU UMF3 MA38 MA41u
Li time: 5210.4 1889.9 542.6 1085.8 385.1
LI flop: 109145.9 109180.0 II -.7i.1 193152.6 83254.0
mem: 663.9 660.8 685.7 12 I 2 676.9
nnz LU: 57910.2 57924.3 56959.6 91050.9 47011.6
Wang time: 1586.7 388.8 106.7 197.1 25.9
WANG4 flop: 32171.2 32171.2 30497.4 40076.3 10472.1
mem: 292.7 255.5 263.1 271.3 135.3
nnz LU: 25460.7 25460.7 23876.1 24309.5 11472.2
Zhao time: 691.5 179.5 49.5 327.1 85.2
ZHAO2 flop: 11546.8 11932.4 11121.2 60281.0 15056.6
mem: 196.8 189.3 155.7 450.6 186.3
nnz LU: 17045.3 17231.3 14967.0 33259.2 17250.9
Ronis time: 4082.7 999.1 290.4 7' 62.1
XENON1 flop: 88811.1 88811.0 88844.1 1.77'i' 26790.6
mem: 702.3 600.1 552.5 955.6 313.2
nnz LU: 61153.7 61153.6 59024.2 67816.8 30118.5
Simon time: 2054.7 645.6 172.6 1767.6 125.9
BBMAT flop: 45069.5 45092.9 37749.5 342827.6 40415.2
mem: 571.8 514.4 378.5 1482.2 450.8
nnz LU: 49795.9 49810.1 44086.8 131470.3 44886.1
Simon time: ;* .' 173.2 49.0 147.0 20.6
RAEFSKY4 flop: 13408.8 13408.8 12888.4 29327.7 8247.9
mem: 245.4 216.9 181.9 256.3 151.7
nnz LU: 21354.5 21354.5 20933.5 23812.3 13336.5
FIDAP time: 496.1 145.9 33.4 274.6 19.8
EX11 flop: 11822.3 11888.1 6756.5 n 7278.7
mem: 217.5 197.6 123.8 389.2 130.5
nnz LU: 18928.0 !I't.1 13983.8 33196.1 12520.6
Table 8: 'i. ii,, ry usage (median bytes per nonzero in L + U)
Test set LU SuperLU UMF3 MA38 MA41u
I.'! !ui!i li, 12.3 12.7 12.6 14.3 19.0
symmetric 12.3 11.9 10.4 12.8 13.8
arrays (but excluding the matrix A) divided by the total number of nonzeros
in L + U. Eight bytes are required to hold the numerical value of each entry
L or U itself; an integer row or column index takes four bytes.
U:.iFPACK3 is more memory-efficient that its predecessor, A.1.38, in
terms of bytes per entry in the resulting LU factors, primarily because of
its use of a post-ordered supernodal column elimination tree. U '.iPACK3
tends to be faster than .1.38 and finds better orderings for most matrices,
with the notable exception of circuit simulation matrices. Tl, -, matrices
often have dense rows, and thus ATA tends to be dense. In this case, the
worst-case upper bound is too pessimistic. 1.A38 uses an approximate-degree
.ii I.,llwitz criteria on the pattern of the active submatrix Ak itself, and does
not consider ATA. This is also true of '.iA41u, which uses an approximate-
degree ordering on the pattern of A + AT (A.il), [1]). Both .1.A38 and
M.iA41u attempt to reduce an optimistic lower bound on fill-in. Tli, perfor-
mance of U':. iPACK3 on these matrices can be improved dramatically when
a non-default symmetric ordering is used instead.
For matrices with unsymmetric nonzero pattern, U- .iFPACK3 typically
uses less memory than .1iA41u. Tli, latter method ran out of memory for
one matrix in the test set. In terms of run time, M.iA41u and U :.iPACK3
are roughly split, in terms of which method is fastest on which particular
unsymmetric matrix. For symmetric-patterned matrices, M.iA41u is nearly
always the fastest method of those considered here, and it also tends to
find a better ordering in terms of fill-in and floating-point operation count.
U '.iPACK3 has less memory overhead per nonzero in L and U, however,
and thus it can still factorize some symmetric-patterned matrices using less
memory than '.iA41u.
Brainman and Toledo [10] use a nested dissection dissection method (re-
cursive graph partitioning) for finding good orderings for left-looking meth-
ods. T111 ii method partitions ATA without forming ATA explicitly, using
wide separators of A + AT. It is particularly well suited to matrices arising
in 2D and 3D problems, finding better orderings than COLA '.ii) for those
matrices. ., -'.1 of the symmetric-patterned matrices reported in Table 4 fall
in this category. Since U'. I PACK3 exploits the same upper bound on fill-in
as left-looking methods, and since U '. FiPACK3 can accept any given column
pre-ordering instead of its default COLA '.i ) ordering, their method can be
applied to U '. I PACK3 to improve its performance on 2D and 3D problems.
5 Summary
Tli, new method presented here, U. IFPACK3, tends to perform better than
left-looking methods (LU and SuperLU) on a wide range of matrices. It typ-
ically uses less memory and finds better orderings. It takes less total time on
the particular single-processor computer used for the comparisons (although
a parallel version of SuperLU does exist). Based on the same criteria, it typ-
ically outperforms its predecessor, 'A.1.38, on all matrices with the exception
of those arising in circuit simulation problems. For matrices with moderate
or highly unsymmetric nonzero pattern, U. iFPACK3 typically finds a bet-
ter ordering that 'A.141u and uses less memory, but it is not always faster.
.1iA41u is clearly superior for matrices with symmetric nonzero pattern (in
terms of fill-in and run time), although U'. iF PACK3 can still factorize many
symmetric matrices using less memory than .1iA41u, even a few very large
ones.
U'-.iFPACK Version 3.2, prior versions of U. iFPACK, COLA:-. I),
SY'. IA. I), and A :. 1) are available at www.cise.ufl.edu/research/sparse, and
as a collected algorithm of the AC'.1 (Algorithm 8xx) [13]. UM.iFPACK3
will also appear as a built-in routine in a future release of .iATLAB as a
replacement for LU and the forward and backlash matrix operator (x =
A\b).
References
[1] P. R. Amestoy, T. A. Davis, and I. S. Duff. An approximate minimum
degree ordering algorithm. SIAM J. Matrix Anal. Applic., 17(4):886
905, 1996.
[2] P. R. Amestoy, T. A. Davis, I. S. Duff, and S. :.I Hadfield. Parallelism
in multifrontal methods for matices with unsymmetric structures. In
Abstracts of the Second SIAM Conference on Sparse Matrices, Snowbird,
Utah, Oct. 1996.
[3] P. R. Amestoy and I. S. Duff. Vectorization of a multiprocessor multi-
frontal code. Intl. J. Supercomputer Appl., 3(3):41-59, 1989.
[4] P. R. Amestoy, I. S. Duff, and C. Puglisi. :'. !i ilrontal QR factoriza-
tion in a multiprocessor environment. Ain,, i.. Linear Algebra Appl.,
3(4):275-300, 1996.
[5] P. R. Amestoy, I. S. Duff, and C. Puglisi. .iA41 subroutine (unsym-
metrized version). Harwell Subroutine Library, Aug. 2000.
[6] P. R. Amestoy and C. Puglisi. An unsymmetrized multifrontal LU
factorization. Technical Report RT/APO/0003, ENSEEIHT-IRIT,
Toulouse, France, Sept. 2000.
[7] '.1 Arioli, J. W. Demmel, and I. S. Duff. Solving sparse linear systems
with sparse backward error. SIAM J. Matrix Anal. Applic., 10(2):165
190, 1989.
[8] C. C. Ashcraft and R. Grimes. Thl influence of relaxed supernode parti-
tions on the multifrontal method. ACM Trans. Math. Softw., 15(4):291
309, 1989.
[9] C. C. Ashcraft and J. W. H. Liu. Robust ordering of sparse matrices
using multisection. SIAM J. Matrix Anal. Applic., 19(3):816-832, 1998.
[10] I. Brainman and S. Toledo. : -i ,i -dissection orderings for sparse LU
with partial pivoting. 2001. www.math.tau.ac.il/~stoledo.
[11] T. F. Coleman, A. Edenbrandt, and J. R. Gilbert. Predicting fill for
sparse orthogonal factorization. J. of the ACM, 33:517-532, 1986.
[12] T. A. Davis. University of Florida sparse matrix collection.
www.cise.ufl.edu/research/sparse.
[13] T. A. Davis. Algorithm 8xx: U:. iFPACK V3.2, an unsymmetric-pattern
multifrontal method with a column pre-ordering strategy. Technical Re-
port TR-02-002, Univ. of Florida, CISE Dept., Gainesville, FL, January
2002. (www.cise.ufl.edu/tech-reports. Submitted to ACM Trans. Math.
Softw.).
[14] T. A. Davis and I. S. Duff. An unsymmetric-pattern multifrontal method
for sparse LU factorization. SIAM J. Matrix Anal. Applic., 18(1):140
158, 1997.
[15] T. A. Davis and I. S. Duff. A combined unifrontal/multifrontal method
for unsymmetric sparse matrices. ACM Trans. Math. Softw., 25(1):1-19,
1999.
[16] T. A. Davis, J. R. Gilbert, S. I. Larimore, and E. G. :... Algorithm 8xx:
COLA i ), a column approximate minimum degree ordering algorithm.
Technical Report TR-00-006, Univ. of Florida, CISE Dept., Gainesville,
FL, October 2000. (www.cise.ufl.edu/tech-reports. Submitted to ACM
Trans. Math. Softw.).
[17] T. A. Davis, J. R. Gilbert, S. I. Larimore, and E. G. :'.. A column
approximate minimum degree ordering algorithm. Technical Report
TR-00-005, Univ. of Florida, CISE Dept., Gainesville, FL, October
2000. (www.cise.ufl.edu/tech-reports. Submitted to ACM Trans. Math.
Softw.).
[18] J. W. Demmel, S. C. E i-, i-,,, J. R. Gilbert, X. S. Li, and J. W. H.
Liu. A supernodal approach to sparse partial pivoting. SIAM J. Matrix
Anal. Applic., 20(3):720-755, 1999.
[19] J. J. Dongarra, J. J. Du Croz, I. S. Duff, and S. Hammarling. A set of
Level 3 Basic Linear Algebra Subprograms. ACM Trans. Math. Softw.,
16:1-17, 1990.
[20] I. S. Duff. On algorithms for obtaining a maximum transversal. ACM
Trans. Math. Softw., 7:315-330, 1981.
[21] I. S. Duff. Thi solution of nearly symmetric sparse linear systems. In
R. Glowinski and J.-L. Lions, editors, Computing methods in applied
sciences and engineering, VI, pages 57-74, Amsterdam N'" York and
London, 1984. North Holland.
[22] I. S. Duff, A. :'-. Erisman, and J. K. Reid. Direct Methods for Sparse
Matrices. London: Oxford Univ. Press, 1986.
[23] I. S. Duff and J. K. Reid. :'iA27 a set of Fortran subroutines for
solving sparse symmetric sets of linear equations. Technical Report
AERE-R-10533, AERE Harwell Laboratory, United Kingdom Atomic
Energy Authority, 1982.
[24] I. S. Duff and J. K. Reid. Tih multifrontal solution of indefinite sparse
symmetric linear systems. ACM Trans. Math. Softw., 9:302-325, 1983.
[25] I. S. Duff and J. K. Reid. Tih multifrontal solution of unsymmetric sets
of linear equations. SIAM J. Sci. Statist. Comput., 5(3):633-641, 1984.
[26] I. S. Duff and J. K. Reid. Til. design of .1iA48, a code for the direct
solution of sparse unsymmetric linear systems of equations. A CM Trans.
Math. Softw., 22(2):187-226, 1996.
[27] I. S. Duff and J. A. Scott. Tih design of a new frontal code for solving
sparse unsymmetric systems. ACM Trans. Math. Softw., 22(1):30-45,
1996.
[28] A. George and J. W. H. Liu. Computer Solution of Large Sparse Positive
Definite Systems. Englewood Clif- '- Jersey: Prentice-Hall, 1981.
[29] A. George and J. W. H. Liu. Tih evolution of the minimum degree
ordering algorithm. SIAM Review, 31(1):1-19, 1989.
[30] A. George and E. G. :';N. An implementation of Gaussian elimination
with partial pivoting for sparse systems. SIAM J. Sci. Statist. Comput.,
6(2):390-409, 1985.
[31] A. George and E. G. :'. Symbolic factorization for sparse Gaus-
sian elimination with partial pivoting. SIAM J. Sci. Statist. Comput.,
8(6):877-898, Nov. 1987.
[32] J. R. Gilbert. personal communication.
[33] J. R. Gilbert, C. :.i, i and R. Schreiber. Sparse matrices in -.iATLAB:
design and implementation. SIAM J. Matrix Anal. Applic., 13(1):333
356, 1992.
[34] J. R. Gilbert and T. Peierls. Sparse partial pivoting in time proportional
to arithmetic operations. SIAM J. Sci. Statist. Comput., 1) ,_'--874,
1988.
[35] S. Hadfield. On the LU Factorization of Sequences of Identically .i, .-
tured Sparse Matrices within a Distributed Memory Environment. PhD
thesis, University of Florida, Gainesville, FL, April 1994.
[36] S. :.1 Hadfield and T. A. Davis. Thi use of graph theory in a par-
allel multifrontal method for sequences of unsymmetric pattern sparse
matrices. Congressus Am,," rantium, 108:43-52, 1995.
[37] B. Hendrickson and E. Rothberg. Improving the runtime and quality of
nested dissection ordering. SIAM J. Sci. Comput., 20 !1,i 1,'), 1999.
[38] G. Karypis and V. Kumar. A fast and high quality multilevel scheme
for partitioning irregular graphs. SIAM J. Sci. Comput., 20:359-392,
1998.
[39] S. I. Larimore. An approximate minimum degree column ordering algo-
rithm. Technical Report TR-98-016, Univ. of Florida, Gainesville, FL,
1998. www.cise.ufl.edu/tech-reports.
[40] J. W. H. Liu. Tih role of elimination trees in sparse factorization. SIAM
J. Matrix Anal. Applic., 11(1):134-172, 1990.
[41] J. W. H. Liu. Tih multifrontal method for sparse matrix solution: T!i -
ory and practice. SIAM Review, 34(1):82-109, 1992.
[42] P. :'.I, 1-1.. i- Sparse QR factorization in iATLAB. ACM Trans. Math.
Softw., 20(1):136-159, 1994.
[43] E. G. :"g and P. Raghavan. Performance of greedy ordering heuristics for
sparse Cl 1..1 -!:y factorization. SIAM J. Matrix Anal. Applic., 20(4):902
914, 1999.
[44] E. Rothberg and S. C. Li-, i-ii1 Node selection strategies for bottom-up
sparse matrix orderings. SIAM J. Matrix Anal. Applic., 19(3):682 i,'",
1998.
[45] : i Yannakakis. Computing the minimum fill-in is NP-Complete. SIAM
J. Alg. Disc. Meth., 2:77-79, 1981.
*
* |