Citation
Application of eigenvalue sensitivity and eigenvector sensitivity in eigencomputations

Material Information

Title:
Application of eigenvalue sensitivity and eigenvector sensitivity in eigencomputations
Creator:
Sarmah, Purandar
Publication Date:
Language:
English
Physical Description:
vi, 115 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Algorithms ( jstor )
Approximation ( jstor )
Diagonal arguments ( jstor )
Differential equations ( jstor )
Eigenvalues ( jstor )
Eigenvectors ( jstor )
Factorization ( jstor )
Matrices ( jstor )
Matrix equations ( jstor )
Permutations ( jstor )
Dissertations, Academic -- Materials Science and Engineering -- UF
Materials Science and Engineering thesis Ph.D
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1993.
Bibliography:
Includes bibliographical references (leaves 112-113).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Purandar Sarmah.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
001948678 ( ALEPH )
31181973 ( OCLC )
AKC5075 ( NOTIS )

Downloads

This item has the following downloads:


Full Text






APPLICATION OF EIGENVALUE SENSITIVITY AND
EIGENVECTOR SENSITIVITY IN EIGENCOMPUTATIONS










BY

PURANDAR SARMAH


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1993











ACKNOWLEDGEMENTS


I am indebted to a number of people,


prepare my dissertation.


his advice and help throughout


directly or indirectly


First, I would like to thank my advisor I


for helping me to

)r. William Hager


the different stages of this dissertation.


I am


grateful to


Kermit Sigmon and Dr.


James


Keesling for working with me on


several algorithms which I used to get some numerical results in my dissertation.


would like to thank Dr.

committee member. Wi


Bruce Edwards for his invaluable advice as my supervisory


thout his advice I would not have been able to push my work


this far.


I am grateful to Dr.


Nilotpal Ghosh,


who was a visiting professor in our


department.


His work on eigenvalue problem had direct


influence on some of my


results.

Finally, I want to express my deep sense of gratitude to my wife for helping me


in different ways.


Without her encouragement and help during the more frustrating


moments of my research, this work would not have come to this final stage.

















TABLE OF CONTENTS


ACKNOWLEDGEMENTS

LIST OF FIGURES .


ABSTRACT


CHAPTERS


INTRODUCTION . .


EIGENVALUES OF UNSYMMETRIC MATRICES .


Block Diagonalization of a Matrix .
A Differential Equation Approach to Eigen
A Differential Equation Approach to Eigenc
Stepsize
Convergence of Block Diagonalization of a
Block Schur Decomposition of a Matrix .
An Algorithm for Block Schur Decomposit
Parallel Processing in Eigencomputations


EIGENVALUES OF SYMMETRIC MATRICES


computations .
omputations with Armijo's


r


* S 4 4 4 5 S S S S 5 4 S
Lct i X* S S S S S S S S S S *
M a ri .


ion of a Matrix


4 4 4 4 9 5 5 5 4 1 8


Diagonalization of a Symmetric Matrix using Armijo's Stepsize. ..
Block Diagonalization of a Symmetric Matrix using Armijo's Stepsize


4 CONCLUSION


REFERENCES


9 44a4108


9 9 4 5 4 4 4 5 5 4 4 5 5 4 5 9 6 4 9 9 S S 6 4 9 5 5 4 9 S S 6 5 9 1 1 2


BIOGRAPHICAL SKETCH


S4 4 5 6 0 4 4 S 114
















LIST OF FIGURES


The path of convergence of the eigenvalues of the matrix A. .

The path of convergence of the eigenvalues of C. .

The direction, in which the elements of G(S, U) are computed, when
the number of processors are greater than half of the total blocks.


The direction, in which the elements of G(S, U) are computed,
the number of processors are less than half of the total blocks.


when











Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

APPLICATION OF EIGENVALUE SENSITIVITY AND
EIGENVECTOR SENSITIVITY IN EIGENCOMPUTATIONS

By

PURANDAR SARMAH


December 1993


Chairman: William Hager
Major Department: Mathematics


In 1987


, in his paper "Bidiagonalization and Diagonalization," William W


Hager


presented an algorithm to diagonalize a matrix with distinct eigenvalues.


ment the algorithm,


First,


To imple-


we need a good approximation to the matrix of eigenvectors.


we present the derivation of his algorithm, and then we show how it can be


generalized to diagonalize a matrix with multiple eigenvalues.

vergence result is established for the generalized algorithm.


A local quadratic con-

When a good starting


guess for the eigenvectors is not known, Hager's algorithm may not converge.


To re-


store convergence, we modified the algorithm by replacing diagonalizations by Schur


decompositions.

complex plane,


We found that if uniformly distributed numbers on a circle in the

which includes all Gerschgorin disks for the matrix, are used to ap-


proximate the eigenvalues, and if the eigenvectors are approximated by the columns


of the identity matrix, then rapid convergence is obtained.


We presented some node


programs which, alone with some modifications to the last algorithm, can be used in











we present additional modifications of Hager's algorithm and the generalized algo-

rithm that take advantage of matrix symmetry and reduce the given matrix into a

diagonal matrix with eigenvalues along the diagonal.

















CHAPTER 1
INTRODUCTION


A be an n x n matrix.


We want


to find an iterative method to compute


a diagonalization


XAX-1


, provided it exists,


where


the columns of


eigenvectors of A,


and A


= diag(A1, A2,.. *


n ) whose diagonal elements are the


eigenvalues of A.


Here we will use a sensitivity result for eigenpairs to diagonalize a


matrix.


We assume eigenvalues of A are all distinct.


In the derivation of necessary


equations to diagonalize a matrix,


we need the continuity of eigenvalues, and


perturbation theory based on Gerschgorin's


theorem.


We give the statement of the


theorem below:


Theorem 1.O. 1


C Cxn


, Pi =


aijd, and


Iz- a l


A(A)


Furthermore,


a set of k


Gerschgorin disks


are isolated


from the


other n k


disks, then their union contains precisely k eigenvalues of A.


Proof of Theorem 1.0.1


For proof, the interested readers are referred to [Atk89,


The-


orem 9.1, page 588].


D = {z


< Pi},












column of


small number e,


Given an


we will show


arbitrary


the next


matrix


theorem


a sufficiently


there exist continuously


differentiable functions Aj(e), and xj(e) with Aj(O) = Aj, and


(A + EE)xj() =


Theorem 1.0.2


(0) = xj such that


That is (Aj(c),xj(c)) is an eigenpair of A + cE.


Suppose X


-'AX


= diag( A,


is a diagonalization of


1. .


Then given


arbitrary n


x n matrix


for a sufficiently


small C,


there


exist


continuously


diferentiable


functions Aj(e),


and xj(e) with A,(O)


(0) = xj such that


the following matrix equation holds:


(A + eE)xj(E) = Aj(e)xj(e).


(1.1)


Proof of Theorem 1.0.2 Let X'- AX


= diag( A


I. .


An ) be a diagonaliza-


tion of A


Let E be an n x n matri


a small number.


Consider the matrix


-1(A + eE)X


Then


for i
for i


where


yfExj,


is the


row


= diag(


and let C = BDB


.. .


then


cfi,


C2

*rs

S


* kf,3
.5. kf2 1


tf'n
L:


tfs2
a
a n


I* 1 n, i lkVL I


Let A be an n xn matrix. Assume the eigenvalues of A are all distinct.


A,(e)zr,(c).


= A j,


,A2,. .


dij = {


X? + fjj
E/ij


1, 1,... e/k,


I








3


and k is chosen so as to make the jth Gerschgorin disk disjoint from the other n 1


disks.


We may choose a value of kc,


which is independent of e in the following way.


Choose k to have the largest value consistent with the inequalities:


1
IA-- I A i


1, ., n.


We achieve the above inequality if


mm
isj


1 n.


With the above value of k, the jth disk will be isolated from the other n 1 disks, the


radius of the jth disk will be of order


, and the perturbation in the jth eigenvalue


will be of order E as e 0.


Next we discuss the effect of perturbations in


the elements


of A on


the eigen-


vectors in its


diagonalization.


As earlier, let X-'AX


= diag


be a diagonalization of A.

eigenvectors. We denote ei


The columns zl,


..., 2 of X


igenvectors of A + eE by xi(e),


form a complete set of


..., xn(), and eigen-


vectors of D


= X-'(A + cE)X by vl(c), V2(j),


* ..


vn(E) such


that xi(c)


= Xvi(c).


Since (A,(e),v1(c)) is an eigenpair of D, and XA(0) = Aj, so we must have vj(O)


which is the jth unit vector. Let vj(c) be


For sufficiently small


for some


normalized so that its


e, claim that the jth element of vj(e) is 1.


j. Since Dvy() = Aj(e)v)(e),


largest element is 1.


If not, let vij(e)


so equating the ith component on both


sides, we get


(1.2)


\j(e)vij()= AiVij(e) + tCfuv(1),
I=1


that is


S t -' A' '


-= ej,


Ikfi I


, 2, n


52(<),











less than NE as e -- 0 for some N


Since Iv (c)I


1, so from (1.2),


we get


IA1(E)- A II|v (c)I


CZIMfi.


A(E) A I


1
-As AsI for i


i = 1,..., n, then Ivij(e)


Ne, where


max
i#j


Thus vij(c) are of order e for


(Ay(c) A\)vij(e) = efi +


The second term on the right is of order e2, so for


1=1
f ---


,... ,n. Also from (1.2), we get


failvj(E).


1,... ,n, we have


V-j()


fij
- Ai


Hence Aj(E

continuous.


and xyj() are differentiable functions of e, and their first derivatives are

This completes the proof of the theorem. O


Now we are ready to derive the necessary equations to find a diagonalization of a


matrix A,


whose eigenvalues are all distinct.


Let (As,


n x n matrix


xj) be a simple eigenvalue, and corresponding eigenvector of A.


E, and for a sufficiently small


Given an


e, let A2(c), and xyj() be differentiable


functions of c such that Aj(0) = Aj, and xj(O)


to c and putting e = 0,


Differentiating (1.1) with respect


we have


Ax4(O) + Ex, = A (0)xj + Aax'(0).


Since the columns


(1.3)


Xi, x2, ..., x, of X form a complete set of eigenvectors of A,


N
N V-.


= X .


=O(e2).











Since a scalar multiple of an eigenvector is again an eigenvector, we can take bjj(c) =

without loss of generality.

Now differentiating (1.4) with respect to e and setting c = 0, we have
n


(1.5)


Substituting (1.5) into (1.3), we obtain


b1 (0)x; +


Exz =


A'(O)x


+ Aj


b j(0)x


bj (0)


Exj =


I = xA(.x


b'(o) (


Aj -


(1.6)


To obtain expressions for A[(0), and bl1(O) we use relations


(1.7)


where yT is the ith row of X-1


So premultiplying (1.6) by y


, and yT


yields


A(O) = y Ex1.


(1.8)


yTExj = b'kj(O)( j Ak)


bkj(o) =


Yk Exz


With these values of b'k(0), (1.5) becomes


x=j0


nyTE


(1.9)


A first order


Tavlor expansion


rives


k -


= k,


=(0)


b',(0)z;.


b:j (0)XjA x


+ (0)x, +











Substituting the values of A)(0), and sx(0) from (1.8),

imations, we get


and (1.9) in the above approx-


xj( )


Aj(0) + eyjfExj,

xj(O) + Z yTE xj
j -3


(1.10)


To develop an algorithm we use (1.10) in the following way.


Suppose XAX"' is


an approximate diagonalization of B.


If we


identify


E in (1.10) with B -


XAX


then to the first order:


Aj(0) + ey (B XAX-1)

Aj(O) + eyBxj eAj(),


yT (B


(0) +


- XAX'-1) j
j -- \ xi
42 A


x 0) +


With


SyjTBxy
v- Bx1


A-


e = 1, these approximations for an eigenvalue, and corresponding eigenvec-


tor give:


\ne"w
3


ynew


Bx(ld
S(yd)T


= xold


,


t L.
'=l


Inew
A]


1 : n


oxd


1: n,


\net


(1.11)


We can use thnew above expressions to formulate an algorithm to compute a diago-
We can use the above expressions to formulate an algorithm to compute a diago-


nalization of a nondefective


matrix A


as follows:


Alqorithm 1.0. I


(Diaqonalization)


Given a nondefective matrix A


E C x"


a matrix


of approximate eigenvectors X0o, and a tolerance tol greater than the unit roundoff,


the following algorithm comnutes a diagonalization A =


XAX-1


x;(e)












C1, C2,...,c, ] is an x n zero matrix.


for


( )T


end

for j = 1:n

for i=1

if i


<-cj


(yiT


end

end


xj 4 xj


j *--


xj/lI xjII


4X-


Dif IICIIF


where II


I IF is the Frobenious matrix norm.


end

Above, we use a normalization for xj to reduce the growth of X.


Example


1.0.1


If Algorithm 1.0.1 is applied to


-1.4


-3.1
-4.5


-4.7


-0.1
-2.3
-0.5


5.0
-1.3


-7.7


-1.4


with Xo = I, and tol = 10-9


, then the eigenvalues converge as shown in the following


table:


C=


+cj;











Iteration


8.0000
7.6170
7.1096
8.7039
7.9836
7.9953
7.9943
7.9943


7.1000
5.0012


4.5657
5.0217
4.9963
4.9964
4.9964


-0.1000
0.9699
1.5900
0.9711
0.8616
0.8889
0.8892
0.8892


-7.7000


-6.71


-6.1843
-6.0978
-6.0530
-6.0571
-6.0571
-6.0571


-1.4000
-0.9700


.0941
.2429


-1.9139
-1.9235
-1.9229
-1.9229


The exponents of convergence p in


I IXk+1


- xIPl


-XlIF


for different values of k are given


in the following table,


where Xk is the matrix of eigenvectors at the kth iteration,


and X is the final matrix of eigenvectors:


3.207
0.430
4.123


1.962


Hence the method appears to converge quadratically.


( N. Ghosh gave a theoretical


proof of the quadratic convergence of the diagonalization method ).


Example


1.0.2 If Algorithm 1.0.1 is applied to


with X0 =


I, and tol =


, then the approximate eigenvalues do not converge to


the eigenvalues of A, which is clear from the following table:


10-8











Iteration


1
100
200
300
400
500
600
700
800
900
1000
1100


55
-1241.77
124807.75
416577.75
539592
685138
-13438605.19
-72503.38
70919.84
436916.5
-2682893.03
-1244105


-4853


-318959.32
230821
-163324.75
422251
-1459224.5
-513144
586343.25
1537.75
42945.63
440503.5


1586.


-368210.46
-768266.74
474518.5
17628
7491094
-3064817
787740.75
635731.38
-901415.25
-11654959.5


29
5100.90
369633.58
-1252454
-868096
-235428.5
14400555
1601468
-886619.25
-511936
-1409781.21


1572


6
-446.15
192874.49
1373474
17506.02
-889446
-6993606
2049060
-558243.75
-562132
4951288.04
-3270228.12


The eigenvalues of A are A


233.3505


27.1460


3339


-43.2339


-62.9287 ].


The above example shows that in general, the identity matrix can not be used

for the starting guess matrix of eigenvectors for a nondefective matrix with distinct

eigenvalues.

If by another method, a matrix of eigenvectors of a matrix A can be determined,

then this method can be used to find the eigenvalues, and corresponding eigenvectors


of a slightly perturbed matrix A + EE,


small scalar.


where E is an arbitrary matrix, and e is a


For example, first reduce the matrix A to an upper Hessenberg form


H by an orthogonal matrix Q, and then apply the QR method with double implicit


shift to H to obtain a quasi-upper triangular matrix.


Next solve (H


-pIl)


find corresponding eigenvector v of the eigenvalue p.


Then x = Qv is the eigenvector


corresponding


to the eigenvalue


Now these eigenvectors can


used as


approximate eigenvectors in the diagonalization method to find the eigenvalues, and

corresponding eigenvectors of the perturbed matrix A + eE.


I 0l Q W0 rnA,.n +b* ni A :I tn 1 I


. n T f + n ni inf arnv I-I n o an k at'


S-, v I


-9











U are A = [233.3505, 27.1460,


-8.3339


-43.2339,


-62.9287].


To find an eigenvector


x corresponding to the eigenvalue pu, let B


-A --ip.


Then solve Bz = 0 for x.


Here is the matrix of eigenvectors:


0.4674
0.4487
0.4730
0.4986
0.3284


0.5909
-0.6005
0.2783
0.1176
-0.4460


0.3002
0.0657
-0.4712
-0.6361
0.5280


-0.7147
-0.1294
0.3346
0.0707
0.5962


0.2386
0.3043
0.5919
-0.4198
-0.5691


Next consider the following matrices AI, and A2,


which


we get


by perturbing the


elements of A:


55.056
47.032
74.078
68.042
42.075


7.065


61.012


1.045


33.087
56.035
73.068


55.56
47.32


68.42
42.75


33.87
56.35
73.68


4.013
59.053
26.086


61.12
21.45
4.13
59.53
26.86


1.021


61.067
79.031
29.046
20.079


51.21
61.67
79.31
29.46
20.79


79.023


60.024


6.097


79.23


60.24
52.57
6.97


If we use Algorithm 1.0.1


2 with X0


= X, and


then af-


ter 3


iterations


we obtain


diagonalizations


Y-1A


A2Z


whereas if we apply the QR method with double impli


cit shift to ST


AIS,


and ST


then after 4 iterations we obtain real Schur decompositions pT(S


AS)Q = F.


The derivation of the results


), and


(1.9) can


be found


n many


Numer-


Linear


Algebra texts.


The derivation of


the formu


which are used


* aI *


- Iw 1


4- '


.... .... I ..1 1 1 I 1. .I E *I S L.*)L I I 1 I.. *1 I I


S)P


4


A1 =


A2 =


A2S,


** I


_ _:_ : _


I_ 1 ..


!- .. II._ A







11


In Section 2.1, we will show how Algorithm 1.11 can be generalized to diagonalize


a nondefective matrix with multiple eigenvalues.


In Section


we will show that


the generalized algorithm converges locally quadratically.

Approach to Eigencomputations is introduced in Section


A Differential Equation


to create an iterative


method


to find


the eigenvalues,


corresponding eigenvectors of a nondefective


matrix A.


A(t)


The differential equations governing the block matrices have the form:


= -A(t) + Diag (X(t)-AX(t)) and X(t) = X(t)F(X(t),A(t)),


*1


where Diag(M)


a block diagonal matrix formed from


the diagonal


blocks of the


block matrix M,


Fjj(X(t), A())


is a square zero matrix, and


Fij(X(t)


A(t))


j is the solution B to the matrix equation BAj(t) -


Ai(t)B


= Y(t)TAXj(t).


Here Y1(t)T


denotes the ith block row of X(t)-1


The Euler approximation to the


differential equations can be expressed


An+l(t) An


+ At (Diag


xAX,) A)


and Xn+(t)= Xn


+ AtXF (X,


In Section 2.3,


we modify (1.12) by replacing the fixed stepsize At in each iteration


by a


variable stepsize


Armijo


rule from optimization theory is used to find a


stepsize s, for which


IA-


Xn+ (


s)A}n+(s)Xn1{(S)I IF


X, AX 1IIF


holds.


Example


is not known,


.3.2 shows that


, when a good approximation to the matrix of eigenvectors


the algorithm in Section 2.3 may not converge.


In Section


discuss the factors sensitivity in the Schur decomposition relative to perturbations

in the coefficient matrix to compute a block Schur decomposition of a matrix. In


Section 2.6,


we present an algorithm to compute a block Schur decomposition of a











from optimization theory.


The starting eigenvalues associated with the Schur decom-


position are uniformly distributed points on a circle in the complex plane, where the
circle includes all Gerschgorin disks for the matrix, and the starting guess unitary

factor is the identity matrix.


In Chapter 3, we discuss how to exploit the structure of a matrix.


matrices are diagonalizable,


Since symmetric


we modify the diagonalization method, and the block


diagonalization method to exploit the matrix symmetry and reduce the given matrix

into a diagonal matrix with eigenvalues along the diagonal.
















CHAPTER 2
EIGENVALUES OF UNSYMMETRIC MATRICES

2.1 Block Diagonalization of a Matrix


Let A be an n


x n nondefective matrix.


Our aim is to find an iterative method to


compute a block diagonalization of the form A = XAX


-1, where A = diag(Ai, A2,...,


At) is a block diagonal matrix, and X


= [X,X2,


..., Xt] is a compatible block column


matrix such that AXj =


Xj Aj.


Here A = diag( Aj


. .


,j ) is an mj x mj scalar


matrix.


can always be arranged so that the eigenvalues of A;


and Aj for i


are distinct.


We will use a sensitivity result for eigenvalues and eigenvectors to block


diagonalize a matrix.


In the derivation of necessary equations to block diagonalize a


matrix, we need the continuity of eigenvalues, and the perturbation theory based on

Gerschgorin's theorem.

Recall that the coefficients of the characteristic polynomial of a matrix B are con-

tinuous functions of the elements of B [Wil65, page 66], and the zeros of a polynomial


depend continuously on its coefficients.


Proof


in [Hen74, page 281


So the eigen-


values of B depend continuously on the elements of B. Also, Rouch6's th

that if f and g are analytic in a neighborhood of a closed disk D = {z:


eorem states


z a


centered at a


, f has no zeros on 9D = {z


* -a a=


and ig(z)


< If(z)


holds


z C OD, then


z) + g(z) have the same number of zeros in D.


Detail


is in [Hen74,


page










then the disk D = {z


v7} centered at A contains precisely k eigenvalues


of A+E.


Proof of Theorem 2.1.1


Proof is given in [Ste90, page 167].


Next we will discuss the effect of perturbations in the elements of A on the eigen-


values, and the eigenvectors in its diagonalization,


which is based on Gerschgorin


theorem.


Let A be an n x n nondefective matrix.


Suppose X-'AX


s a block


diagonalization of A, where A is a block diagonal matrix.


onal block of A


If Aj denotes the jth diag-


, then we have AXj = XjAj, where Xj is the jth block column of X.


Given an arbitrary matrix E, and for a sufficiently small e, we will


show in the next


theorem that there exist continuously differentiable functions Aj(e), and Xj(c) with


Aj(O) = A- and Xj(O) = Xj such that (A + eE)X,(e) =


X,(c)A,(e).


Theorem


2.1.2


Let A


be an n


x n nondefective matrix.


Suppose X- AX


= A is


block diagonalization of A, and if Aj


is the jth diagonal block of A, then AX, =


where XA


eigenvalues.


the jth block column of X.

Then given an arbitrary n


Suppose Ai, and Aj for i


x n matrix E


have distinct


, and for a sufficiently small


, there exist continuously differentiable functions A,(e), and Xj(e


) with A(0))


= A


and X(0O) =


Xj such that the following matrix equation holds:


(A + eE)Xj,() = Xj(c)Aj(e).


Proof of Theorem


2.1.2


Let X


-1AX


= diag( A1, A2,..., At


) be a block diagonaliza-


tion of


where A1


= diag


Aj ) is a


scalar matrix


, and let mrn


= dim(A3).


Suppose min


> 1. Let E be an n x n matrix, and e be a small number.


Consider the


Xj A,












matrix D = X-1(A + eE)X.


The form of this matrix is illustrated below for t = 5:


where Fij = YTEXj,


YT is the ith block row of X


Let B


= diag


where B1


= diag -
P


, .


P


for i


= 2,..., t, and


dim(Bj)


1,..., t.


Let C = BDB


-1. then


cFU1

pF21
PF31
pF41
pFas


F,2
P
EF22
EF32
eF42
eF52


2
-F13
P
eF23
EF33
EF43
F53


2
-F14
P
eF24
eF34
EF44
eF54


--F15
P
cF25
EF35
cF45
EF55


Next,


U-1FllU


E be


a diagonalization


where


diag(


'1,02,


ami ) with some of ai's


may be


identical.


Let W


= diag( U


, B2, Bt ).


Then


W-1CW


pF21U
pF31U
pF41U
pF51U


2
SU- F1
p
EF22
F32
F42
F52


6 U-1 F1 3
P
EF23
eF33
F43
EF53


2
SU-1 F14
p


We may choose a value of p,


which is independent of E such that the first mi disks


are disjoint from the other n mi disks. So for a proper value of p, mi eigenvalues of


. .


P


, B2, ... Bt),


= m ,








16


Next we discuss the effect of perturbations in the elements of A on the eigenvectors


in its block diagonalization. As earlier, let X-1 AX


= diag( A, A2,...,At) be a block


diagonalization of A,


where


= diag( A3,.


.., Aj ) is a scalar matrix, and mj


dim(A1).


Clearly the columns x


, X, of X form a complete set of eigenvectors.


We denote eigenvectors of A + eE by x1(e),


a2(),..


and eigenvectors of D =


. ,n(c),


-'(A + cE)X by ux(e),u2(c),...


,un(c) such that x;(e) =


Xui(c).


Illustrate, we consider a matrix


D with t


= n-2


such that A1 = diag(A1


Aj= "j+2,




D =


where fi; = yTjExj, and yT is the ith row of X


First consider the simple eigenvalue


. Since


(A4c), U4(C))


an eigenpair of


A4(O)


- A4,


so we must


have


u4(0)


, which


the fourth


vector.


normalized


so that


largest element is 1


For sufficiently small e, claim that the fourth element of u4(e)


If not, let ui4(c)


1 for some i


Since Du4(


= A4()u4(c), so equating the


th component on both sides, we get


4{()Ui4(E)= A XiU4() +


(2.2)


that is


A4() -


Ai=E


tjnuj4(e).


j 2 n
3 i***?-


22, .


S 1),


= e4


f;/u.4(e),


1 1


* 1* II 1~ ~











all less than Me as e -* 0 for some M


Since IUj4(E)


1, so from (2.2), we get


4() Au4(e)


cEI f~slI


Xt(e) A.I


A4 Ai for i


4, i = 1,...,n, then Iu4(e)lI


Me, where


max
id4


Thus un4(c) are of order e for i


1, ..., n.


Again from (2.2), we obtain


(A4(e) Xi) ui4(e) = ef;4 + e


The second term on the right is of order e2, so for i


fijUj4(c).


1,..., n, we have


Ui4(c) -


A4-


= O(e2).


Now we turn to eigenvectors corresponding to the multiple eigenvalue A1. Suppose

ul(E) is a normalized eigenvector of D corresponding to the eigenvalue AX(c). Using


a similar argument as in the case of the simple eigenvalue 24(e),


we can show that


the largest element of ui (e) can not be


4,...,


n. In fact, these elements


are of order E.


Therefore the normalized ui (e) must have one of the following forms:


1

P2(e)


pl(e)
1
P2 (E)


pi(e)
p2(E)
1

,(E)


u;i(E),











three degrees of freedom to normalize ul(c), two degrees of freedom to normalize

u2(e), and one degree of freedom to normalize u3(c). Hence we take


ux1() =


1
0
0
u41(c)


, u2(c) =


1
0
"42(<0)


- uf2 ()


u3(c) =


0
0
1
U43(t)

t"f3(e)


As in the


case


of the simple eigenvalue A4(c), we can show that


= 0(2),


i = 4, ..., 72


1,2, 3,


where A1 = A2


= A4.


Hence Aj(e),


and X3(e) are continuously differentiable functions


This completes the proof of the theorem.


Now we are ready to derive the necessary equations to obtain a block diagonal-


ization A =

element of A


XAX-' of a nondefective matrix A.


Let Aj be the jth block diagonal


, and X, be the corresponding block column vector of X


matrix E, and for a sufficiently small e


E R, let A3(c), and Xj(e) bt


Given an n x n

e differentiable


functions of


E such


Aj(0)


= A1, and Xj(O)


= Xi


. Differentiating (2.1)


with


respect to e and then putting e = 0, we obtain


AX>(0) + EX, =


X (0) A +


XjA (0).


(2.3)


Since the columns of X


are linearly independent, so we can express XA(e) in the


following way:


(2.94)


where Bo,(c)


is an mi x mn matrix.


We normalize (2.4) by taking B3,(c) = I.


Differ-


uj({) -











Substituting (2.5) into (
t


we obtain


XiB~j(0) + EX1


=


XiB'j(0)A1 +


x OA(0)


XA B


,(0) + EX6


=z


XB~j(0)A


XjA (0)


=-


X, (B


(0)Aj


- A.B(O)) +


XjA (0).


(2.6)


Since


YiXk


for i
for i


(2.7)


where YT


th block row of X


Premultiplying


.6) by


, and YT


respec-


tively, yields


AJ(0)


= YTEXJ,


- A.iBb(0)


Equation (


a linear equation


in the unknown Bi(0O)


Here


we mention two well-


known methods to find B3j(0).


To derive these two methods


, we need the following


two theorems.


Theorem


.3 [H


, page 317]


E Cp


and C


e Crx'


are nondefective


matrices, then the


solution


to th


matrix equation


(2.10)


given by


=VWU


ere B


EU-1


and C


are diagonalizations


of B, and C


respectively,


and W


- (wej


is arx p mat


with wij


/-1RU)
aj wi


Proof of Theorem i


Let B


= UEU-1


and C


= VflV-1 be diagonalizations of B,


= yiTE


=U


S_ _


BA'(0)Aj


= R


= V~V-1


I


M











Let W


= V-1ZU


and D = V-'RU


, then the last equation reduces to WE


Comparing the (i,j) entry on both sides of the preceding relation, we have


wij (aj w) di,


-1R )


1,2,... ,p,


wij =


--wi


and with this W,


VWU-1


Theorem 2.1.4


E RPXp


and B


E Rrx'r


then the solution


to the matrix equation


AZ ZB = C


(2.11)


given by


Z=U


PVT


where A = URUT


and B = VSVT


are real Schur decompo-


sitions of A, and B respectively, and P

RP -


is the solution to the Sylvester equation


PS = UTC V.


Proof of Theorem 2.1.


Let A = URUT


, and B = VSVT be real Schur decomposi-


tions of A, and B respectively. 1

URUT


RUT


7hen (2.11) becomes


- ZVSVT
- T zvS


Let P =


and D = UTCV


Then the last equation reduces to RP


- PS=


which is a Sylvester equation. For the detailed solution of a Sylvester equation readers

are referred to [Gol90, page 387]. 0

Next we will show how to use the above two theorems to derive two methods to


find the unknown B (0))


n (2.9).


Method 1 ( Theorem 2.1.3):


Let P = B,(0), and R = YITyEX, then (2.9) reduces


DA AD ?


19 1Q\


S0.


UTCV.


- flW


1,2,...,r


DA._A.D--D











Method


Theorem 2.1.4


-= B(0),


.=- YTEXj,


then


(2.9)


becomes


PAj AiP = C.


(2.13)


Solving (


.13) for P,


we obtain P


= VZUT


, where Aj = URUT


A, =


VSVT


are real Schur decompositions of A and A; respectively, and


the solution to the


Sylvester equa


tion SZ -


ZR=


W with W


-- vTCU.


Combining the analysis of the effect of perturbations in the elements of a non-

defective matrix A on the eigenvalues, and corresponding eigenvectors in its diago-


nalization, and the results from (2.5),

Expansions of Aj(e), and Xj(e):


), and (


.9) we get the following


Taylor


Theorem


2.1.5


a nondefective


matrix,


-= diag(Aj


...,A)


a scalar


matrix,


where A,


is an eigenvalue of A of multiplicity dim(Aj)


let Xy


be the matrix


of linearly independent


eigenvectors of A corresponding to the eigenvalue


A, such that


Xj Aj.


and Xj(c)


Given an arbitrary matrix E, and for a sufficiently small


functions of


e, let Aj(e),


Xj(e)Aj(e), Aj(0) = A, and X,(0) = Xy


. Then


Aj() = A +eYjTEXj + O(c
t


Xj() = Xj +


the solution to the matrix equation PAj AiP = YiTEXj.


We use


Theorem 2.1.5 to develop an algorithm in


the following way.


Suppose


nA f 1 A 10 ann r\-rrwmS ,f D


T ...-


AX,


be continuously differentiable


e such that (A + eE)Xj(E)


where P = Pi=


XiPg + 0(















xj( )


Aj + CYf BXj eAj,
t
X1 + ZX;Psj,


where P = P,1 is the solution to PA, AiP


With e


, these approximations lead


to the following block version of


Algo-


rithm 1.11:


A51CW =(


BXOJd
I


3=1


Xnew
i


=Xld
_ Y^


XoldPid


j=l


(2.14)


ynew (Xnew)-


where P = P,j


the solution to PAew"
2


- AWP = Yod
-A, r- ri


B.?,"3


With a matrix


of approximate eigenvectors X0o, we can use the above expressions


to compute the block diagonalization of a nondefective matrix A in the following way.


Alhorithm


(Block Diaqonalization)


Given an


n x n nondefective matrix A


matrix of approximate eigenvectors X0, and


a tolerance tol


greater than


the unit


roundoff, the following algorithm computes a block diagonal


zation A =


XAX-1


Dif=


So0 =-


...


= x 11,1 ,


until Dif


< tol


is a n x n zero matrix.


A
n *


)


ak


. .


YTBXj


C -=n
' .-<-U


+


.. 1











Cj=Cj
C.7=03

end

end

xj = xj + cj


X =


/I Iz II


for l= 1


IJ, .3x
"25 in3


is column partitioning.

Sis column partitioning.


Dif = IICIIF


end

Above, we normalize Xj to control the growth of X


The main


problem in


this method is


how to choose a matrix of approximate


eigenvectors for the starting


guess.


If by another method, a matrix of eigenvectors of


a matrix A can be determined, then this method can be applied with X as the starting

matrix of eigenvectors to find the eigenvalues, and corresponding eigenvectors of a


slightly perturbed matrix A + eE,


where E is an arbitrary matrix, and e is a small


number.


For example,


first reduce the matrix A to an upper Hessenberg form H by


an orthogonal matrix Q, and then apply the QR method with double implicit shift


to H


to obtain a real Schur decomposition STH


To obtain an eigenvector y


corresponding to the eigenvalue y let B = H


- upI.l


Next solve By = 0 for y in the


following way. Put y(i) =


, and then solve [B(:,


1) B(:,


i+1: n)]z =


i) by


using the QR factor


zation technique to solve an overdetermined system of equations.


Put y(l


1), and y(i + 1


n)=


:n-1).


Then x = Qy is


the eigenvector of A corresponding to the eigenvalue $i. Now these eigenvectors can


-r n 1 1


1


S =


:i -


= X-l ;


1)


r. r n I











Example 2.1.1


Consider the matrix


We reduce A


, first to an upper Hessenberg form H


, and then apply the QR method


with double implicit shift with tol


= 10-6


After 8 iterations


, we get a real Schur


decomposition


STAS


=U.


The eigenvalues of


are A


4.0394,


-1.0197


0.4450i,


-1.0197 0.4450i


Next, we use the technique discussed in the last para-


graph to find corresponding eigenvectors.

eigenvalues, we solve the equation By = I


Here for each pair of complex conjugate

3 only for one, and then take the real and


the imaginary parts of y as two vectors corresponding to the complex conjugate pair


of eigenvalues.


The corresponding matrix is:


0.5921
0.7387
0.3222


0.2979
0.1265
-0.9462


0.7927
-0.2726
-0.5453


Next consider the following matrix Ai,


which we get by perturbing the elements of


1.010


-1.007


2.002
2.005
3.009


1.003
1.006
-1.008


If we use Algorithm 2.1.1 to A1


with


and tol = 10-6


then after 2 iterations


we get a block diagonal


double implicit shift to ST


zation Y-1AY


then after 2


i, and if we apply the QR method with


terations we obtain a real Schur form


R as well.


Examlpie 2.1.2 Consider the matrix


=D


A =-


QT(S'


AIS)Q =











If we apply the QR method with double implicit shift with tol


after


= 10-6 to B, then


1 iteration, we obtain a real Schur form STBS = U, and the eigenvalues of U


are A


= [-1,


]. Next, we use the technique discussed prior to Example 2.1.1


to find corresponding eigenvectors. Here is the matrix of eigenvectors:


0.7559


-0.3780
-0.3780
-0.3780


-0.5774
0.5774
0.5774


0.6547


-0.4364
-0.4364
-0.4364


-0.40
-0.40


0.8165


Next consider the following matrix B1,


which we get by perturbing the elements of


-9.9931
6.0059
6.0093
6.0085


.9947


5.0009
6.0065
6.0042


-2.9930
3.0091
2.0076
0.0026


-5.9995
3.0074
3.0033
5.0063


If we apply


Algorithm


2.1.1


to B1


with


= 10-6. then after


iterations we obtain a block diagonalization Y-1B1Y


= Di,


whereas if we apply the


QR method with double


mplicit shift to STB1


S, then after 3 iterations we obtain a


real Schur form QT(STB1S)Q = R.

Example 2.1.3 Consider the matrix


B1=


-X







26




2 -5 7 7 9 4 5 8 -3 -5
5 -9 7 4 7 9 8 3 -5 7
-7 6 9 3 4 8 5 2 3 4
5 3 -9 9 -7 5 4 3 9 5
6 7 7 10 9 2 3 4 5 6
6 8 -3 5 5 10 7 6 9 6
7 9 2 2 3 9 7 2 7 4
4 1 6 3 5 5 10 1 2 3
9 4 1 9 4 1 9 -9 2 -7
2 7 9 3 9 5 5 6 8 8
C-
3 5 5 1 4 6 9 4 7 9
5 7 7 2 9 9 9 5 1 5
2 10 2 6 6 0 0 8 7 3
6 2 2 -8 4 10 9 4 3 7
8 1 4 6 2 8 2 10 3 2
6 2 3 7 1 8 5 6 6 3
5 3 6 5 4 3 4 4 4 5
9 5 9 3 5 5 7 4 6 10
9 3 10 2 8 7 5 8 -2 2
0 5 -8 2 9 8 7 4 -1 6

5 7 -1 4 -7 4 7 6 9 8
3 6 8 10 4 2 10 7 8 7
2 5 9 9 -6 9 5 5 3 10
5 9 -5 8 8 8 1 -2 7 9
8 4 8 3 4 5 5 3 2 2
8 2 2 7 1 -9 3 0 4 -3
4 5 1 6 8 6 10 6 1 10
6 1 7 6 8 2 5 4 2 -3
4 3 1 8 5 3 5 8 9 7
5 10 6 4 8 7 7 10 10 9
7 8 7 7 -2 9 5 5 -7 7
10 2 4 3 5 9 4 5 8 4
4 7 7 2 8 7 8-9-8 8
3 8 8 4 3 2 -1 2 10 2
1 2 6 7 8 7 8 7 6 -6
7 1 9') Q R 7 3 1











If we apply the QR algorithm


with


double implicit shift


with


= 10-6


to C


then after 27


iterations we get a real Schur form STCS


= U, and the eigenvalues


of U


are A


= [92.2615,


-20.8808


13.5193 + 10.7480i


13.5193 -


10.7480i


0.2459 +


15.2499i


0.2459 -


15.2499


6.3218 + 13.5313


6.3218


13.5313,


-9.3504 + 7.8883i,


-9.3504 -


7.8883i, 9.0619 + 8.7751i, 9.0619


-8


.7751i, 1.5718 + 10.4593i,


1.5718


10.4593i


,8.0296,


-7.3844 +


1.4578i,


-7.3844 -


1.4578i, 0.6072,


-3.1993, 2.2102


Next,


we use


the technique discussed


prior to


Example 2.1.1


to find


correspond-


ing eigenvectors.


Here for each pair of complex conjugate eigenvalues,


we solve the


equation Bx = 0 only for one, and then take the real and the imaginary parts of x as


two vectors corresponding to the complex conjugate pair of eigenvalues.


the matrix of vectors by


We denote


Next consider the matrix


I -- -


1I


P








28


whereas if we apply the QR method with double implicit shift to STCaS, then after

27 iterations we obtain a real Schur form QT(STCIS)Q = R.

2.2 A Differential Equation Approach to Eigencomputations


In the previous section,


we found that the main problem in using the block di-


agonalization method is how to choose a matrix of approximate eigenvectors for the


starting guess.


Here we will study,


whether the Euler method can be used to block


diagonalize a nondefective matrix A with the identity matrix as the matrix of approx-


imate eigenvectors.


In numerical ordinary differential equations, one way to obtain


the Euler method is to drop the 2nd order error term in


Taylor Series of the


given function.


Theorem 2.1.5 dropping the 2nd order error terms,


expressions for Aj(c), and X1(c), which can be used


n Euler's method.


we get


Now we will


discuss how to implement the Euler method to find the eigenvalues of a nondefective


matrix A.


Euler's


method for the differential equation


z(t)) has the form:


Zn+l =- Zn


+ Atf(z,),


where


z, is the approximation to z(nAt), and At


the constant time step.


Here


f are vector functions of t in general.


In the block diagonalization procedure,


we attempt to generate a block diagonal matrix A, and an


asso


ciated block matrix


such that


= X A.


If Aj denotes the jth diagonal


block of A


then


we have


AXA=


AXA,.


The differential equation governing the block matrices has


the form:


A(t) = -A

where Diag(M)


(t) + Diag ((t)-'AX()) ,


a block diagonal


matrix formed from the diagonal


blocks of the


X(t) = X(t)F(X(), A(t)),











= X(t).


Typical starting guess is A(0)


Diag(A), and X(0)


=I.


Hence the


Euler approximation to the differential equations can be expressed


A+l -= An


+ At (Diag (X-,AX,) A) ,


Xn+l = Xn


+ AtXF (X,, A,)


(2.15)


use the


values of


Xn+l


from


(2.15)


to create an


terative


method to compute a block diagonalization of an n x


n real matrix A.


But first,


we will discuss how to compute a block diagonalization of an upper triangular ma-


trix.


For any upper triangular matrix T, its eigenvalues are the diagonal elements.


We will show


how to compute a


block diagonalization


y-1TY


= D of T


where


= diag( Dn, D22, ..., D ) with


the property that each


diagonal


block is upper


triangular, and the eigenvalues of Di, and Dj for


j are distinct. Let Y


= In, and


ai-yi = \tii


- til.


Given a tolerance tol, if aOj


> tol, then define


be equal to the identity matrix except for the (i,


j) element,


ti -- t
I'll tJJ


which is z.


. Let Wij

Then the


product


W ITWi


is an upper triangular matrix, whose (i, j) element is zero.


Next


update Y by ynew


= yoldWij.
t3*


We use this technique to zero tij,


whenever


An efficient way to zero tij, when a


from the bottom right corner of the matrix.


n, is to start


Zero tij in the decreasing order of the


row index i until i


1. If aij


< tol is encountered, then momentarily stop zeroing


on the jth column and go to the (j


1)th column.


Continue this process all the


way to the second column.


Once the second column is done, then in the increasing


order of the column index j go to those columns, where some of the tij are not zeroed


in the first round, even though the corresponding aij are greater than the tol.


This


time also zero t;; in the decreasing order of the row index i until i


= 1.


whenever


aj >


An+l,











In fact


, the above technique can be extended to a k x k quasi-upper triangular


matrix T


Rmxn


In


this case


is equal


to the k x k


block


identity matrix


except for the (i,


j) block, which is


where Z is the solution to the matrix


equation


TiiZ -


ZTfj =


We summarize


the block diagonalization of T in the following


algorithm.


Algorithm


2.2.1


Given a k


x k quasi-upper triangular matrix T


E Rx" "


and a coa-


lescing tolerance tol greater than the square root of the unit roundoff, the following

algorithm computes a block diagonalization of T.


S is the n x n identity matrix.


row -


i] col =


-1:


3 --


1; minim = 2tol


while minim


> tol


Q = min{ Je w


if ai3


:a E A(Ta), and wE A(T7j)


< tol


row


row ]; col =


j col


{ row, and col are two arrays,


whose elements are the row index i,


and the column index j respectively, of an entry


Tj of T


which will


not be replaced by a zero matrix.

minim = Gij

else


Solve T.


ZT=j -


-Tii for


and then form I'V,.


t rN I Pn p


f^-


F1 _











end

end

m = length(row)

for 1 = 1 : m


i = row(1)

while i >

if ij ~;


- 1; j3 = col(Z)

1


W-'TW,,;
ii t&7 y ,~


S = SWij


i=i


end

end


Let l


, and suppose NT


is an array of k elements whose ith element is the


dimension of the ith block of T


along the diagonal.


For i


< j, define a;j as in the


above code.


< tol, then merge blocks Ti and Tjj to form a single block using


permutations of columns and rows of T.


Use those column permutations to


merge


Update ko"d


and NVTota to obtain kfc"


and NT"""


STry the above


merging technique for all possible combinations of 1 < i < j3 1.

Now we are ready to compute a block diagonalization of an n


x n nondefective


matrix A using the following procedure:


Alaorithm


2.2.2


(Dunamical Eiaencomputations )


Given a nondefective matrix A


RanXn


, a tolerance


greater


than


unit roundoff,


a coalescing tolerance


greater than


the square root of the unit roundoff,


a constant stepsize At smaller


r











Define tol8 =


100toll


Next we break the sketch of the algorithm into several steps.


Step 0:


Take X


= I, A = Ao, Y


=X-1


=n(


the number of blocks


), and


, 1,..., 1 ]


an array of n elements whose ith element is the dimension of the


ith diagonal block of A.


Step


1: Let m


For i


< j, define DIF


= min{


a uW


a E A(Aj)


w E (A.)


If DIF


< toll


to form a single block


using permutations of columns and rows of A.


Use those column permutations to X


to merge Xi


and Xj,


and those row permutations to Y


to merge Y;T


and yT


Update


NBold


, and NSold to get


lynew


NSA"ew


Try the above merging technique for


all possible combinations of 1


Using column and row permutations,


arrange blocks of A


n the decreasing order of sizes along the diagonal.


those


column permutations to arrange block columns of X in the decreasing order of


sizes,


those row permutations to arrange block rows of Y in the decreasing order of


SIZes.


Step 2:


Construct F(X, A) as follows:


Fo (X,A) =


fori
for i


where P = P, is the solution to the matrix equation PAj AiP = Y)T


AX,.


Step 3:


for i


Suppose p = [Am+i,


= 1, 2,


. .


Am+2, ., AaNB


m, and dim(Ai)


where 0


for i = m + 1, m +


m < NB with dim(Ai)


Define d


, *


diag(Y AX)(1


: n) p, where t


= 1 +


(i), and diag(AM)(t


is a vector formed


from t through n elements of the vector diag(M).


Our aim is to find the smallest


s (0,1) with the property /; + sdi = yj + sdj.


To obtain this


s we do the following.


, then merge blocks Ai and A,












and define s = min{ ri


step of size


s instead of the normal time step At.


1 }. If s < At, then take a time

Otherwise take the normal time


step At.


Step 4:


Update A, X, and Y


as follows:


+t((Yold)T


AXr ld
J


1,2,..., NB,


Xold (I + AtF (Xold


(Xnew)


Step 5:


Consider the block Aj


with 1


, where m is defined as in Step


= QUQT


be a Schur decomposition of A,.


Then


where


D = diag


Ull, U22, ..., Ug ), and N is the strictly upper triangular part of U.


either a 1 x


1 matrix, or a 2


matrix. Let NU be an array whose ith element is the


dimension of the block Usi.


to reduce U


Use Algorithm


1 with tol2 as the coalescing tolerance


to a block diagonal matrix, to update the array


blocks 1, and to get the invertible matrix S.


Replace Aj by U


and the number of


XjQS, Yj


S-1YQTy


NB by


NB + I- 1, and the size of the block Aj in NS by


Try the


above decoupling procedure for all j such that 1


< m.


Step 6:


Compute f


= IIA-


XAYI\IF.


Goto Step 1 until


< tol.


Example 2.2.1


If Algorithm


2.2.2


is applied to


with


tol = 0.01. toll


= 10-4


and At = 0.05


, then after 130 iterations the diagonal matrix


Xne"


ynew


,Aod))


U2 is


1, 2, ..., n t, O < r7


- A^a ,
/*? /


Anew
^j


= D+N,


AO =


.













with


tol = 0.01, toll


= 10-4


, and At = 0.05, then it takes 131 iterations to the diagonal


matrix Ao to converge to


D= diag


-37.00078
-58.11556


27.61726
42.99974


-1.99896


The eigenvalues of D are A =


2.99948 + 2.22976i, 2.99948 2.22976i,


-1.99896


Example


2.2.


If we apply


Algorithm 2.2.2 to the matrix


with Ao = diag( 1,


-3), tol = 0.01,


= 10-4


, and At = 0.01,


then it takes


terations to the diagonal matrix Ao to converge to


diag (


-109.0896
-193.3755


62.4219
110.5633


15.3313
13.3030


-35.8551
-28.3276


-1.2449


0.1837


-196


10.7697


6.8342,


-4.8365


The eigenvalues of D are


0.7369 +3.0020i


0.7369


- 3.0020i,


-6.4982 + 0.6740i,


-6.4982


- 0.6740i,


4.7624 + 0.2597


i, 4.7624 0.2597i, 6.8342,


-4.8365 i.


This method has the following disadvantage.


the size of the given matrix A


A =


AO =







35


2.3 A Differential Equation Approach to Eigencomputations with Armijo's Stepsize

In the Differential Equation Approach to Eigencomputations, we found that with


a constant time step At,


which is usually a small number, too many iterations are


required to achieve a few digits accuracy in the results.


So here we will modify the


Differential Equation Approach to Eigencomputations by varying the time step At


in each iteration.


We plan to achieve this by using Armijo's rule from optimization


theory.


Let A be an n


x n real nondefective matrix.


We will find an iterative method to


compute a block diagonal


zation of the form A =


..., At ) is a block diagonal matrix, and X


XAX-l


= [ x,


, where A = diag( A, A2,

Xt ] is a compatible block


column matrix such that AXj


= XjAj.


As in Section


, for the block eigenvalue


problem, the differential equations that we solve are:


A(t) =


-A(t)+ Diag (X(t)-'AX(t)),


X(t)= X(t)F(X(t),A(t)),


where


Diag(M)


a block diagonal matrix formed from


the diagonal


blocks of a


block matrix M


, F(X(t), A(t)) is a square zero matrix, and Fi(X(t), A(t)) for i


is the solution


B to the matrix equation BAj(t) Ai(t)B


= Y(t)


AXj(t).


Here


denotes the ith block row of X(t)-1


. Hence the Euler approximation to the


differential equations can be expressed as:


An+1 = A=


+ At (Diag (X,-AXn) A) ,


Xn+l


+ AtXF (X, An)


(2.16)


Euler


method


the normal stepsize At is constant.


Here we will


vary the


s tnsize At in each iteration. Let


s be a positive Darameter, and let Q2(


s), and Z(


X2, ,


y(t)


= X,


SI











So n(O) = An,


Z(0) = X,, and R((At)


= An+,


Z(At) = Xn+1, the matrices gener-


ated by a Euler step.


Define


f(s) =


IG(s)IIF,


where G(s) =


A Z(s)Q(s)Z(


Hence f(0) = IA-


XA, X,1'F, and f(At)=


IIA Xhn+IAl+,XRnylIF.


If the starting guess block diagonal matrix A0o,


and the starting guess invertible


matrix Xo are good approximations of A, and X


then we must have f(At) < f(O).


n the factorization A


Here our goal is to find an s, 0


X AX-


for which


f(0) holds.


To this end,


we use Armijo's rule from optimization theory.


Armijo's rule, we determine


s in the following way.


Evaluate


f(s) at s = 1, 1...


stopping when


Simplifying the above inequality we get


f(O)


1
- -f(o).
2


It turns out ([H


ag88,


page


178 80]) that to use the above rule,


we must have


f'(0)


--f(O).


So our next aim is to determine


f'(0),


when


f(s) =


G(s) lr, and


G(s)= A- Z(


s)Q(s)


1. Suppressing the subscripts of A., and Xn in (2.17), we


s (Diag (X-'AX) A)


Before we find


+sXF(X, A).


f'(0), we want to prove the following fundamental result.


Lemma


2.3.1


Let G(s) be an nx


n complex matrix, whose elements are differentiable


functions of s.


If we define


f(s) = IIG(s) p, then


f(0)f'(0) = trace (G(O)"


(2.18)


:G(s),=o).


(1 ) f(O).
2/


s) -


Z(s)










Differentiating (2.19) with respect to s, we have


trace (d (G(s)H) G(s) + G(s)H G(s)


d \H
trace -dG(s)
\ds


d
G(s) + G(s)H G(s)


d
= 2trace G(s)H G(s)


After simplification setting s = 0, we get

d
f(0)f '(O) = trace G(0)dG ( s)1=o


Now with OL(s) = A +s (Diag (X-AX) A),


Z(s) =


+ sXF (X, A)


A- Z(s)l(s)Z(s)-1


, and f(s)


= IG(s) IF and the result of the above lemma, we


in a position to show that f'(O)= -f(O).


Theorem 2.3.1


Let A


E R xn be a nondefective matrix.


Suppose


= XAX-1


block diagonalization of A, where A = diag( A1, A2,..., At ) is a blo


diagonal matrix


such that Ai, and Aj for i


is a compatible block column m


j have distinct eigenvalues,


atrix.


If we define L(s)


IX = [X, X2,..., Xt]

s (Diag(X-1 AX) A),


+sXF(X,A), G(s)= A


-Z(s


)Q(s)Z(s)-1


, and f(s)


IG(s)|lF, where


Diag(M)


is a block diagonal matrix formed from the diagonal blocks of the block


matrix M


, F (X, A)


zs a square zero


matrix, and Fij (X, A) for


is the solution


P to the matrix equation PA1 AiP = YTAXj, where YiT


denotes the


th block row


, then f'(O) = -f(O).


Proof of


Theorem 2.3.1


The result (2.18) of Lemma 2.3.1


gives


2f(s)f'(s)


are


ofX-1


s) =


=A+











Differentiating the expression for G(s) with respect to


we have


d
G(s)
ds


d
- -
ds


(s)) Q(s)Z(s)-1


d
-d (Z(s))\


d
(s) (n(s)) 2
d
d5


d
'(s) O(s) -s (Z(s)-1


(s))Z(s)-1


(2.21


d
G(s)18,=o
ds


d
- Z(


s)18=on(0)Z(O)-1


d
- Z(O)j- (s)I.=oZ(O)-
d6s


(2.22)


Differentiating nf(s),


and Z(s) with respect to


s and then evaluating the derivatives


= 0, we get


d ( =o
-f(s)Is=0O
ds


= -A + Diag (X-'


AX)


d
Z(s)1,o = XF(X,A).
ds


Using these values and values of f(0), and Z(0) in (2.22), we have


G(s),=0o


-XF(X,A)AX-'


- X (Diag (X-'AX) -


A) X-1


+XAX-' (XF (X, A))X-'


- X(F (X, A) A


-AF(X,A))X-1


-XDiag (X


-'AX) X-1


+ XAX-'


(2.23)


Next, let D = F(X, A)A


- AF(X, A).


Then


(2.24)


FPA, AXF,
1JI^* &


where F1i


= F(XA).


According to our assumption for i


is the solution P


r t vnntin PA -- A-P


gn P.A


- VT A V .


d
S)Z5(s)-1_


d
Evaluating -G(s) at s = 0, we obtain
ds


(o)-d
+t Z(0)n(0) Z(0)-" Z(s) S-o Z(0)-x


- VTAY


(s)-


()-1


+Z(s)(l


Dij -


A 7 .


Hen e


it )th matrev


!-----










Using (2.25) into (2.23), we have


s)1,=o


-X (X-1AX


- Diag (X-'AX)) X-1


-XDiag (X- AX) X-


XAX-1


-(A-


XAX-1 .


(2.26)


Since G(0)


XAX-1


, so (2.26)


gives


d
ds


s)ls=o =


-G(0), and with this


value (2.20) reduces to


f(o)f'(O)


Hence f'(O) =


trace (G(O)T (-G(O)))


-f(O), provided f(0)


The above theorem implies that Armijo's rule can be applied to Algorithm


2.2.2


mentioned in the Differential Equation Approach to Eigencomputations.

A modified algorithm to find the eigenvalues and corresponding eigenvectors of a

nondefective matrix A can be obtained as follows:


Algorithm 2.3.1


Given a nondefective matrix A


R nxn


, a tolerance tol greater than


the unit roundoff,


a coalescing tolerance toll


greater than


the square root of the


unit roundoff, an invertible matrix Xo, and a block diagonal matrix A0, the following

algorithm uses (2.15) to compute a block diagonalization A = XAX-1


Define tol2


100toll


Next we break the sketch of the algorithm into several steps.


Step 0:


Take X


- Xo,


= A0.


denote the number of


diagonal blocks of A,


and let NS denote


an array of NB elements whose ith element


d
ds


(o)n(o)Z(o)-1


SA


SA -


- f(0)2


= X-











permutations of columns and rows of A.


Use those column permutations to


merge Xi and X1,


and those row permutations to Y


to merge yT


and YT
Sr


Update


NBold


, and NSo'd to get NB"e"


and NS"n"


Try the above merging procedure for


all possible combinations of 1


Using column and row permutations,


arrange blocks of A in the decreasing order of


sizes


along the diagonal.


those


column permutations to arrange block columns of X in the decreasing order of


sizes,


and those row permutations to arrange block rows of Y in the decreasing order of

sizes.


Step


Construct F(X, A) in the following way.


Take F3j (X, A) = 0,


which is an mj x mj zero matrix, and for i


Fij (X,A) can


be determined by using the following loop:


forj = 1

for i


:NB


=1


:NB

3


Solve PAj AiP =

F ,(X, A) = P.

end

end

end


AXj for P


Step


where G(s)


X(s)A(s)X(s)-'


, X(s)


X(I+


sF),


and A


s) = A+


s(Diag(X


-'AX) A).


Evaluate f(s) at s = 1, ,


stopping when


ifi /


S) ().


(2.27)
x .w











Step 4: Suppose p = [Am+l, Am+2,..., ANB]T


, where 0


m < NB with dim(Ai)


1 for i


diag (YAX) (t


...,m, and dim(A,)

: n)--p, where t = 1


= 1 for i


+ l,m + 2


SNS(i), and diag(M)(t
t=1


, .


,NB.


Define d


: n) is a vector formed


from t through n elements of the vector diag(M).


Our aim is to find two indices il,


and iu as follows.


Let k be an array with the property that Pk(i)


I#k(i+1).


Next


form the ratio:


Pk(i+1)


dk(i+l)


and let rio


= min{ ri


n-t, 0


Let il=t 1


min{k(io),


k(io + 1) }, and iu


Step 5:


=-L 1


Update A, X, an


+ max{ k(io), k(io + 1) }.

id Y as follows:


+z ((Y,"o)T


AX"d
.7


1,2,...,NB,


X new


zF (X"


, Aod))


Y new


(Xnew)


Step 6:


Consider the block Aj


with


, where m is defined as in Step


= QUQT be a Schur decomposition of


Then


where


D = diag( Ul, U22, ..., U** ), and N is the strictly upper triangular part of U


. Ui is


either a 1


x 1 matrix, or a


matrix. Let NU be an array whose ith element is the


dimension of the block Uii.


Use Algorithm


1 with tol2 as the coalescing tolerance


to reduce U to a block diagonal matrix, to update the array


and the number of


blocks l, and to get the invertible matri


Replace Aj


by U


X1Q


S-'Qryj


NB by


NB + 1- 1, and the size of the block Aj in NS by


. Try the


above decoupling procedure for all j with 1


< m.


:i z




- Pk(i)


- A ,
^j


A ,ew
1"J"


X'ld (I +


=D+











Step 8:


Compute f(0) = IA -


XAYIIP.


Goto Step 1 until


f(0)


< tol.


Example 2.3.1


If we apply


Algorithm 2.3.1


with


Xo = I, tol = 10-6


and toll


= 10-4


, then after 42 iterations the diagonal matrix Ao


converges to

D are diag

The eigenvalues of D are A= [-


-6.01310
-5.98585


1 + 1.41421i


4.53255
4.01310


-1 1.41421i, 0].


Example 2.3.2 If Algorithm 2.3.1 is applied to


with


Xo


after


, the tolerance, and the coalescing tolerance are as in Example


2.3.1, then


196 iterations the diagonal matrix Ao becomes


D196 = diag ( -1.04694840, 0.00224879,


-0.95530039)


In the following 4 successive iterations Die changes to


-1.04676152, 0.00223123,


-1.04657539, 0.00221380,


-0.95546971


-0.95563841


diag


-1.04639002


, 0.00219651,


-0.95580649 )


diag (-1.04620540,


0.00217935,


-0.95597395 )


i


Ao =


Ao =


D197


diag


Dis


D199


D200oo











Example 2.3.3 Consider the matrix


If we apply the QR method with double implicit shift with tol


= 10-6 to A,


then


after


iterations


we obtain a real Schur form STAS


= U


the eigenvalues


of U


are A =


10.8425 + 7.9032i


10.8425 7.903


.6168 +


7556i,


-2.6168 -


7556i, 3.5414 +


1.2899i, 3.5414


- 1.2899i, 8.4658 ].


Next,


we use the


technique


discussed prior to Example 2.1.1


to find corresponding eigenvectors.


Here for each


pair of complex conjugate eigenvalues,


then


we solve the equation Bx = 0 only for one,


take the real and the imaginary parts of x as two vectors corresponding


to the complex conjugate pair of eigenvalues.


Consider the following


We denote the matrix of vectors by


g perturbed matrix A1, which we obtain by perturbing the


elements of A:


A-=


-4.98
-5.97
-3.91
-1.93
7.02


3.03
3.06
-0.99
7.07
2.06


-1.94
4.08


-1.96
3.01


-2.9


7.05
-1.96
4.02
7.00
1.09


1.09
-1.96
3.01
-0.91
6.01


-4.99


If we use


Algorithm 2.3.1


to A1


with Xo


= X-1AX


= 10-6


= 10-4


, then after 3 iterations we get a block diagonalization Y-'AIY


zb n rF ,/ft^ I nrn 'xnnv 1 7 +1r:. flfl rA *,t, I4 r vhln Irnl4 ^ fn .i A C 4 + n ,rSn r


, and


-X


, Ao


= Di,














A = Ydiag( 1, 2, 1, 3, 2, 1,


-1 )Y-1


where Y


= Qdiag( 1, 2, 3,


-3 )QT


,and Q is an orthogonal matrix.


If we


apply the QR method with double implicit shift with tol


= 10-6 to


A, then after


4 iterations,


we obtain a real Schur form STAS


Next,


we use the technique


discussed prior to Example


2.1.1 to find corresponding eigenvectors.


We denote the


matrix of eigenvectors by


Consider the matrix


If we use Algorithm 2.3.1


to the perturbed matrix


= A + P/1000 with Xo =


X-1AX


, tol


_ 10-6


, and


= 10-4


then


after


iterations


we obtain


a block diagonalization


whereas if we apply the QR method with


double


mplicit shift to STA1


then after 6 iterations we obtain a real Schur form


A1S)Q, = R,


Although this method can be applied to find eigenvalues of some matrices ( Exam-


pie 2.3.1, ) in general it does not work for all matrices.


equal eigenvalues, or the size of the matrix is


For example, if a matrix has


arge, then the method does not work


Example


In all of these cases, two 1


x 1 diagonal blocks


approach each other,


but as the iterations continue, the rate at which they approach each other gets


slower


.* ,. n


= U


-'A


= D,


Ql( ST


IT ... ..: : 1 .. .... *f > :...,_ ** I, ,l I l, j I,.,* w -,I, ,*k*, *r A *


- I 1 _! ...











perturbed matrix A + eE,


where E is an arbitrary matrix, and


e is a small scalar (


Example


.3.3 and 2.3.4.


2.4 Convergence of Block Diagonalization of a Matrix


In Section 2.1,


matrix.


we developed Algorithm 2.14 to block diagonalize a nondefective


Here we will examine the speed of convergence of Algorithm 2.14.


Let A be an n


x n nondefective matrix, and let A


= YAY-1


be a block diag-


onalization of A


where A


= diag


A1, A2,..., Ak ) is a block diagonal matrix,


Y1, Y2,


. .


Yk ] is an associated block column matrix such that AYk = YjA.


Define X


SY (I+E)


Xnew'


= X(I+ F),


where E is an n


x n


perturbation


matrix such that


Diag (X


-1AX)


= diag( D1


D2, .


, Dk ) is an approximate


block diagonalization


have distinct eigenvalues, and


= (Fij) is a k x k block matrix such that Fjj is a square zero matrix, and Fij for


j is the solution P to the matrix equation PDj


. Assume X is close to Y


DiP


We will show that


= (X-1AX).,.


Xnew


-y II=o(


Clearly

El2) =-


O( lX


- YI2), and j)Diag(X-1AX)


-Aj


O(llX


That means the block


diagonalization method converges locally quadratically.


Theorem


2.1.1


Let A


E R~nxn


be a nondefective matrix and


suppose A


= YAY-1


a block diagonalization of


where A


= diag( A1, A2,


..., Ak )


is a block diagonal


matrix


and Y


Yi, Y2,


Y ]


an associated block column


matrix such that


AYj=


YjAj, and Ai


, and A1 for


have distinct eigenvalues.


Suppose the pertur-


nation matrix E


E Rn" Xn


is sufficiently small to ensure that X


is close


to Y


in the


equation X


= Y (I + E), and Diag(X-


'AX) =


D is an approximate block diagonal-


= D


, Di,










Then


IIDiag (X-'AX) Aj|


O (|IEI2),


SIXrne


Proof of Theorem 2.4.1 Assume that if ipj


qij for i, j = 1,2,...,n, then I|P|I


Now


X-'AX


(I+ E)-1 Y-AY(I+ E)


(I + E)-A (I + E).


Since (I + E)- =

X-'AX


I E + E2(I + E)-


[I E + E(I+E)-] A(I + E)


= A+ AE- EA- EAE+(I + E)-1A(I+E).

Let L = -EAE+ E2(I+ E)-1 A(I+ E). Then


- EAEII+ lIE2 (I + E)


211A


IIE II


A(I+E)II


E112 (1 +


+ AE EA + L, so


Diag (X-'AX) = A + Diag(AE EA) + Diag(L).


Next normalize Y


(2.28)


such that the diagonal blocks of Y-'X are identity matrices.


That is Diag(Y-1X) I


= 0. Let


then from the relation E = Y-iX I,


we have


0
7~T V


Since X -AX


I l


-Y l| = iE|l[ .


Eij =-


i 4-i







47


Thus Diag(AE EA) = Diag(C) = 0, and from (2.28) we have Diag(X-x AX) A =

Diag(L). Hence


IDiag (X-AX) All


|Diag(L)


o(j1E,12),


(2.29)


which proves the first part of Theorem 2.4.1.


Since Diag(X- AX) = D = diag( D1, D2,..., Dk ), and F = (Ff1) is a k


matrix, where Fjj = 0, and for i f j,

BD, DiB = (X-1AX)i, so G = FD


k block


Fij is the solution B to the matrix equation

- DF, where


F.j Dj Dn Fj
F*J"J-LlAt


(X-1AX) j


Hence FD DF


= G = nondiag (X-AX, where nondiag(M) is a block matrix


formed from the block matrix M by replacing the diagonal blocks by zero matrices.

Since X-1AX = A+AE EA + L, hence nondiag(X-1AX) = nondiag(A+AE EA


For z


That is


(2.30)


j, taking the (i,j) block on both sides of (2.30), we obtain


FjD DYFij
U tj U~ -^Ai


- (EijAj AiEij) + Lij.


(2.31)


Let Hij = EijAj AiE1y, and let Dj = UEU-1


izations of Dj, and Di respectively.


, and Di = VR-V1 be diagonal-


Substituting these values in (2.31), and then


1',r 1


+L).


FD DF = nondiag (AE EA + L).











Let W= V-'FiU


Then the above equation reduces to WE


- flW = -V-'HUijU +


V-'LijU.


Now solving for W


= (wim), we get wim


=- -Ptm


+ qim, where pim


(V-' LaU),m


-W-


= 1,2,...,mi, and m


- 1,2,...,


= (Pin), and Q = (qpm).


Then simplifying the relation W


= P + Q, we obtain


F = -VPU-1 + VQU-1


(2.32)


Next, let A, = UiE1,U-


1, and A1 = ViQ1 Vj- be diagonalizations of Ai


respectively. Using (2.29), we have lAt Dil _

exist O1,2, 1r, and F2 such that U1 = U + 01


o (1[E|2), where t


and A,


= zi,. So there


S= 0+Z+ F2, and IiO tl


O(||El|2) and lFi,


O(||EjI2), where


t=1,


Now


(UE+ ur


1 + OE + O1Ti) U-' (I + 1U-1)-


(usUl


+UFrlU- + 10EU-l


+ OFU-) (I OU-' (I + -1)


UEU-1


where


UrFU-1


+ eOU-'


+ xe,rU-'


- (UEU-'


+ UF U-' + ,EU-L


+ 01FiU-') OlU-1 (I + 0hU-1)-'


Hence


IA I


IIUF17U-1' + IiO1 U-1'


+ IO U-elrlu-l II UEU-1


+ UF U- +6 ,sEU-'


+ OiFU-'


IOU-1 I


I + OeU-')


O (l ll2).


Similarly, A = VOV-1 +


with IAl II


Eij (U U-'


+A,)


0 (| Ej 2). Since Hi, = EjA, AEA,,


- (VnV-' + A2) .E,


V-IH I I


V-1IErIrU


AAF1 JK Jft 9*-- *


- V1E.;U + V-' (E.Ai, A,9 ) U.


(2.33)


-- i


, Ei


, Qim


- E+ F,


= V+-2


(U + o1) (+ r)(U +o,61)-


+ AI,


*











After simplifying the last equation, we get ptm =


(V-1HijU)im = (V-lEijU)
a v -m V-1EU
dT faT \r /77


+Stm,


where sim


(V-1 (EijAI


-A2E-) U)tm


- W


Let


=- (sm),


then P


and with this value of P (2.32) reduces to


-VV-'EijUU-1 + V(-S + Q)U-1


- E1ij + R j,


where Rij


= V(-S+ Q)U-1


Since s-im


(V-1 (E;jA


- A2E^ ) U)tm


, and qim


- U.


(V' L~1 U)im


-U).


AIlIA +


IRj II


mini am


II IIEII)I IUI IV-ll
minl


O (IIE 2)


Thus F


= -E + R, where R = (Rij) with IIRII =0 (|E|l2). Next


X"ne


X(I+F)


Y (I + E) (I E + R)

Y (I E2) +Y (I + E)R

-YE2 + +Y(I+ E)R.


Hence IIX"ew


- Y = 0 (E12). This proves the second part of Theorem 2.4.1. ''


The result of the above theorem implies that the block diagonalization of a non-

defective matrix converges locally quadratically.


2.5 Block Schur Decomposition of a Matrix


XneW


-a ]/


= V-'EgjU +


|V--l(lE I


IlLr ilm IIUII U
11m-- IM P) I







50


approximation to the matrix of eigenvectors, and in some methods a good approxi-

mation to the eigenvalues as well. Here we will find an iterative method to compute

a block Schur decomposition of a matrix of A.


Let A be an n x n complex matrix. Let A


= SUSH be a block


Schur decomposition


of A, where U is


x k block upper triangular matrix whose (i, j) block is an mi x m


matrix, and


, S2,. *


Sk ] is a compatible block column unitary matrix such


that ASj


Uij. It can always be arranged so that the eigenvalues of Ul/, and Ujj


j are distinct.


Before we derive the necessary equations to find a block Schur


decomposition of a matrix A, we discuss the effect of perturbations in the elements of


A on the columns of the unitary matrix


and the elements of the upper triangular


matrix


ts Schur decomposition S AS = U


. That is given an arbitrary matrix


E, and for a sufficiently small e,


we will show in the next theorem that there exist


continuously differentiable functions


and U(c) with


S(0)


, and U(0) = U


such that (A + cE)S(c) =


S(c)U(e).


Theorem 2.5.1


Let A


be an n


X n


complex matrix.


Suppose ST A


a block


Schur decomposition of A such tha


the diagonal blocks LUi,


and Ujj for i


have


distinct eigenvalues.


sufficiently small e,


Then given an arbitrary nx n


there exist continuously differentiable


complex matrix


functions


, and


S(c),


for a


and U(c)


with S(0


)= S


and U(O0) = U


such that the following matrix equation holds:


(A + cE) S(c) = S()U(c).


2.34)


First we discuss the effect of perturbations in the elements


of A on the columns of the unitary matrix


S in its Schur decomposition SHA


Proof of Theorem











and let Q(c)-1DQ(e)


= U(c) be the triangular decomposition of D by the invertible


matrix Q(c).


Then we have


s;(e) = Sq(c),


1, ...,n.


The form of the matrix D


is illustrated for a 7


x 7 matrix


with k = 5, such that dim(Uin) = dim(U22) = 2, and


dim(Ujj) = 1, j = 3,4,5:


U17
U27
U37
U47
U57
"67
U77
U?7 a


where gij


= s[Esj.


Equating the jth column on both sides of DQ(e) = Q(c)U(


we obtain


uijqzj(e) +


gilqlj (


(2.35)


2i, =


..., n. Since uj(O)


= uj, so we must have qj(O)


which is the jth unit


vector.


Next


and taking


we normalize qj(e), j =

as the largest element.


For sufficiently small e,


claim that qgn(e) =


which is the 1st component of ql(e). If not, let q,ni() = 1.


Then from (


.35) we get


"U11()-- "u = e


gniqfl(e).


Now letting


we find u11 un = 0, a contradiction.


Next we will show that


Iq l()


0, for some Mnl


Since Iqil(e)


so from (2.35),


obtain


Iul((e U,


Ign, I


Let lut(&) u..I


1
-lul -- u..


then Iai (e'l < M


r,1. where


= ej,


,,,J -


e --0,


D =


qil(c)uhj(e),


1,..., n by setting qij(e) = 0,


Mn, e


qI~(e))l












Using the above arguments to the components qil,


i 3,..


1 in the decreas-


ing order of the row


index


we can show that qi1(c)


1, and


q.(e)I


Min~, where


Again from (


.35), we have


(u11(c) -- u)qn(c) =


ulqi (c) +


t=1
6 1= ~ c


(u11(e) -- u)921()=


n
U2Iqn(I()+ g2qn(().
1=1


To make the above two equations consistent we take q21(E) =


Using similar argu-


ments, and considering the elements in the decreasing order of the row index i,


can show that jqij(e)


for some Mi


n -


j+l


, ., n.


To show


Iqjj(c)


< AI~1e,


we use the result


qit(c)


= 1,


Mt,, 1


To keep the consistency, we take q43(


c) =0.


max


Then


q;j(e)


34


=t -


V ,n


Thus qij(c) are of order


c for


1,...,


j + 1,..., n, except q21(c), and


q43(e), which are equal to zero.


Next from (2.35), we have


ujj(c) u)qj(c) +


) gij -= e


I

1=i+l


l=j+1


The term on the right is of order


ij(fc) -
Vu)'->


ujj Ui


l= +1


utqij(0) -


j--
>q(O0)ut,
I=1


=-O(


Next we discuss the effect of perturbations in the elements of A on the elements


of their runner trianpuilar matrix


Uf in its


Schtir decomnosition


SHA


. As in the


* V V


j =


MAlif,


Iqj(c)


= i -+ 1...n.


...,n- 1


qu(c)uij(e)


g;tQfj(E).


uitq{j(


, -- I











same 7x7 matrix D with k = 5


Here we will use the results of perturbations of the


columns of Q(e). Putting j


1, and


j=2,


in (2.35),


we obtain


nuilq(c) + f glqi(e) = ut1(),
/=1


ulq12(e) + e


u21ql2(e) +


gljql2(E) = "12(0),


g21iq2(e) = U1(E).


After simplifying the above equations, we get


u1( -- ull +t- gl +


U12() (U12 + 6 12 +


u11(e) (u11 +


+E
/=3


uliq2(e))


n
1=3
n

1=3

=Ef
1=3


glfqin (e),


gllql92(),


921qI2(t)


For sufficiently small


let lunl(E) uii


1
IUl ~u^\'


= 3,5,...,n.


Then


terms on the right of the above equations will be of order c2, and we have


u11() (ull + g11 +


2() u12 + g12z +


sl(e) (ill +


ullq,2(0)


u21q12(0)


= O(e2),


Thus perturbations in sll, and u12 are of order e as


Due to


Theorem 2.1.1, it


is always possible to find a sufficiently small c, such that Iu1(A) ua


1
2u\I u IS
_ AiiL"! J&


- 1,3,5,...,n.


So after simplifying (2.35),


we can show that for


=0(











Hence


uj() uj + C/ij + C


i-1
uqj=(O) Q(O)uj
l=1


Thus the perturbation


is of


order


E as e


-- 0.


Hence U(e)


S(c) are


differentiable functions of


and their first derivatives are continuous.


This completes


the proof of the theorem. C

Now we are ready to derive the necessary equations to find a block Schur decom-


position A


= SUSN
^U


Given an n x n complex matrix E, and for a sufficiently small


and U() be continuously differentiable functions of e such that S(0) = S,


and U(O)


Differentiating (


.34) with respect to e and then putting


obtain


AS'(O) +


ES=


S'(O)U + SU'(0)


SHAS'(O) + SH


ES=


(O)U+U'(O).


(2.36)


Since


S"A = US"


, so (2.36) gives


USHS'(O) + S"


ES=


SHS'(O)U + U'(O).


(2.37)


Taking the jth column block on both sides of (2.37),


we have


US" s (o) + ES = SS'(O) + (0).


(2.38)


Next consider the following expressions for S,(c):
k


S,(c) =


SiBij(),


(2.39)


where B ,()


is an mi


x mj matrix.


We normalize (2.39) by taking Bj(c)


Rn f


In;ifforn nit.;ntr (9


-QfQ\ vutbth r cnort t n r znrl then cot tl.na


l-j+1


= U


f -I I


=0


--fnilr











Since S is unitary, so


SHSt'=


(2.41)


Using (2.40), and (2.41) we get SHS'(0)


= B'(0).


So (2.38) reduces to


UB5(O) + SHES1 = B'(0)U- + VJ(0),

where BJ(0) is the jth column block of B'(0). Hence for each j, we have the following

two matrix equations:


k

1=-j+1


uB'1(O) + s+


ES =


B,(O)U1j + U' (O)


for i


1,2,...,


U.B ,(0) + sH


ES =
Z7


B, (O) Ui


The last two equations can be rewritten


UJ:(0) = S3ES, +


l=j+l


UitBI,() -


SB,(O)Uj


for i


1, 2,..., k.


(2.42)


Bj.(0)Ujj UBJ(0)


=SHES +


Ua1Bf1(0)


1=i+1


Solution of (


-=1
l=1


B1,(O)Ulj for i


.42) is dependent on the solution of (2.43).


the solution of (2.43).


we need entries below


From the right hand side of (2.43)

Btj(0) on the jth block column;


> j, j


= 1,...


k-1.


(2.43)


So we focus on how to find

t is clear that to find Btj(0),


namely


Bf+1i(0),


. .


Bkj(0),


and entries on the left of B y(O) on the ith block row; namely Bi.(0),..., Bj_(0)

Pictorially these can be described with zero suppressed in Bi(0) as:


Bd' -_


27.


= 1l, k- 1.











That is to determine B ,


we need entries just above the arrowheads directing towards


the left, and just on the right of the downwards arrowheads.


Here we will give a sketch to find


Let C


= S,"ESj.


< k, then C


k

1=i+1


Un BB, and ifj


> 1,then C


-z B,'Ui. Next we need to solve the matrix


equation B' -j UjBf = C for B'.
yUI t3 tJz


To solve this equation, let P = Bt


- U11,


Then it becomes GP


- PF


where R =


since F


and G are


upper triangular matrices, so using the following method (


detail is


n [Gol90, page


387]), we can solve the matrix equation GP


-PF


= R for P. Let R = [ri


2, *


rm, ,


and P =


Pl P2, Pm,


be column partitionings, then solve


(G fitI) pt = rt +


i-1
/ fmtPm
m=1


for Pt, 1 = 1,...,mnj.


Once we obtain P, then B'


= P for i


2.6 An Algorithm for Block Schur Decomposition of a Matrix


In Section


we derived the necessary equations to find a block Schur decom-


position of a matrix.


Here we plan to develop an algorithm using Armijo's


rule from


optimization theory to find a block Schur decomposition of a matrix.


A be an n


x n


complex matrix.


Suppose


= SU


is a block


chur de-


composition of A,


where U is a k x k block upper triangular matrix


such that Ut,


and Uj for i $ j have distinct eigenvalues, and S

block column unitary matrix such that AS, =


_-[ S,


S2,. .,Sk


is a compatible


Given an arbitrary complex


matrix


, and for a sufficiently small


e, let S(c),


and U(e) be analytic functions of e,


such that


,U(0)= U


( A+ eE)S


(c)= S(c)U(E).


(2.44)


= R,


12 k 1


0) =







57


To develop an algorithm, suppose SUSH is an approximate block Schur decompo-


sition of B.


We identify A, and E in (


44) with SUSH


, and B


- SUSH respectively.


Let S'(0)


where Gij(


is an mi x my zero matrix for i


j, and for


U) is the solution P to the matrix equation PUjj UViP


= C, where


c = S (B SUSHI)


E
1=i+l


UiGij(S,,


j-1
- Git(S,U)UIJ.
1=1


From (2.37),


we have


USHSI'() + SHE'S


= SHS'(O)U+U'(O).


Next substituting the values of E, and S'(0) in the above equation and then simpli-

fying, we obtain


U'(0) = UG(S,


U) + SHBS U G(S,


U)U


Now substituting the values of U'(O), and S'(0) into (2.45) and suppressing zeros in


S(0),


and U(0), we get


S(e)
U(c)


S (I + eG(S,
U + e (UG(5


U) + SHBS


U)U) (2.46)


Taking


e = 1 in (2.46),


we obtain


Snew


(I+G(


Sold, Uold))


UoldG (Sold


,Uold) (


old H


BSold


- G (Sold


,Uold) Uold


If the starting guess unitary matrix So is not a good approximation of Q, where


= QRQH


is a Schur decomposition of B,


then in


the update


Sne"w


= Soda (I+


U""


-U-G(


1 ~L











steady change in the values of S"e"


and U"ew"


we need a small increment in each


iteration.

parameter.


To achieve this,


we redefine each iterate, and introduce a small positive


Let t be a positive parameter; define S(t) = Sold (I + IG (Sald


Uotld))


and U(i) = UoId+t (UOtdG (Sold


,u,,) + (sold)H


BSold


_ Uold


,Uod) UOid)


Then


S(0)


- Sold
-


, S(1)


= Sne"


, U(0)


-= Uold


U(1)


=U- U


Define


IIZ(t) lF,


where Z(t)


Then


f(O)


= IB -


s(0)u(o)s(o)"


f(1) =


IIB -


When the starting guess So, and Uo are good


approximations of Q, and R in


the factorization


= QRQH


then


we must have


f(1)

holds.


f(O).


As earlier, here our goal is to find a t,


To this end


1, for which f(t)


, we use Armijo's rule from optimization theory.


In Armijo's


f(0)
rule,


we determine t in the following way.


Evaluate


f(t) at


t=l,2


,..., stopping when


- I) f().


in Section 2.3


, to use the above rule, we must have


f'() =


-f(o).


So our next


aim is to determine


when


f(t) =


IIZ(ti) 7, and


(t) =


Suppressing the superscripts of


and U"d


n the definitions of S(t)


and t


t) we


(it)
U(t1)


S(I+tG(L
U+t (UG(


With the above definition of


f(t),


SHBS -U --G(


we will


(2. 47


,U)U)


show in the next theorem that


-f(O).


Theorem 2.6.1


Let A


s C"x"


and suppose A = SUS"


is a block Schur decomposition


where U


ak x k


block upper triangular matrix such


that U i,


and Ul, for


, 4 t **


-.I 1 1 I


C.. ...........3f. .. ..A 5-> -II -


=B


/"B


- G (Sold


S(t)U(t)S(t)-1


S(1)U(1)S(1)- IIF.


s(t)U(t)s(t)-"


/'(O),


,U)+


f'(0) =


1.. ^ .-U-. **


J ..... J J ,ll


* h


s










Proof of Theorem 2.6.1


The result (2.18) of Lemma 2.3.1 gives


Differentiating the expression for

derivative at t = 0, we have


(o)H


Z(t) with respect


(t) t=o)


to t, and


(2.48)


then evaluating the


d
Z(t) Ito=
di


d


t=oU(0)


+s(o)u(o)S(o0)-


d
- S(O) U(t)It=oS(O)-1


S(t)I =oS(O -


(2.49)


Next differentiating U(t), and


S(t) with respect to t, and then evaluating the deriva-


tives at t = 0, we obtain


d
U(t)t=o = UG(
dt


U) + SHAS -U-G(


U)U


-S(t)t=o = SG(S,
dt


Using these values and values of U(0), and S(0) in (2.49),


we get


d ()l
di I


-SG(S, U)USH


-s(UG(


U) + SHAS U G(S, U)U) SH


+ SUSHSG(S, U)SH

- (A SUSH)

-Z(O).


d
Hence -
dt


(t)It=o = -Z(0), and with this value (2.48) reduces to


f(o) df(t)l,=o


trace (

-f (O)


(o) (- Z(0)))


dn


- f(fl provided f(O1


* 1-


f(0)df(t)

s(0)-1











Alaorithm 2.6.1


Sxn"n


(Block Schur Decomposition) Given


a tolerance


greater than the unit roundoff, a coalescing tolerance toll


greater than


the square


root of the unit roundoff, an upper triangular matrix


Uo, and a unitary matrix So,


following algorithm uses (2.47) to compute a block Schur decomposition A = SUSH


Define tol8


100ltoll


We break the sketch of the algorithm into several steps.


Step 0:


Take


U= Uo.


Let NB denote the number of diagonal blocks of


, and let NS denote an array of NB elements whose ith element is the dimension


of the ith diagonal block of U.


2m =


define ayj


= min {


C" Wo


: a A(L


If aij


, then merge blocks Uii and Ujj


to form a


ngle block using


a unitary transformation to U.


Postmultiply S by the same unitary transformation.


Update


NBold


, and NSOad to get NBle"C


and NS"


Try the above coupling procedure


for all


possible combinations of 1


Using a unitary transformation


arrange diagonal blocks of U


n the decreasing order of


sizes.


Postmultiply


S by the


same unitary transformation.


Construct G(S, U)


as follows:


For i


take Gij (


, which is an m; xmj zero matrix, and for i


can be determined by using the following loop.

B = AS SU


for j =


:NB


for =


:J+1


S =


S(Ujj


, /VP~












C=C


-1
- Gk(S,


U)Uk


end

Solve PUjj UiiP = C for P


U) =P


end

end


Step 3:


(tl) F,


where


Z(t)


SA-


, S(t)


= S(I


+ tG(S, U)), and U(t) = U + t (UG


U) + SHAS- U -G(


U)U)


Evaluate f(t)


-,..., stopping when


(1-


Let p be the first value of t for which (


-) f(O).
2\~n


(2.50)


.50) is true.


Step 4:


Suppose p = [ Um+lm+l,..


*, UNBNB


, where 0 _< m


< NB with dim(U/)


for i


dim(Uii)


, ,


Define d


diag (UG(U

diag(M) (t


, U) + SHAS


-U -G


where


: n) is a vector formed from t through n elements


= 1 + NS(i), and
of the vector diag(M).
of the vector diag(M).


aim is to find


two indices il


u as follows.


be an


array with


property that


Itk(i+I)


Next form the ratio:


/Pk(i+1)


- Pk(i)


and let


= min{


i= 1, 2, n -t,


Re(r,)


<1}.


Let il


=t--


1+ min{ k(io), k(io + 1) }, and i


u=t--


1+ max{ k(io),


k(io + 1) }.


Step 5:


Update U


S as


follows:


,2, m,


: n),


Gij(S,


S(t)U(t)S(t)-1


= m +


U)U)(t


dk(i) dk(i+l)


>0 &











S""ew


Sold (I + pG (old


, UoI))


Snew new (


Snew


Step 6:


Consider the block Uj j


with 1


m, where m is defined as in Step


Let Ujj


= QRQH be a Schur decomposition of U1j. Find a unitary matrix P such


that PH RP


=D+N


, where D = diag( Vi1,


V227, .


Vkk ),


N is the strictly


upper triangular part of V,


mi, and min{


a -wl


: rE A(Vi,


and w A(ut)


> to12.


Replace Ujj by V


SJQP.


Update NBOJa


, and NSod to get NBnC


NSn""


Try the above decoupling procedure for all j such that 1


Step


Create a 2


diagonal block by merging Una and


UU.,iu


by a unitary


transformation.


Postmultiply


S by the same unitary transformation.


Let Sold


= QR be a QR-factorization of Sold


Then take


Snew,


U new


Q= ZQ,


where


is the matrix from Step 5.


Step 9:


Compute f(0) = hA -


]F and goto Step


1 until


f(0)


< tol


Example


If we apply the real


version of Algorithm 2.6.1


to the matrix A in


Example


with U0 = diag( 0,


0),So= I


tol= 10-6


and toll


= 10-4


then


after 9 iterations the upper triangular matrix Uo converges to


-0.88991
-0.73813


.72597


-1.11009


1.63393


2490


The eigenvalues


of U


are A


-1+


1.41421


- 1.41421i, 0 ].


we use the


QR method with double implicit shift


to A


with


the same tolerance,


then after


iterations we get a real Schur decomposition A = QRQT


w here


n -


_ F1*--_


Z =


SV


h*


=Q


_. **r -i


iJi


-1-











Next consider the following matrix A1,


which we obtain by perturbing the elements


of A:


0.001
-2.002


1.002
-2.003
-1.002


1.001
-1.001
0.001


If we apply the real version of Algorithm 2.6.1


tolerance, and


to A1


with So


the coalescing tolerance are the same, then after


iteration we get


a block Schur decomposition PTA1P = 0,


whereas if we use the QR method with


double implicit shift to QT


AiQ,


then after 5 iterations we obtain a real Schur form


VT(QTAIQ)V


=r.


Example


2.6.2


If the real version of Algorithm


is applied to the matrix A in


Example


with Uo = diag (


-1),


o = I, the tolerance, and the coalescing


tolerance are as in Example


, then after 10 iterations the upper triangular matrix


Uo converges to


-1.00104
0
0


-3.13124
0
0


-1.64159


-1.22


-0.99896


If we use the QR method with double implicit shift to A


with the same tolerance,


then after 6


terations we obtain a real


chur form:


-1.04199
0.00051
0


-3.46359
-0.95802
0


-0.69221
1.23323
0


Example 2.6.3 Consider the following matrices A, and P:


-80
236
-164


AQ=


= Q,


Ai=


= R,











If we apply the QR method with double implicit shift to


A with tol


= 10-6


then


after

A =


iterations we get a real Schur form STAS


2, 5, 1, 1,


= U. The eigenvalues of U


1]. Now if the real version of Algorithm


.6.1 is applied to


the perturbed matrix A


1 =-


A + P/5000 with Uo= U


= 10-4


So =


, and the


tolerance is


above, then after 3 iterations we obtain a block Schur decomposition


Q'AiQ


whereas


if we apply the QR method


with


STA\S, then after 10 iterations we obtain a real Schur form


double implicit

VT(STAS)V :


shift


Although the real version of Algorithm 2.6.1 works for all matrices A with a diag-

onal matrix formed from the diagonal elements of A, as the starting upper triangular


matrix U0, and So =


I, it has some disadvantages.


For example, if the size of the


matrix is large, then we need to take a very large coalescing tolerance compared to


= 10-4 through several iterations at the beginning.


lescing tolerance,


However with a large coa-


the sizes of some diagonal blocks of U become large compared to the


size of A, which undermines the main goal of this algorithm; namely keep the size of


the diagonal blocks of U as small as possible.


In Example 2.6.1, and 2.6.3 we noticed


that if by another method, a real Schur form A = SUST

can use U as the starting upper triangular matrix, and S


can be computed, then we


as the


starting orthogonal


matrix to obtain a block Schur decomposition of a perturbed matrix A + eE


very rapid convergence,


with


where E is an arbitrary matrix, and e is a small scalar.


the following table for different values of n we give the number of iterations


required


to obtain block Schur decompositions of A +eE by the real version of Algorithm 2.6.1


with Uo


, and So


=S,


and real Schur forms of ST(A + rE)S by the QR method


with double


cit shift,


where ST


AS = U is a real


chur form of the matrix A.


= t,


=U











number of iterations number of iterations for
size for our algorithm the QR algorithm with double shift

A + eE A + eE S (A +eEE)S
10 2 15 19 10-2
20 2 27 28 10-2
30 2 42 45 10-2
40 2 53 56 10-2
50 2 79 70 10-2


Example 2.6.1


If Algorithm 2.6.1


s applied to the matrix


with Uo


= diag(


9.5 + 16.4544


-9.5 + 16.4544


-19,


-9.5 16.45448i


9.5-


16.45448i),


where the elements of Uo are the uniformly distributed points on a circle,


which includes all Gerschgorin disks for


= I,


= 10-6


= 10-4


then after


13 iterations we obtain a block Schur decomposition


SHA


where


= diag


-2.25911


- 3.43367i, 1.83571


- 3.95


-2.25911


3.43367i,


-7.52532


1.83571


+ 3.9588


,7.37213 ),


is the strictly upper tri-


angular part of U. Figure 2.1 shows how the initial eigenvalues of Uo converged to

the eigenvalues of A. If we use the QR method with double implicit shift to A with


the above tolerance, then after 9 iterations we obtain a real Schur form QTAQ


Next consider the following matrix A1,


=R.


which we get by perturbing the elements of


4.0022
0.0068


Snn01


-4.9949
-3.9940
A Anna


0.0063


.9956


_9 QQ1i


3.0095
-4.9915
n nn09


-1.9959
-0.9942
> nnaR


1.0072
2.0080


-9 QQ090


= U,


D +














20r

15

10-

5

0

-5

-10

-15

-20"
-20



Figure 2.1.


-15 -10 -5 0 5 10 15


The path of convergence of the eigenvalues of the matrix A.


If we use Algorithm 2.6.1 to Al with Uo = U


S, o = S, the tolerance, and the coalescing


tolerance are as above, then after 4 iterations we get a block Schur decomposition


PHAIP


=9


whereas


f we apply


the QR method


with


double implicit shift


A1Q with the same tolerance,


then after 8 iterations we obtain a real Schur form


VT(QT


AiQ)V


=F .


Example 2.6.5 If Algorithm 2.6.1 is applied to the matrices











with


diagonal matrices,


= diag( 19, 3 +


-13, 3 -


16i ),


= diag( 7


E = diag(


+ 30i,


-32,


- 30i ), and fl = diag( 1, 0.5 +


0.8660i


-0.5+0.8660i


-0.5 0.8660i, 0.5 0.8660i )


as Uo, where the elements


of A, e,


and (2 are the uniformly distributed points on circles, which include all


Gerschgorin disks for B,


, D, and E respectively, So =


I, tol =


10-6


, and toll


, then after 16, 112, 12, and 11 iterations we obtain block Schur decompositions


B, C


respectively.


If we use the QR method


with


double implicit


shift, then after 6, 14, and 1 iterations we obtain real Schur forms of B, C, and D

respectively, whereas E diverges.


Example 2.6.6 If we apply


Algorithm 2.6.1


Xdiag( 1, 1, 1, 1,


3, 3, 0,


-5, 4, 5,


-4 )X-1


where X


= Pdiag(1,


, 5, 6, 7,


are orthogonal


matrices


with


= diag(


43.6939


41.5811+


13.3395i


35.4496+


5.3733i, 25.8996+34.9233i,


13.8658+41.054


0.5263+43.1676i


-12.8132+41.0548


-24.8470 +34.9233i


-34.3970 +25.3733i,


-40.5285+ 13.3395i


-42.6413


.5285-


13.3395i,


-34.3970-25.3733i,


-24.8470-34.9233i,


-12.8132-


41.0548i, 0.526


43.1676i


, 13.8658-41.0548i, 25.8996-34.9233i,


35.4496-25.3733i,


41.5811


-13.3395i), where the elements of Uo are the uniformly distributed points on a


circle, which includes all Gerschgorin disks for A, So = I, tol = 10-6


and toll


S10-4


then after 29 iterations we obtain a block Schur decomposition SHA


where


=D+N


, D = diag(


3, 3,


0, 4,


and N is the strictly upper triangular part of U.


If we use the QR method


-1, 1, 1, 1, 1,


, 5),








68


the tolerance, and the coalescing tolerance are the same, then after 4 iterations we


obtain a block Schur form PHAIP


double implicit shift to QTAIQ,


AzQ)V


whereas if we apply the QR method with


then after 19 iterations we obtain a real Schur form


=rF.


Example 2.6.7 If Algorithm 2.6.1 is applied to the matrix C in Example


Uo = diag( 139.50, 132.6969+42.9537i,


1.3 with


112.9537+81.7022i, 82.2022+112.4537i, 43.4537


+132.1969i


0.50 + 139.00i


-42.4537 + 132.1969i,


-81.2022 + 112.4537i,


-111.9537 +


81.7022i


-131.6969+ 42.9537i,


-138.50,


-131.6969 42.9537i,


- 111.9537


1.7022i,


-81.2022-

-112.4537i


112.4537i,


112.9537


-42.4537-132.1969

- 81.7022i, 132.6969


i, 0.50-139.0i, 43.4537-132.1969i, 82.2022

- 42.9537i ), where the elements of Uo are


the uniformly distributed points


on a circle, which includes all Gerschgorin disks for


I, tol


= 10-6


and toll


= 10-4


then after 26 iterations we obtain a block


Schur form SN


CS=


Figure 2.2 shows how the initial eigenvalues of Uo converged


to the eigenvalues of C.


If we use the QR method with double


mplicit shift to C,


then after 27 iterations we obtain a real Schur form QTCQ =


Next consider the


perturbed matrix C1


= C + P/1000,


where


the matrix


from


Example 2.1.3.


If we apply


Algorithm 2.6.1


to C1


with


the tolerance,


coalescing tolerance are the same, then after 2


terations


we obtain a block Schur


form pHc1p = f,


whereas


f we apply the QR method with double


implicit


shift to


QTCtQ, then after 27 iterations we obtain a real Schur form VT(QTC1Q)V


In the following table for different values of n we give the number of iterations

required to obtain block Schur decompositions of A by Algorithm 2.6.1 with diagonal


= '*,


, So =-


= U,


= S


VT(QT


I


t


I
































-150'-
-150


Figure 2.2.


-100 -50 0 50 100


The path of convergence of the eigenvalues of C.


where ag,


are real numbers and


< 1,


the tolerance


coalescing tolerance toll


= 10-4:


number of iterations number of iterations for
for our algorithm the QR algorithm with double shift
20 25 27
40 44 58
60 64 79
80 75 110
100 80 134
120 93 178
140 103 193

Algorithm 2.6.1 can be applied to all matrices A with So = I, and diagonal ma-

trices as Uo, where the elements of Uo are the uniformly distributed points on circles,


= 10-6











convergence.


To prevent it,


we take one fourth of the stepsize from iteration 1


iteration 2 or 4.


If the size of the matrix is very large, then we may have to take one


fourth of the stepsize from iteration 1 to iteration 6 or 10.


the following.


The reason of doing this is


If all diagonal entries of the upper triangular matrix at the beginning


become real,


then


the approximate eigenvalues are clustered on a segment of the


real line.


When two elements are close to each other, but not within the coalescing


tolerance, then in the updating process we are dividing by a small number (


which


is their difference


is the case when the size


If this happens in almost all iterations at the beginning,


of a matrix is greater than 10,


which


then it brings instability in


the updating process

not work very well,

the given matrix is u


. In fact for that reason the real version of Algorithm 2.6.1 does

when a diagonal matrix formed from the diagonal elements of

ised as the starting upper triangular matrix.


2.7 Parallel Processing in Eigencomputations

We already discussed the derivation of five algorithms in the previous sections to


find al


eigenvalues and corresponding eigenvectors of a matrix.


To implement each


algorithm, we need either a matrix of approximate eigenvectors,


or all approximate


eigenvalues, or


both.


But except for the last one,


we have not been able to come


with


ways


to choose either


a matrix


approximate eigenvectors,


or all


approximate eigenvalues for the starting guess. So


n this section, we will only


discuss


the implementation of the last algorithm in a parallel computer.


Most of the algorithms,


which are available to find all eigenvalues of an unsym-


metric matrix are serial algorithms. For example, consider the famous QR algorithm,


which is used to find al


eigenvalues of an n x n unsymmetric matrix


To make it












form.


The algorithm, which reduces an n x n matrix A to a Hessenberg form H


the algorithm which uses double implicit shift to H are both serial algorithms.

The motivation behind the derivation of various algorithms in this chapter is to

find all eigenvalues of an unsymmetric matrix by making use of a sensitivity result

for eigenvalues and eigenvectors. Then try to determine whether an algorithm can be

implemented in a parallel computer. We already mentioned that except for the last

algorithm, all other algorithms described in this chapter have some shortcomings. So

we will give a sketch, how and where to modify the last algorithm such that it can be


implemented in a parallel computer.


For clarity, we will assume that n = rp,


where


p is the number of processors.


We will also use the notation Proc(i) to denote the


ith processor.


In Step 1, with p processors,


we can accelerate the merging of diagonal blocks in


the following way.


For a fixed


< NB


and NB


p, we can find aj =


min{ ja wI


: cr E


A(Uii), and


wE A(Ujj


=i+l


,...,NB,


in separate


processors.


If aj


< toll


, then we can merge Ujj


with Ua in the increasing order of


If NB i > p, then we have to use some processors more than once.


Proc(() will


compute ct,


where l C


+i :


p: NB.


Parallel pro


cessors


can also be used to arrange


diagonal


blocks of U


the decreasing order of


sizes.


[(] denote the smallest


integer such that x


If U has k blocks along the diagonal,


then not


more than


swaps among the diagonal blocks are necessary to arrange them


in the decreasing order of


times to finish


sizes.


swaps.


use m processors, where m


In a single processor,


we have to execute


But in a parallel computer with p processors,


pifp


and m =


< p to finish


k


we can
:(k- 1)


I I I I


_-- I 1


v


c








72



there are two processors, then we can swap the elements of NS in the following way:


a 1
2
a3
04
a5


a2
a'
a4
a3
a5


a2
a4
al

a3


a4
a2
Cs
a
03


04
a5
02
a3
a'


Here we use


one processor to swap blocks enclosed by a brace on the right, and the


other processor to swap blocks enclosed by a parenthesis on the right.


Each processor


does


comparisons, and


In Step


swaps.


to compute the strictly lower triangular matrix G(


we can use


parallel processors.


Suppose A


= [A1,.


S., Ap],


= [ ., S], and U


are column partitionings of A,


S. and U


respectively, where each block column has


width


To compute B


first we find D =


SU, and then find B = AS


- D.


Since Uj


[= U, .


,U-1


,,o. .T


SCrxr


so Dk


-z


Thus each Dk can be computed in a separate


processor.


nce U is upper


triangular,


so D, involves much more work than


To load


balance,


we assign


Proc(


) the computation of


: n= SU(


, :p : n)U(l


as suggested in [Gol90, page 291].


Next with


SJ


, Sij


e Crr


Bk = ASk -Dk


- Dk,


k = 1,... ,p.


So p processors can be used to compute B,


where D can be overwritten by


we mentioned


n Section


to compute G;,


we need entries below G1i on the jth


block column;


narnel


Gi+l, ,GNBj


entries on


left of


on the ith


*.. A


= I[U, t ]


.. .


p:n,


I _I 1 *


= SUk


Sj Ujk,


: p :n),


:n)=
































Figure 2.3.


The direction


, in which the elements of G(


U) are computed, when the


number of processors are greater than half of the total blocks.


Suppose p


Then


2NB
.2


we use


Proc(1)


to compute GNBj,


- 1, ,


- 1 in the increasing order of the column index j, and the other p 1 processors


to compute the remaining elements of


G(S, U) in


the decreasing order of the row


index i


as we explain below


. We use Proc(2) to compute Gn,


Proc(3) to compute Gi2,


NB 1,...,3.


Continuing the assignment in this way,


we use


Proc(p)


to compute Gip-1,


-NB -


1, ..., p.


NB -


then


will continue this


ascending order of assignments to those processors except Proc(1).


That means


, Proc(2)


to compute Gip,


= NB ,..,p+ 1,


Proc(3) to compute


= NB-


and in


1, ...


this order if necessary


Proc(p)


to compute


Gi2p-2,


NB-1,...,


p- 1. Pictorially it can be presented by arrow heads, where


,...,2,


Gip+l,


,p+





























-1p


Figure 2.4.


The direction, in which the elements of G(S, U) are computed, when the


number of processors are less than half of the total blocks.


Next, suppose p


[NBt
+ 1.
2 JI


This time we will assign each processor to compute


the elements in the decreasing order of the row index


Proc(1) to compute G1,


S. 2,


Proc(2) to compute Gf2


NB,


signing the columns


. S


in this


order,


we use


Proc(p)


to compute Gip,


= NB


Next


we will repeat


the order of


assignment as


in the


strategy.


Proc(1)


to compute Gip+1,


, Proc(2) to compute Gip,+2


, *,


i = NB,... ,p+3, and in this order Proc(p)


to compute Gip,


NB,...,


2p+l


. If necessary,


we will repeat it.


Pictorially it


can be shown by arrow heads,


where the elements are computed towards the arrow


heads as in Figure 2.4.


both strategies,


we can not start all processors simultaneously.


the first


ft ratieoev


hofnrr


Prnctfi.


= ..... n. commutes an element G;;,


we need to make


.,p+


s











algorithm to compute G(S, U) is a complicated function of the block
5
all blocks are 1 x 1 matrices, then the algorithm requires -n3 flops.
3


sizes


of U


In Step 3,


to evaluate f(t) for different values of t, we can use parallel processors.


To compute the lower triangular matrix L,


where Li,


= tGij


we can use one of the strategies discussed earlier to compute


G(S, U), depending on


the number of processors p.


Assign Proc(s),


to take L


where


0, 1,..., and j


We can use the ideas discussed in the analysis


of Step 2,


to find matrix products.


Next we will give


a sketch to find the inverse of


a matrix in a distributed memory multiprocessor,


ring.


where processors are set up in a


An efficient way to find the inverse of a matrix A is to find the LU factorization


of A with partial pivoting first. Let piv(1


- 1) = 0 be a zero vector.


Consider the


following node program,


which is due to G.


H. Golub and C. F


Van Loan,


where (


is the identity of the


th processor:


pivot = piv((l


:n-1)


L = length(col)


while q


if j = col(q) A j


{ Find the permutation index piv(j), and Gauss vector A(j + 1 : n,j).


Determine k with j


so IB(k,q) =


B(j : n, q)


pivot(q)


,1: L)


B(k, 1


B(j + 1 :n, q)


= B(j +1 : n,q)/B(j,q)


Send pivot(q), B(j + 1: n, q) to processors on the right.


{ Update local columns.


= I,


= .k;


pa+(,


:n,~


: p :n;













j+1; q=q+1


else


Receive


piv(j),


and A(j + 1


: n, j) from the left, and if the processor on


the right is not the processor which computed A(j + 1


:n,), and j is


less than the last column


ndex of the processor on the right,


then send


piv(j), and A(j + 1 : n,j) to the right.


{ Update local columns.


B(j, 1


B(piv(j)


: L)


B(j +1


:n,q


: L)= B(j + 1


:n,q


: L)- A(j +1


: n,j)B(j,q


j=3+1

end

end

After each processor in a p-processor ring finishes the execution of the above code,


we will get the LU factorization of A; Proc(4) will house A(1


array B, a

A is of the


nd piv("(


form PA = LR.


1) in a local vector pivot. Thb

After simplifying, we get A-1


: n) in a local


e above LU factorization of


SL-' P, and therefore


to obtain A


1 we need to determine L-1


and R-1L-'


First


we will discuss


how to find the


inverse B of a lower triangular matrix L,


where L is overwritten by


Consider the following node program:


D = [L(k


: R, ik)],


col=


j=1


:p:n


length(col)


while j


: n, (


; N=


< n


: L) +-*+













Update local columns.


if q >


1 :q -1)= D(j


D(j+1 :n, 1


,1:q


:q-1)


- D(j + 1


- 1)/D(,q)


= D(j +1 : n, 1 q -1)

q)D(j, 1 : q -1)


end

end

D(j,q) = 1/D(j,q)


D(j + 1 : n,q) = -D(j + 1 : n,q)D(j,q)

end


+l1


else


> col(l)

Receive L(j : n,j) from the right, and if the processor on the left is


not the processor which houses L(j


: n,j), and j is greater than the


first column index of the processor on the left, then send L(j : n,j)

to the left.


Update local columns.


D(j, 1

ifj <


1 q)/L(j,j)


-\ s- I -1 \ I -




: q) =


:n,


rr I r











if col(q) + p= j


q+1


end

else


j+1


end

end

end

If each node in a p-processor ring executes the above code, then we will get the


inverse of L, and upon completion Proc(() will house L(k


local array


: n in a


: n,k) for k =


Our next move is to find the inverse B of an upper triangular matrix


R, where R is


overwritten with B.


A node program can be structured as follows:


D = [R(1


=7n


col =


= length(col)


q=N


while j > 1

if j = col(q)


Send D(1


q) to processors on the right.


Update local columns.

a

q+1


: N) = D(j, q + I


:N)/D(j, q)


: k, k)],












D(j, q) =


1/D(j, q)


: j- 1,q)= -D(1 j- 1,q)D(j,q)


j -1


< col(N)


Receive R(1


: j, j) from the left, and if the processor on the right is


not the processor which houses R(1


:j,j), and j is less than the last


column index of the processor on the right, then send R(1


:j,j) to the


right.


Update


local columns.


: N) = D(j, q


N)/R(j,j)


-1,q


: N)= D(1


:j -1,q :N)- R(1 j-


1,j)D(j,q: N)


end

j=j-


if col(q) -


q=q-1

end

else


j-1











If each processor in a p-processor ring executes the above node program,


upon completion Proc(() will house R(1

In the LU factorization of A, R can be s


then


: k, k) for k = : p : n in a local array D.

tored in the upper triangular part of A, and


the unit lower triangular matrix L can be stored in the strictly lower triangular part


of A.


Then we can use the above code to find the inverse of the upper triangular part


of A


, and after a slight modification of the node program,


which finds the inverse


of a lower triangular matrix,


we can use it to find the inverse of the strictly lower


triangular part of A.


Since A-1


= R-RL-'P


, so our next goal is to create a node


program to find the product R-'L


1. Consider the following node program:


: p :n)


1; col


length(col); q = 1


while j

if j


= col(q)


Send B(1


:j, q) to processors on the left.


{ Update local columns.


1 :q -1)


+ B(l:j,q)B(j,1 :q-1)


aq-l


dimensional row zero vector.


j+1> col()
j > colqi)


= A(1 : n,


: p :n;


j-1











the left.


{ Update local columns.


-1, 1 :q)


+A(1 :j,j)B(j,:q)


j+l


if col(q) + p


q+1


end

else


j+l


end

end

end

After each processor in a p-processor ring executes the above code, we will obtain


the product


R-1L-1


which overwrites A, and upon


: n) in a local array


completion Proc(()


Since A is overwritten by


R-1L-


will house


, and the


permutation matrix P is encoded in the (n 1) dimensional vector piv, so


(R- L-1)P can be computed in the following way:


for k


piv(k)


A(1 : n, k)


A(1 : n, piv(k))


end

end


Next


, if we store the upper triangular matrix U in the upper triangular part of a


:q)=


j,1


: n,(











product G(


U)U in a more efficient way. After computing UG(S, U), S-1


S-1A


G(S,U)U,


we can


take advantage of


parallel


processors


to compute U(t)


U + t(UG(


S-'AS


U)U).


Next after Z(t) = A


S(t)U(t)


is calculated


we can find


trace(Z(t)HZ(t)) in the following way.


Let x(1


= 0 be a zero vector.


To compute ZH


we use the


following procedure:


for i


= 1 r


y = y + [B(1


: n,i)]HB(1


: n,i)


end

Here each processor can execute the above code simultaneously, and upon com-


pletion Proc(t) houses xz() in a local variable y.


Then


f(t) = v/1 +


t..+ XL.


In Step 4,


f NB m


< p, then we can not use all processors to find i


and iu.


Suppose NB


p. Using a similar technique, as d


1, we can get an array k such that ipk(i)


II k(i + 1)


described in the analysis of Step


holds.


To determine


I = min{


.,NB


--m--1


, Re (ri)


can use p processors as follows.


-m]
2


Define


/tk(1:s+1)
Ik(s+1:2s+l)


/lk((p-2)s+l:(p-1)s+l
ILk((p-1 )s+1:NB-m)


dk(l:s+l)
dk(s+1:2s+1)


dk((p-2)s+l:(p-l)s+l)
dk((p-1l)s+1:NB-m)


we


=Z


B


y= x


:


S --


,U) +


-U-G(


f(t) =


:n, 1 +((


/tk(i+1)


- ik(i)


dk(i) dk(i+l)


>0&












vf= ()
for i =


L= length(f); r(1: L -1)= 0


1 :L


- gi+x


/-=1;


for i=1


if Re(ri)


> IOAr I


end

end

ifj =0


6=0

else


1+j


end

If each processor executes the above code, then upon completion Proc(C) houses


-= and


Next the desired minimum is u(0o)


= min{


1,...


,p }, and if the corresponding v(0o)


then define i


= t 1 + min{k(v($o)),


k(v(Co) + 1)}, and iu = t 1 + max{k(v(o)),


k(v(o) + 1)}


In Step


we can use parallel processors,


as described in the analysis of Step


and Step 3 to obtain Unew


Snew


Z = Snew


Unew (


Snew


In Step 6, if m


then we can assign a separate processor to implement the code


to each diagonal block Ujj,


< m.


If m > p, then assign Proc(t) to implement


= b.


< p,


u(t),











a = col(q); Implement Step 6 to U00

end


In Step


we can use parallel processors to find a QR factorization of S.


Here we


will discuss how to find a QR factorization of a matrix A in a distributed memory


multiprocessor,


where processors are interconnected in a ring.


Let b(1


:n-l)= 1 0


be a zero vector.


C.F


Consider the following procedure, which is due to G. H. Golub and


Van Loan:


B = A(1 :n,


1; col =


while q

if j


{ Find Householder vector A(j + 1


B(J: n, q) j2


if a = 0


B(j : n, q) = 0;


quit


else


f B(j, q)


/3=


aB(j, q)/ B(j, q)j


else



end


B(j + 1


: n,q) = B(j + 1


B(j,q) = -o


SI I


:n-1)


length(col)


= col(q)


Sn,j).


q)+ +- f;


9


/9 = B(j


: nq)/













B(j :n,q+ 1


: L)= B(j


: n,q+ 1


: L) + c(q)vvHB(j :n,q + 1


j+1; q=q+1


else

Receive b(j), and A(j + 1: n,j) from the left, and if the processor on the

right is not the processor which formed A(j + 1: n,j), and j is less than

the last column index of the processor on the right, then send b(j), and


A(j + 1


: n,j) to the right.


{ Update local columns.


[ A(j+ 1
:n,q: L)


:n, j)
=B(J


: n,q: L) + b(j)vvHB(j


: n, q : L);


j+1


end

end


After each processor in a p-processor ring executes the above code,


QR factorization of A, and Proc(() will house A(1


and b( : p : n 1) in a local vector c.


we will get a


: n) in a local array


The upper triangular part of A is overwritten


by the upper triangular part of R, and components j + 1: n of the jth Householder


vector are stored


A(j +


: n, j),


Here we need an explicit form of the


unitary matrix Q.


We find Q using the following procedure, which uses Householder


vectors A(j + 1


:n,j)


where j


< n, and the vector b to form Q explicitly, and zeros


subdiagonal elements of A.


Let Q = I,:


B = A(1 :n,


I=Q(1


:n-1)


: n, (


c=b((


: ni,(











Send c(q), and B(j + 1 : n, q) to procesors on the left.


{ Update local columns.


B(j + 1


: n,q)


: n,q)


: n,q : L)


= I(j


:n,q: L) + c(q)vvHI(j


: n,q


:L); j= -1


else


Receive b(j), and A(j + 1


: n, j) from the right,


and if the processor on the


left is not the processor which houses A(j + 1


: nj),


and j is greater than


the first column index of the processor on the left, then send 6(j), and

A(j + 1 n,j) to the left.


{ Update local columns.


A(j + 1


Sn, j)


:n,q


= I(j: n,q


: L)+ b(j)vvHI(j


:n,q


j-1


if col(q) p = j

q= q-1

end

end

end

If each processor in a p-processor ring executes the above code, then upon com-


pletion


Proc(5)


will house


: n in a local


array


B, and


: n) in a local array


t we can use parallel processors,


as described


n the analysis


of Step


to find the product Unw'


QHZQ.


Finally in

3 to evaluate


tep 9, we can use multiprocessors, as described in the analysis of Step


f(O) = IA- ZIF.


B(j + 1


: L);

















CHAPTER 3
EIGENVALUES OF SYMMETRIC MATRICES


3.1 Diagonalization of a Symmetric Matrix using Armiio's Stepsize


be an


nx n


symmetric matrix.


Suppose


the eigenvalues of


are all


distinct. In this section we will find an iterative method to compute a diagonalization


= QAQT


where A


= diag


l, A2, *.., An ), and


s an orthogonal matrix such


that the diagonal elements of A are the eigenvalues of A, and the columns of Q are


eigenvectors of A.


Consider the following algorithm to compute the eigenvalues, and


corresponding eigenvectors of a matrix


Detail is given in [Hag


Spage 304 14],


or in Chapter 1:


\new

new
a


(yoldT


Aold
I


to n,


" AxPtd
n ew
~ ^a


to n,


(3.1)


Y new


(Xnew -1


Algorithm in (3.1) can be rewritten in the following way:


A new
Xnew


where Diag(M) = diag( mn,


Diag((Xold)


Xd (I + F (X


m22, (,0

S0


AXold),
)d Anew))


(3.2)


mnn ), and


,A new) =


((Xold


-1AX od)


f/j (Xold


TTM"9












In (3.2),


if we put


ynew"


= I + F (Xd


, A"ew), and AOd


(xo")


AXOt"d


then it


reduces to the form:


Diag((y"d)-1
xold y new


Aotdyoud)


Hence


n each iteration


, we can update the eigenvalues,


and corresponding eigenvec-


tors in the following way:


A new

y new
AT


(yold)


old yold


Diag (A "),
I + F(Anew)
Sold ynew


(3.3)


where


fi (Anew)
Jii A


new
a


new
aL


fori


new
- aii


updating process in (3.3) does


, in the update Anew


S(yold)


not restore the


Aotdyyotd


symmetry of the given matrix


In each iteration to restore the symmetry


of A, and the orthogonality of X


we do the following. Let ynew,


= Q~uR"f"r be


aQR


factorization of Y n"1


.Now


if we take An"f


(QoldjT
= Q


AoldQold


old snewr


then Ane"


will be symmetric, and X


new


will be orthogonal.


With these modifi


cations


(3.3) becomes:


Anew

A new

XTneux


(Qold T


AoldQold


Diag (A"new),
I + F(Anew),


Qn""


Rnew


Xold new


( QR factorization )


(3.4)


where


F AneW\


-) new
j -yUCw
"""" / -


for t


Anew


ynew


,c ( Anew "












the increment


F (AnefC)


may


too large.


However


with


a large


increment,


may


likely


have instability


updating


process,


algorithm


may


verge.


To restore the convergence of the algorithm,


to have a steady


change


value of y"e"w


we need


a small increment in


each


iteration.


achieve


this,


we redefine each


a positive parameter, and


iterate, and introduce a small positive parameter.


define Z(s)ne"


Let s


= I, and


= I + sF (Ane).


_ ynew


, the matrix generated


(3.4)


. Define


IG(s)| IF,


where


= lotr ((


s)old)


lotr(B) is a strictly lower triangular matrix


formed from the strictly lower triangular part of B. Suppressing the superscripts in


Z(s)new


we have


= I + sF(A)


= lotr(


(s>)-AZ(


Hence f(O)-=


ljlotr(A)


IF, and f(1)


Ilotr (Y-


1AY)


With a


starting guess Xo, if the algorithm in (3.4) converges,


then we must have


< f(O)


. As


in Chapter


here our goal is to find an s,


for which


holds.


To this end


, we use Armijo's rule from optimization theory.


Armijo


rule, we determine


s in the following way.


Evaluate f(


s) at


s=l, 1
'2


stopping when


- f(o).


To use the above inequality, f(s) must satisfy


f'(0) =


-f(O).


ce our next goal is


to evaluate


f'(O), when


f(s) = IIG(


s)llF with G(


s) = lotr(


s)-IA


(s)).


Before we


f'(0), we will prove the following basic result.


Lemma 3.1.1


E IR7x n is a symmetric matrix, and B


e Rnxn"


a skew symmetric


matrix, then AB BA is a symmetric matrix.


(0)new


(1),new


G(s),


_ __


_


I_ 1 _











- A'BT


(-B)A -A(-B)

AB-BA


Hence AB BA is a symmetric matrix.


With the above definition of


f(s), now we are ready to find


f'(O), and will show


f'(O) =


- f().


Theorem 3.1.1


e RnXn


a symmetric matrix.


Suppose


its eigenvalues


all distinct.


Define


+ sF(A),


= lotr ((Z(


where


fj A) =


ajj -- a


lotr(B)


a strictly lower triangular matrix formed from the strictly lower triangular


part of B.


Then


f'(0) =


-f(O).


Proof of Theorem 3.1.1


Using the result (2.18) of Lemma 2.3.1, we have


d
f(O)f'(0) = trace G(O)T G(s
ds


Differentiating G(s


) s=o


) with respect to s, and then evaluating the derivative at s = 0,


we obtain


d
-G(s) I= o
ds


= lotr(


d
s) 1=o (0Z(o)-A Z(O) + Z(0O)A- Z(s)I
ds


Differentiating


s) with respect to


d
s, we have -Z(s) = F
ds


d
SHence
ds


s)1So = F














Since G(0) = lotr(Z(O)-lAZ(0)) = lotr (IAI)


= lotr(A), so (3.5) becomes


f(0)f'(0) = trace ((lotr(A))T lotr (AF


- FA)).


(3.8)


Because A is symmetric, so (lotr (A))T


= uptr(A)


, where uptr(A) is a


strictly upper


triangular matrix formed from the strictly upper triangular part of A.

reduces to


Hence (3


f(0)f'(0) = trace (uptr (A) lotr (AF


-FA)).


(3.9)


Next, for


aj-
s3


-a"


Since


symmetric,


fi= fji.


That is F is a skew symmetric matrix.


Hence by Lemma 3.1.1,


-FA


is symmetric.


Now


-FA=


all- aii
n

Si=l2
ittz


n a

i=a1 22
ig2


Z2iail


liai2
--a


E-Q


n
--2 a

;=2a11


n-1l



n-l

i=1


- aii


ann --ii

a2iain


ann


- aii


n


a aliain

all ali

a2iain


i=l 22
t12


n
v -^ u;1


niail
-- a


n-1

+ n-
i-1- ana -


22 ii


n-l1

i=1 ann


tiai2
-a


- a i


Therefore (3.9) gives:



f(0)f'(0) =


akiail


- aji


akiail


--a


trace


akiai2


na


99, -a M,


2iail
-a


akiai2


arj -- aQ_ I


n a
i=211l


2 -


liai2


aliain


-- aii


- aii


n-1 2
i=la-
:--1 a~n


aniai2




















an-ln


n-in-1 -


n-1
t" afni

i=1 ann


Gin-1
-- an


where X


are numbers.


Simplifying the right hand side of the above expression, we


f(0)f'(0)


akiail


all -- aii


1=1


- aii


k=3


a22 ii


akiai2
akk ai

a22a21


12 (


- a22
a33a32


+ -- n-ln


aniain-1


-In--1


a21a11 + .Q a,
a22 -- all all


- ann
no, n2


n--1



+ 2
i=l ann

anlal
ann -


S,_II
-- aji


1
i)


afn2022


22 a33


a33 a22


)+ + a 2n (

ann-ln-ln-l


22 -- nn


--a22
1

-<33


n-1l


- all


-- 22


- a22


1
n-2n-1 n-2n n-In
an-2n-2- ann


an-ln-i -- n-2n-2


- an-2n-2


0n-2n-2 n-ln-1


1
an -- n-n-1
Onn --Cln--ln1


f (0) = -f(O), provided


Hence


Theorem 3.1.1 implies that Armijo's rule can be applied to the algorithm


-+a23 (a


Sann-an-1.
+" "-an-ln --
\Un-ln-1 --nn


a22 -- 33


a22 -- all


S+ +


E
*K

ann


aniain-1


akiail


akiai2


(


a11


n anl


232a22


~a12a13l23 --
11n


--On-I


an-ln-i -ann


- f(O)2











Algorithm 3.1.1


(Symmetric Diaconalization)


Given a symmetric matrix A


e R"x"


whose eigenvalues are all


distinct,


an orthogonal


matrix Xo,


a tolerance


greater than the unit roundoff, the following algorithm computes a diagonalization


= A, and A is overwritten with the diagonal matrix A.


Step 0:


Take X


= Xo.


Compute Ane" = XTAOldX


f() = II lotr (A) hI.


Step 1:


Construct F


= (fij), where


ij -


= IIG(s)IIF, where G(s) = lotr ((I + sF)-<


A(I+ sF))


Evalu-


ate f(s) at s = 1,


* -*, stopping when


) f(O).


(3.10)


Let t be the first value of


s for which (3.10) holds, and let Y


= I+tF


Let Y


= QR


be the QR factorization of Y.


Step 3:


Update A, and X


as follows.


Anew = QTAodQ, and Xne"


= XoldQ.


Step 4:


Evaluate


f(0) =


lotr (A)


Go to Step 1


until


f(O) < tol, where tol is


the tolerance for the desired accuracy of the eigenvalues.


Example 3.1.1


If Algorithm 3.1.1 is applied to the matrix


with X0


= I, and


= 10-6


, then after 6 iterations we obtain a diagonalization


r'T A r S S S n -


< 1-


ajj aii











which we obtain by perturbing the elements of A:


A =


If we use Algorithm 3.1.1 to A1


2.002
-1.001
3.003
with Xa


-1.001
1.001
2.002


3.003
2.002
-1.004


and tol


= 10-6


then after 2 iterations


we obtain a diagonalization YTA1Y


= A, whereas if we apply the QR method for a


symmetric matrix with single


mplicit shift to XT


A1X


then after 3 iterations we get


a diagonalization ST(XTA1X)S = D.

Example 3.1.2 If Algorithm 3.1.1 is applied to the matrix

B = Qdiag(1,2, 3, 4, 5, 6)QT,


where Q is an orthogonal matrix, with Xo = I, and tol =


10-6


, then after 7 iterations


we obtain a diagonalization XT BX


= diag( 5, 6, 2,


, 4, 3), whereas if we apply the


QR method


a symmetric matrix with


single implicit shift


to B,


then after


iterations we get a diagonalization PTBP = diag


Example 3.1.3 If Algorithm 3.1.1


6, 5, 1, 4, 2, 3).


applied to the matrix


with Xo

xTCX.


- 10-6


then after


7 iterations we obtain a diagonalization


If we apply the QR method for a symmetric matrix with single implicit


shift


to C,


then


after


iterations


we get


a diagonalization


QTCQ


Next


consider the following matr


which we obtain by perturbing the elements of C:


6.004
4.006


4.006
4.007


2.007
1.004


2.006
2.009


-1.002
-2.003


3.007
-3.004


= X


-D




Full Text

PAGE 2

$33/,&$7,21 2) (,*(19$/8( 6(16,7,9,7< $1' (,*(19(&725 6(16,7,9,7< ,1 (,*(1&20387$7,216 r %< 385$1'$5 6$50$+ $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$

PAGE 3

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

PAGE 4

7$%/( 2) &217(176 $&.12:/('*(0(176 /,67 2) ),*85(6 $%675$&7 &+$37(56 ,1752'8&7,21 (,*(19$/8(6 2) 816<00(75,& 0$75,&(6 %ORFN 'LDJRQDOL]DWLRQ RI D 0DWUL[ $ 'LIIHUHQWLDO (TXDWLRQ $SSURDFK WR (LJHQFRPSXWDWLRQV $ 'LIIHUHQWLDO (TXDWLRQ $SSURDFK WR (LJHQFRPSXWDWLRQV ZLWK $UPLMRfV 6WHSVL]H &RQYHUJHQFH RI %ORFN 'LDJRQDOL]DWLRQ RI D 0DWUL[ %ORFN 6FKXU 'HFRPSRVLWLRQ RI D 0DWUL[ $Q $OJRULWKP IRU %ORFN 6FKXU 'HFRPSRVLWLRQ RI D 0DWUL[ 3DUDOOHO 3URFHVVLQJ LQ (LJHQFRPSXWDWLRQV (,*(19$/8(6 2) 6<00(75,& 0$75,&(6 'LDJRQDOL]DWLRQ RI D 6\PPHWULF 0DWUL[ XVLQJ $UPLMRfV 6WHSVL]H %ORFN 'LDJRQDOL]DWLRQ RI D 6\PPHWULF 0DWUL[ XVLQJ $UPLMRfV 6WHSVL]H &21&/86,21 5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+ P

PAGE 5

/,67 2) ),*85(6 7KH SDWK RI FRQYHUJHQFH RI WKH HLJHQYDOXHV RI WKH PDWUL[ $ 7KH SDWK RI FRQYHUJHQFH RI WKH HLJHQYDOXHV RI & 7KH GLUHFWLRQ LQ ZKLFK WKH HOHPHQWV RI *68f DUH FRPSXWHG ZKHQ WKH QXPEHU RI SURFHVVRUV DUH JUHDWHU WKDQ KDOI RI WKH WRWDO EORFNV 7KH GLUHFWLRQ LQ ZKLFK WKH HOHPHQWV RI *68f DUH FRPSXWHG ZKHQ WKH QXPEHU RI SURFHVVRUV DUH OHVV WKDQ KDOI RI WKH WRWDO EORFNV ,9

PAGE 6

$EVWUDFW RI 'LVVHUWDWLRQ 3UHVHQWHG WR WKH *UDGXDWH 6FKRRO RI WKH 8QLYHUVLW\ RI )ORULGD LQ 3DUWLDO )XOILOOPHQW RI WKH 5HTXLUHPHQWV IRU WKH 'HJUHH RI 'RFWRU RI 3KLORVRSK\ $33/,&$7,21 2) (,*(19$/8( 6(16,7,9,7< $1' (,*(19(&725 6(16,7,9,7< ,1 (,*(1&20387$7,216 %\ 385$1'$5 6$50$+ 'HFHPEHU &KDLUPDQ :LOOLDP +DJHU 0DMRU 'HSDUWPHQW 0DWKHPDWLFV ,Q LQ KLV SDSHU f%LGLDJRQDOL]DWLRQ DQG 'LDJRQDOL]DWLRQf :LOOLDP : +DJHU SUHVHQWHG DQ DOJRULWKP WR GLDJRQDOL]H D PDWUL[ ZLWK GLVWLQFW HLJHQYDOXHV 7R LPSOHn PHQW WKH DOJRULWKP ZH QHHG D JRRG DSSUR[LPDWLRQ WR WKH PDWUL[ RI HLJHQYHFWRUV )LUVW ZH SUHVHQW WKH GHULYDWLRQ RI KLV DOJRULWKP DQG WKHQ ZH VKRZ KRZ LW FDQ EH JHQHUDOL]HG WR GLDJRQDOL]H D PDWUL[ ZLWK PXOWLSOH HLJHQYDOXHV $ ORFDO TXDGUDWLF FRQn YHUJHQFH UHVXOW LV HVWDEOLVKHG IRU WKH JHQHUDOL]HG DOJRULWKP :KHQ D JRRG VWDUWLQJ JXHVV IRU WKH HLJHQYHFWRUV LV QRW NQRZQ +DJHUfV DOJRULWKP PD\ QRW FRQYHUJH 7R UH VWRUH FRQYHUJHQFH ZH PRGLILHG WKH DOJRULWKP E\ UHSODFLQJ GLDJRQDOL]DWLRQV E\ 6FKXU GHFRPSRVLWLRQV :H IRXQG WKDW LI XQLIRUPO\ GLVWULEXWHG QXPEHUV RQ D FLUFOH LQ WKH FRPSOH[ SODQH ZKLFK LQFOXGHV DOO *HUVFKJRULQ GLVNV IRU WKH PDWUL[ DUH XVHG WR DSn SUR[LPDWH WKH HLJHQYDOXHV DQG LI WKH HLJHQYHFWRUV DUH DSSUR[LPDWHG E\ WKH FROXPQV RI WKH LGHQWLW\ PDWUL[ WKHQ UDSLG FRQYHUJHQFH LV REWDLQHG :H SUHVHQWHG VRPH QRGH SURJUDPV ZKLFK DORQJ ZLWK VRPH PRGLILFDWLRQV WR WKH ODVW DOJRULWKP FDQ EH XVHG LQ D GLVWULEXWHG PHPRU\ PDFKLQH ZKHUH SURFHVVRUV DUH LQWHUFRQQHFWHG LQ D ULQJ 1H[W

PAGE 7

ZH SUHVHQW DGGLWLRQDO PRGLILFDWLRQV RI +DJHUnV DOJRULWKP DQG WKH JHQHUDOL]HG DOJRn ULWKP WKDW WDNH DGYDQWDJH RI PDWUL[ V\PPHWU\ DQG UHGXFH WKH JLYHQ PDWUL[ LQWR D GLDJRQDO PDWUL[ ZLWK HLJHQYDOXHV DORQJ WKH GLDJRQDO YL

PAGE 8

&+$37(5 ,1752'8&7,21 /HW $ EH DQ Q [ Q PDWUL[ :H ZDQW WR ILQG DQ LWHUDWLYH PHWKRG WR FRPSXWH D GLDJRQDOL]DWLRQ $ ;$;aO SURYLGHG LW H[LVWV ZKHUH WKH FROXPQV RI ; DUH HLJHQYHFWRUV RI $ DQG $ GLDJ $" $ $Q f ZKRVH GLDJRQDO HOHPHQWV DUH WKH HLJHQYDOXHV RI $ +HUH ZH ZLOO XVH D VHQVLWLYLW\ UHVXOW IRU HLJHQSDLUV WR GLDJRQDOL]H D PDWUL[ :H DVVXPH HLJHQYDOXHV RI $ DUH DOO GLVWLQFW ,Q WKH GHULYDWLRQ RI QHFHVVDU\ HTXDWLRQV WR GLDJRQDOL]H D PDWUL[ ZH QHHG WKH FRQWLQXLW\ RI HLJHQYDOXHV DQG WKH SHUWXUEDWLRQ WKHRU\ EDVHG RQ *HUVFKJRULQfV WKHRUHP :H JLYH WKH VWDWHPHQW RI WKH WKHRUHP EHORZ 7KHRUHP ,I $ e &Q[Q WKHQ ?^$f F 8" $ )XUWKHUPRUH LI D VHW RI N *HUVFKJRULQ GLVNV 'W DUH LVRODWHG IURP WKH RWKHU Q f§ N GLVNV WKHQ WKHLU XQLRQ FRQWDLQV SUHFLVHO\ N HLJHQYDOXHV RI $ 3URRI RI 7KHRUHP )RU SURRI WKH LQWHUHVWHG UHDGHUV DUH UHIHUUHG WR >$WN 7KHn RUHP SDJH @ /HW $ EH DQ Q [ Q PDWUL[ ZKRVH HLJHQYDOXHV DUH DOO GLVWLQFW DQG OHW ;a$; GLDJ $M $ $M $Q f EH D GLDJRQDOL]DWLRQ RI $ 7KHQ $[M $M;M ZKHUH

PAGE 9

;M LV WKH MWK FROXPQ RI ; *LYHQ DQ DUELWUDU\ PDWUL[ ( DQG IRU D VXIILFLHQWO\ VPDOO QXPEHU H ZH ZLOO VKRZ LQ WKH QH[W WKHRUHP WKDW WKHUH H[LVW FRQWLQXRXVO\ GLIIHUHQWLDEOH IXQFWLRQV $MHf DQG ;MHf ZLWK $Mf $DQG eMf [M VXFK WKDW $ H(f[MFf $MWf[MHf 7KDW LV $Hf[MHff LV DQ HLJHQSDLU RI $ H( 7KHRUHP /HW $ EH DQ Q[Q PDWUL[ $VVXPH WKH HLJHQYDOXHV RI $ DUH DOO GLVWLQFW 6XSSRVH ;aO$; GLDJ $c$$Q f LV D GLDJRQDOL]DWLRQ RI $ 7KHQ JLYHQ DQ DUELWUDU\ Q [ Q PDWUL[ ( DQG IRU D VXIILFLHQWO\ VPDOO H WKHUH H[LVW FRQWLQXRXVO\ GLIIHUHQWLDEOH IXQFWLRQV $Hf DQG ;MHf ZLWK $Mf $M DQG ;Mf ;M VXFK WKDW WKH IROORZLQJ PDWUL[ HTXDWLRQ KROGV $ H(f[MHf $ MWf[M^Hf 3URRI RI 7KHRUHP /HW ; $; GLDJ $L $ $M $Q f EH D GLDJRQDOL]Dn WLRQ RI $ /HW ( EH DQ Q [ Q PDWUL[ DQG H EH D VPDOO QXPEHU &RQVLGHU WKH PDWUL[ ;aO$ H(f; 7KHQ IRU L IRU L ZKHUH M \M([M \M LV WKH ]WK URZ RI ; /HW % GLDJ HN f DQG OHW & %'%aO WKHQ $L D HQ O f f f NILM f f f WI?Q f  f f f f N \r M f f f f IQ f f f f f f f f f N -O f r f f f f ff A f f fff f f f f§I NML f $f f f WIQ? f f P f f f f f E I f f f r-QM f f WIQQ $SSO\LQJ 7KHRUHP WR & ZH ILQG WKDW WKH MWK HLJHQYDOXH OLHV LQ WKH FLUFXODU GLVN Q ZLWK FHQWHU $M HMMf DQG UDGLXV U f§ A _MW_ SURYLGHG H LV VXIILFLHQWO\ VPDOO r L L

PAGE 10

DQG N LV FKRVHQ VR DV WR PDNH WKH MWK *HUVFKJRULQ GLVN GLVMRLQW IURP WKH RWKHU Q f§ GLVNV :H PD\ FKRRVH D YDOXH RI N ZKLFK LV LQGHSHQGHQW RI W LQ WKH IROORZLQJ ZD\ &KRRVH N WR KDYH WKH ODUJHVW YDOXH FRQVLVWHQW ZLWK WKH LQHTXDOLWLHV :H DFKLHYH WKH DERYH LQHTXDOLW\ LI N PP r M n9" D ID L  M r :LWK WKH DERYH YDOXH RI t WKH M WK GLVN ZLOO EH LVRODWHG IURP WKH RWKHU Q f§ GLVNV WKH UDGLXV RI WKH M WK GLVN ZLOO EH RI RUGHU DQG WKH SHUWXUEDWLRQ LQ WKH M WK HLJHQYDOXH ZLOO EH RI RUGHU H DV H f§! 1H[W ZH GLVFXVV WKH HIIHFW RI SHUWXUEDWLRQV LQ WKH HOHPHQWV RI $ RQ WKH HLJHQn YHFWRUV LQ LWV GLDJRQDOL]DWLRQ $V HDUOLHU OHW ;aO $; GLDJ $L $ $M $Q f EH D GLDJRQDOL]DWLRQ RI $ 7KH FROXPQV [? [ [Q RI ; IRUP D FRPSOHWH VHW RI HLJHQYHFWRUV :H GHQRWH HLJHQYHFWRUV RI $ H( E\ ;?Hf[Hf [QFf DQG HLJHQn YHFWRUV RI ;aO$ H(f; E\ WTHfAHf YQHf VXFK WKDW [Hf $7WHf 6LQFH $MHf YMHff LV DQ HLJHQSDLU RI =f DQG $Mf $M VR ZH PXVW KDYH 8Mf HM ZKLFK LV WKH M WK XQLW YHFWRU /HW 9MHf EH QRUPDOL]HG VR WKDW LWV ODUJHVW HOHPHQW LV )RU VXIILFLHQWO\ VPDOO H FODLP WKDW WKH M WK HOHPHQW RI 9MHf LV ,I QRW OHW 9LMHf IRU VRPH  L A M 6LQFH 'YMHf $MHfXMHf VR HTXDWLQJ WKH WK FRPSRQHQW RQ ERWK VLGHV ZH JHW Q NYLMWf H-IXYL$HfL /f WKDW LV Q $ HMIQYLMHf 1RZ OHWWLQJ H f§! ZH ILQG $ M f§ $ L A M D FRQWUDGLFWLRQ 6R IRU VXIILFLHQWO\ VPDOO 9MMHf 1H[W ZH ZLOO VKRZ WKDW WKH RWKHU FRPSRQHQWV RI 9MHf DUH DOO

PAGE 11

OHVV WKDQ 1F DV H f§! IRU VRPH M9 6LQFH _YMHf_ VR IURP f ZH JHW Q O$MHf n0.Hf H7?8 /HW _$MHf f§ $ M,$ $_ IRU L A M L Q WKHQ _XMHf_ 1H ZKHUH Q 1 PD[ 'f, L  M _$M $ 7KXV YLMHf DUH RI RUGHU F IRU L M L Q $OVR IURP f ZH JHW Q ;MHf ;LfYLMHf HIWM H A IXYLMHf 7KH VHFRQG WHUP RQ WKH ULJKW LV RI RUGHU H VR IRU L A M ] Q ZH KDYH mL Hf $ $ Ff +HQFH $MHf DQG [Wf DUH GLIIHUHQWLDEOH IXQFWLRQV RI DQG WKHLU ILUVW GHULYDWLYHV DUH FRQWLQXRXV 7KLV FRPSOHWHV WKH SURRI RI WKH WKHRUHP ’ 1RZ ZH DUH UHDG\ WR GHULYH WKH QHFHVVDU\ HTXDWLRQV WR ILQG D GLDJRQDOL]DWLRQ RI D PDWUL[ $ ZKRVH HLJHQYDOXHV DUH DOO GLVWLQFW /HW $M [Mf EH D VLPSOH HLJHQYDOXH DQG FRUUHVSRQGLQJ HLJHQYHFWRU RI $ *LYHQ DQ Q [ Q PDWUL[ ( DQG IRU D VXIILFLHQWO\ VPDOO OHW $MFf DQG ;MHf EH GLIIHUHQWLDEOH IXQFWLRQV RI H VXFK WKDW $Mf $ M DQG ;Mf ;M 'LIIHUHQWLDWLQJ f ZLWK UHVSHFW WR H DQG SXWWLQJ H ZH KDYH $[nM2f ([M $nf[M $M[nf f 6LQFH WKH FROXPQV [? r [Q RI 9 IRUP D FRPSOHWH VHW RI HLJHQYHFWRUV RI $ ZH FDQ H[SDQG ;MHf LQ WKH IROORZLQJ ZD\ Q L f

PAGE 12

6LQFH D VFDODU PXOWLSOH RI DQ HLJHQYHFWRU LV DJDLQ DQ HLJHQYHFWRU ZH FDQ WDNH EMMHf ZLWKRXW ORVV RI JHQHUDOLW\ 1RZ GLIIHUHQWLDWLQJ f ZLWK UHVSHFW WR H DQG VHWWLQJ F ZH KDYH Q [nf L O m 6XEVWLWXWLQJ f LQWR f ZH REWDLQ f Q Q D EnL$rf[L ([M .Mrf[L f L L m Q Q ([ Drf[M 6 .Mrf[M[ W L m L O m Q ([M $nI2.AL82+$M$2[ 7R REWDLQ H[SUHVVLRQV IRU $nf DQG Af ZH XVH UHODWLRQV \M[N IRU L A N IRU L N f f ZKHUH \M LV WKH ]WK URZ RI ; 6R SUHPXOWLSO\LQJ f E\ \M DQG MA \LHOGV :LWK WKHVH YDOXHV RI Af f EHFRPHV [Mrf + L L m $ ILUVW RUGHU 7D\ORU H[SDQVLRQ JLYHV $Hf P $f H$nf [MHf a rf FVMf f

PAGE 13

6XEVWLWXWLQJ WKH YDOXHV RI $nf DQG ]nf IURP f DQG f LQ WKH DERYH DSSUR[n LPDWLRQV ZH JHW 02 ;M^Wf 0rf H\M([M Q Y7 )U } L $r fr! ;L f 7R GHYHORS DQ DOJRULWKP ZH XVH f LQ WKH IROORZLQJ ZD\ 6XSSRVH ;$;a[ LV DQ DSSUR[LPDWH GLDJRQDOL]DWLRQ RI % ,I ZH LGHQWLI\ ( LQ f ZLWK % f§ ;$; U WKHQ WR WKH ILUVW RUGHU 02 $Mf H\M %;$;A;M 02 0rf F\M%[M F$M2f ;ƒfHs\PA ;f[M L O Q ?M $ ; 0rf m( \M%[M L L $L ;L $ n :LWK H WKHVH DSSUR[LPDWLRQV IRU DQ HLJHQYDOXH DQG FRUUHVSRQGLQJ HLJHQYHFn WRU JLYH ?QHZ £M 7 \IGf %[IG IRU M Q ROG?7 ; QHZ ; Q 9L L \ QHZ ; n ?U L O $M QHZ %rI [0 QHZ L IRU M Q f :H FDQ XVH WKH DERYH H[SUHVVLRQV WR IRUPXODWH DQ DOJRULWKP WR FRPSXWH D GLDJRn QDOL]DWLRQ RI D QRQGHIHFWLYH PDWUL[ $ DV IROORZV $OJRULWKP 'LDJRQDOL]DWLRQf *LYHQ D QRQGHIHFWLYH PDWUL[ $ e &Q[Q D PDWUL[ RI DSSUR[LPDWH HLJHQYHFWRUV $ DQG D WROHUDQFH WRO JUHDWHU WKDQ WKH XQLW URXQGRII WKH IROORZLQJ DOJRULWKP FRPSXWHV D GLDJRQDOL]DWLRQ $ $ $$ 'LI ; $7 > b f f f f M M A $ > FO f f f 9Q @ XQWLO 'LI WRO

PAGE 14

& ^& IRU M Q &L & FQ @ LV D Q [ Q ]HUR PDWUL[ HQG IRU M Q IRU L Q < $n 'LI f§ __&__I ^ ZKHUH I LV WKH )UREHQLRXV PDWUL[ QRUP ` HQG $ERYH ZH XVH D QRUPDOL]DWLRQ IRU [ WR UHGXFH WKH JURZWK RI ; ([DPSOH ,I $OJRULWKP LV DSSOLHG WR $ ZLWK ; DQG WRO WKHQ WKH HLJHQYDOXHV FRQYHUJH DV VKRZQ LQ WKH IROORZLQJ WDEOH

PAGE 15

,WHUDWLRQ A D A $ $ L 7KH H[SRQHQWV RI FRQYHUJHQFH S LQ 9r N ;S ) ; ;?? ) IRU GLIIHUHQW YDOXHV RI N DUH JLYHQ LQ WKH IROORZLQJ WDEOH ZKHUH $7 LV WKH PDWUL[ RI HLJHQYHFWRUV DW WKH $rWK LWHUDWLRQ DQG ; LV WKH ILQDO PDWUL[ RI HLJHQYHFWRUV N 3 6 +HQFH WKH PHWKRG DSSHDUV WR FRQYHUJH TXDGUDWLFDOO\ 1 *KRVK JDYH D WKHRUHWLFDO SURRI RI WKH TXDGUDWLF FRQYHUJHQFH RI WKH GLDJRQDOL]DWLRQ PHWKRG f ([DPSOH ,I $OJRULWKP LV DSSOLHG WR ZLWK ; DQG WRO WKHQ WKH DSSUR[LPDWH HLJHQYDOXHV GR QRW FRQYHUJH WR WKH HLJHQYDOXHV RI $ ZKLFK LV FOHDU IURP WKH IROORZLQJ WDEOH

PAGE 16

,WHUDWLRQ $L D A D $ L 7KH HLJHQYDOXHV RI $ DUH $ > f§ f§ f§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f§ SOfY WR ILQG FRUUHVSRQGLQJ HLJHQYHFWRU Y RI WKH HLJHQYDOXH 7KHQ [ f§ 4Y LV WKH HLJHQYHFWRU RI $ FRUUHVSRQGLQJ WR WKH HLJHQYDOXH S 1RZ WKHVH HLJHQYHFWRUV FDQ EH XVHG DV DSSUR[LPDWH HLJHQYHFWRUV LQ WKH GLDJRQDOL]DWLRQ PHWKRG WR ILQG WKH HLJHQYDOXHV DQG FRUUHVSRQGLQJ HLJHQYHFWRUV RI WKH SHUWXUEHG PDWUL[ $ W( ([DPSOH :H UHGXFH WKH PDWUL[ $ LQ ([DPSOH ILUVW WR DQ XSSHU +HVVHQEHUJ IRUP DQG WKHQ DSSO\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW ZLWK WRO $IWHU LWHUDWLRQV ZH REWDLQ D UHDO 6FKXU GHFRPSRVLWLRQ 67$6 > DQG WKH HLJHQYDOXHV RI

PAGE 17

8 DUH $ > @ 7R ILQG DQ HLJHQYHFWRU [ FRUUHVSRQGLQJ WR WKH HLJHQYDOXH OHW $ f§ IL, 7KHQ VROYH %[ IRU [ +HUH LV WKH PDWUL[ RI HLJHQYHFWRUV ; 1H[W FRQVLGHU WKH IROORZLQJ PDWULFHV $L DQG $ ZKLFK ZH JHW E\ SHUWXUELQJ WKH HOHPHQWV RI $ $? $ ,I ZH XVH $OJRULWKP WR $? DQG $ ZLWK ;T ; DQG WRO WKHQ DIn WHU LWHUDWLRQV ZH REWDLQ GLDJRQDOL]DWLRQV < 0MO 'X DQG =a[$= G ZKHUHDV LI ZH DSSO\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WR 67$?6 DQG 67$6 WKHQ DIWHU LWHUDWLRQV ZH REWDLQ UHDO 6FKXU GHFRPSRVLWLRQV 376n $L6f3 IL DQG 4767$O6f4 U 7KH GHULYDWLRQ RI WKH UHVXOWV LQ f DQG f FDQ EH IRXQG LQ PDQ\ 1XPHUn LFDO /LQHDU $OJHEUD WH[WV 7KH GHULYDWLRQ RI WKH IRUPXODV ZKLFK DUH XVHG LQ $On JRULWKP FDQ EH IRXQG LQ >+DJ66@ 7KH UHVXOW LQ f KROGV RQO\ ZKHQ WKH HLJHQYDOXHV RI $ DUH DOO GLVWLQFW 6R D QDWXUDO TXHVWLRQ LV WR DVN ZKHWKHU ZH FDQ GHULYH HTXLYDOHQW UHVXOWV IRU PDWULFHV ZLWK PXOWLSOH HLJHQYDOXHV

PAGE 18

,Q 6HFWLRQ ZH ZLOO VKRZ KRZ $OJRULWKP FDQ EH JHQHUDOL]HG WR GLDJRQDOL]H D QRQGHIHFWLYH PDWUL[ ZLWK PXOWLSOH HLJHQYDOXHV ,Q 6HFWLRQ ZH ZLOO VKRZ WKDW WKH JHQHUDOL]HG DOJRULWKP FRQYHUJHV ORFDOO\ TXDGUDWLFDOO\ $ 'LIIHUHQWLDO (TXDWLRQ $SSURDFK WR (LJHQFRPSXWDWLRQV LV LQWURGXFHG LQ 6HFWLRQ WR FUHDWH DQ LWHUDWLYH PHWKRG WR ILQG WKH HLJHQYDOXHV DQG FRUUHVSRQGLQJ HLJHQYHFWRUV RI D QRQGHIHFWLYH PDWUL[ $ 7KH GLIIHUHQWLDO HTXDWLRQV JRYHUQLQJ WKH EORFN PDWULFHV KDYH WKH IRUP ƒIf f§$f 'LDJ ;Wf $;Wff DQG ;Wf ;Wf) ;Wf $rff ZKHUH 'LDJ0f LV D EORFN GLDJRQDO PDWUL[ IRUPHG IURP WKH GLDJRQDO EORFNV RI WKH EORFN PDWUL[ $ )MM;Wf $Lff LV D VTXDUH ]HUR PDWUL[ DQG )M;f $ff IRU L A M LV WKH VROXWLRQ % WR WKH PDWUL[ HTXDWLRQ %$MWf f§ $WWf%
PAGE 19

IURP RSWLPL]DWLRQ WKHRU\ 7KH VWDUWLQJ HLJHQYDOXHV DVVRFLDWHG ZLWK WKH 6FKXU GHFRPn SRVLWLRQ DUH XQLIRUPO\ GLVWULEXWHG SRLQWV RQ D FLUFOH LQ WKH FRPSOH[ SODQH ZKHUH WKH FLUFOH LQFOXGHV DOO *HUVFKJRULQ GLVNV IRU WKH PDWUL[ DQG WKH VWDUWLQJ JXHVV XQLWDU\ IDFWRU LV WKH LGHQWLW\ PDWUL[ ,Q &KDSWHU ZH GLVFXVV KRZ WR H[SORLW WKH VWUXFWXUH RI D PDWUL[ 6LQFH V\PPHWULF PDWULFHV DUH GLDJRQDOL]DEOH ZH PRGLI\ WKH GLDJRQDOL]DWLRQ PHWKRG DQG WKH EORFN GLDJRQDOL]DWLRQ PHWKRG WR H[SORLW WKH PDWUL[ V\PPHWU\ DQG UHGXFH WKH JLYHQ PDWUL[ LQWR D GLDJRQDO PDWUL[ ZLWK HLJHQYDOXHV DORQJ WKH GLDJRQDO

PAGE 20

&+$37(5 (,*(19$/8(6 2) 816<00(75,& 0$75,&(6 %ORFN 'LDJRQDOL]DWLRQ RI D 0DWUL[ /HW $ EH DQ Q [ Q QRQGHIHFWLYH PDWUL[ 2XU DLP LV WR ILQG DQ LWHUDWLYH PHWKRG WR FRPSXWH D EORFN GLDJRQDOL]DWLRQ RI WKH IRUP $ ;$; ZKHUH $ GLDJ$L $ $rf LV D EORFN GLDJRQDO PDWUL[ DQG ; >;? ; ;W? LV D FRPSDWLEOH EORFN FROXPQ PDWUL[ VXFK WKDW $;M ;M$M +HUH $M GLDJ $M $M $ f LV DQ UULM [ QLM VFDODU PDWUL[ ,W FDQ DOZD\V EH DUUDQJHG VR WKDW WKH HLJHQYDOXHV RI $ DQG $ M IRU L A M DUH GLVWLQFW :H ZLOO XVH D VHQVLWLYLW\ UHVXOW IRU HLJHQYDOXHV DQG HLJHQYHFWRUV WR EORFN GLDJRQDOL]H D PDWUL[ ,Q WKH GHULYDWLRQ RI QHFHVVDU\ HTXDWLRQV WR EORFN GLDJRQDOL]H D PDWUL[ ZH QHHG WKH FRQWLQXLW\ RI HLJHQYDOXHV DQG WKH SHUWXUEDWLRQ WKHRU\ EDVHG RQ *HUVFKJRULQfV WKHRUHP 5HFDOO WKDW WKH FRHIILFLHQWV RI WKH FKDUDFWHULVWLF SRO\QRPLDO RI D PDWUL[ % DUH FRQn WLQXRXV IXQFWLRQV RI WKH HOHPHQWV RI % >:LO SDJH @ DQG WKH ]HURV RI D SRO\QRPLDO GHSHQG FRQWLQXRXVO\ RQ LWV FRHIILFLHQWV 3URRI LV LQ >+HQ SDJH @ 6R WKH HLJHQn YDOXHV RI % GHSHQG FRQWLQXRXVO\ RQ WKH HOHPHQWV RI % $OVR 5RXFKfV WKHRUHP VWDWHV WKDW LI DQG J DUH DQDO\WLF LQ D QHLJKERUKRRG RI D FORVHG GLVN ^] D 5` FHQWHUHG DW D KDV QR ]HURV RQ 2' ^] D "` DQG ?J^]f? ?I^]f? KROGV IRU ] e '' WKHQ I]f DQG I]f J]f KDYH WKH VDPH QXPEHU RI ]HURV LQ 'HWDLO LV LQ >+HQ SDJH @ 7KHRUHP /HW $ EH DUW HLJHQYDOXH RI $ RI DOJHEUDLF PXOWLSOLFLW\ N 7KHQ IRU DQ\ Q R UP DQG DOO VXIILFLHQWO\ VPDOO Uc WKHUH LV D VXFK WKDW LI __en__

PAGE 21

WKHQ WKH GLVN ^] e & ?] f§ $_ ` FHQWHUHG DW $ FRQWDLQV SUHFLVHO\ N HLJHQYDOXHV RI $ ( 3URRI RI 7KHRUHP 3URRI LV JLYHQ LQ >6WH SDJH @ 1H[W ZH ZLOO GLVFXVV WKH HIIHFW RI SHUWXUEDWLRQV LQ WKH HOHPHQWV RI $ RQ WKH HLJHQn YDOXHV DQG WKH HLJHQYHFWRUV LQ LWV GLDJRQDOL]DWLRQ ZKLFK LV EDVHG RQ *HUVFKJRULQfV WKHRUHP /HW $ EH DQ Q [ Q QRQGHIHFWLYH PDWUL[ 6XSSRVH ;aO$; $ LV D EORFN GLDJRQDOL]DWLRQ RI ZKHUH $ LV D EORFN GLDJRQDO PDWUL[ ,I $M GHQRWHV WKH MWK GLDJn RQDO EORFN RI $ WKHQ ZH KDYH $;M ;M$M ZKHUH ;M LV WKH MWK EORFN FROXPQ RI ; *LYHQ DQ DUELWUDU\ PDWUL[ er DQG IRU D VXIILFLHQWO\ VPDOO F ZH ZLOO VKRZ LQ WKH QH[W WKHRUHP WKDW WKHUH H[LVW FRQWLQXRXVO\ GLIIHUHQWLDEOH IXQFWLRQV $MHf DQG ;MHf ZLWK $Mf $ M DQG $Af ;M VXFK WKDW $ H(f;MHf ;MHf$MHf 7KHRUHP /HW $ EH DQ Q [ Q QRQGHIHFWLYH PDWUL[ 6XSSRVH ;aO$; $ LV D EORFN GLDJRQDOL]DWLRQ RI $ DQG LI $ M LV WKH MWK GLDJRQDO EORFN RI $ WKHQ $;M ;M$M ZKHUH ;M LV WKH MWK EORFN FROXPQ RI ; 6XSSRVH $ DQG $ M IRU L A M KDYH GLVWLQFW HLJHQYDOXHV 7KHQ JLYHQ DQ DUELWUDU\ Q [ Q PDWUL[ ( DQG IRU D VXIILFLHQWO\ VPDOO Hf WKHUH H[LVW FRQWLQXRXVO\ GLIIHUHQWLDEOH IXQFWLRQV $ MHf DQG ;MHf ZLWK $Mf $ M DQG ;Mf ;M VXFK WKDW WKH IROORZLQJ PDWUL[ HTXDWLRQ KROGV $ H(f;MHf ;MHf$MHf f 3URRI RI 7KHRUHP /HW ;a[ $; GLDJ $L$$I f EH D EORFN GLDJRQDOL]Dn WLRQ RI ZKHUH $ M GLDJ $ M?M f LV D VFDODU PDWUL[ DQG OHW UULM GLP$Mf 6XSSRVH PM /HW ( EH DQ Q [ Q PDWUL[ DQG H EH D VPDOO QXPEHU &RQVLGHU WKH

PAGE 22

PDWUL[ ; O$ F(f; 7KH IRUP RI WKLV PDWUL[ LV LOOXVWUDWHG EHORZ IRU W f§ n $L n )Q ) )O ) ) n $ ) ) ( )D )6 $ H I ) ) ) I D I I ) ) I $ ) ) ( )V ) ZKHUH )cM ZKHUH %? GLDJ f f S S %W W /HW & %'% ? WKHQ & $L $ $ $ $ EORFN URZ RI ; /HW % f§ GLDJ L M IRU L DQG GLPL"Mf I H H)Q )D ( f§ ) ) 3 3 3 3 S) W) W) H) H) S)L I) eA H) H) S)L H) W( F) F) S) H) H) W)V H) r1M 1H[W OHW 8 )??8 ( EH D GLDJRQDOL]DWLRQ RI )? ? ZKHUH ( GLDJ M? M 7P f ZLWK VRPH RI FUfV PD\ EH LGHQWLFDO /HW : GLDJ 8 % %W f 7KHQ :an&: $L $ $ $ $ H( X[I 8aO)O /))L H8n) 3 S 3 3 S)?8 W) I W) H) S)?8 W) F) H) H) 3)8 A H) H) H) S)VLL W)V H) H) H) :H PD\ FKRRVH D YDOXH RI S ZKLFK LV LQGHSHQGHQW RI H VXFK WKDW WKH ILUVW PL GLVNV DUH GLVMRLQW IURP WKH RWKHU Q f§ PL GLVNV 6R IRU D SURSHU YDOXH RI S PL HLJHQYDOXHV RI : ;&: OLH LQ WKH XQLRQ RI WKH PL GLVNV ZKLFK DUH FHQWHUHG DW $LHFU M PL 7KH UDGLXV RI HDFK GLVN LV RI RUGHU H DQG WKH SHUWXUEDWLRQ LQ HDFK HLJHQYDOXH $L LV RI RUGHU H DV H

PAGE 23

1H[W ZH GLVFXVV WKH HIIHFW RI SHUWXUEDWLRQV LQ WKH HOHPHQWV RI $ RQ WKH HLJHQYHFWRUV LQ LWV EORFN GLDJRQDOL]DWLRQ $V HDUOLHU OHW ;a[ $; GLDJ $L $ $rf EH D EORFN GLDJRQDOL]DWLRQ RI $ ZKHUH $ GLDJ $$ f LV D VFDODU PDWUL[ DQG GLP$Mf &OHDUO\ WKH FROXPQV [M [ [f rI ; IRUP D FRPSOHWH VHW RI HLJHQYHFWRUV :H GHQRWH HLJHQYHFWRUV RI $ H( E\ ;LHf [Hf [fHf DQG HLJHQYHFWRUV RI ;aO$ W(f; E\ 8LHf XHf XQHf VXFK WKDW [Hf $Xcf 7R LOOXVWUDWH ZH FRQVLGHU D PDWUL[ ZLWK W Qf§ VXFK WKDW $ GLDJ$L $ $f $M $M M Q $L IQ O O IO ‘ f f LQ $L  ff n Q $L H \r \r fn f Q $ f f ,D f O f I f f fn LQ f f f $Q f f IQ f f IQ f f IQ f f IQ f f r QQ ZKHUH X \M ([M DQG \M LV WKH IWK URZ RI $7 )LUVW FRQVLGHU WKH VLPSOH HLJHQYDOXH $ 6LQFH $Hf XHff LV DQ HLJHQSDLU RI DQG $f $ VR ZH PXVW KDYH Xf H ZKLFK LV WKH IRXUWK XQLW YHFWRU /HW XHf EH QRUPDOL]HG VR WKDW LWV ODUJHVW HOHPHQW LV )RU VXIILFLHQWO\ VPDOO H FODLP WKDW WKH IRXUWK HOHPHQW RI XFf LV ,I QRW OHW XHf IRU VRPH L L I 6LQFH =fXHf $HfXHf VR HTXDWLQJ WKH WK FRPSRQHQW RQ ERWK VLGHV ZH JHW Q $HfLWWHf $XOHf H 9 ILM8MHf f M L WKDW LV Q $ & f $ M f§ & A A ILM n8M & f  1RZ OHWWLQJ H f§ ZH ILQG $ $L A D FRQWUDGLFWLRQ 6R IRU VXIILFLHQWO\ VPDOO F W[Hf 1H[W ZH ZLOO VKRZ WKDW WKH UHPDLQLQJ FRPSRQHQWV RI XHf DUH

PAGE 24

DOO OHVV WKDQ 0H DV H f§! IRU VRPH 0 6LQFH ?XMHf? VR IURP f ZH JHW /HW _$Hf f§ $ Q _$Hf $__XLHf_ HA_WM_ _$ f§ $ 0 PD[ L A IRU L  WKHQ _WHf_ 0H ZKHUH Q ( O\ M O _$ $ 7KXV XQHf DUH RI RUGHU H IRU L A  $JDLQ IURP f ZH REWDLQ Q $Hf $f LLIHf HIL H A ILMXMHf A 7KH VHFRQG WHUP RQ WKH ULJKW LV RI RUGHU H VR IRU  A L ZH KDYH XrHf IL $ $ Hf 1RZ ZH WXUQ WR HLJHQYHFWRUV FRUUHVSRQGLQJ WR WKH PXOWLSOH HLJHQYDOXH ;? 6XSSRVH LHf LV D QRUPDOL]HG HLJHQYHFWRU RI FRUUHVSRQGLQJ WR WKH HLJHQYDOXH $[Hf 8VLQJ D VLPLODU DUJXPHQW DV LQ WKH FDVH RI WKH VLPSOH HLJHQYDOXH $Hf ZH FDQ VKRZ WKDW WKH ODUJHVW HOHPHQW RI [Hf FDQ QRW EH [Hf L ,Q IDFW WKHVH HOHPHQWV DUH RI RUGHU H 7KHUHIRUH WKH QRUPDOL]HG [Hf PXVW KDYH RQH RI WKH IROORZLQJ IRUPV n 3LHf 3LHf SL 3Hf 3ƒHf 3Hf mLHf f RU mHf f f RU mHf f f f mQOHf f 8PHf f 8QOHf ZKHUH ?SW^Hf? L 8VLQJ WKH VDPH WHFKQLTXH DV LQ WKH FDVH RI LLHf ZH FDQ VKRZ WKDW Hf DQG Hf DOVR PXVW KDYH RQH RI WKH DERYH IRUPV 6LQFH WKH GLPHQVLRQ RI WKH GLDJRQDO EORFN $L LV WKUHH DQG $L LV D VFDODU PDWUL[ VR ZH KDYH

PAGE 25

WKUHH GHJUHHV RI IUHHGRP WR QRUPDOL]H XcHf WZR GHJUHHV RI IUHHGRP WR QRUPDOL]H XHf DQG RQH GHJUHH RI IUHHGRP WR QRUPDOL]H XHf +HQFH ZH WDNH n n ‘ ‘ XOHf 8O f XHf XHf f DQG XHf fHf f f f 8QO f f XmHf f f fQHf $V LQ WKH FDVH RI WKH VLPSOH HLJHQYDOXH $Hf ZH FDQ VKRZ WKDW fRFf m X $ $ Hf L Q M ZKHUH ;? $ $ +HQFH $MHf DQG ;MHf DUH FRQWLQXRXVO\ GLIIHUHQWLDEOH IXQFWLRQV RI W 7KLV FRPSOHWHV WKH SURRI RI WKH WKHRUHP ’ 1RZ ZH DUH UHDG\ WR GHULYH WKH QHFHVVDU\ HTXDWLRQV WR REWDLQ D EORFN GLDJRQDO L]DWLRQ $ ;$;B RI D QRQGHIHFWLYH PDWUL[ $ /HW $  EH WKH M WK EORFN GLDJRQDO HOHPHQW RI $ DQG ;M EH WKH FRUUHVSRQGLQJ EORFN FROXPQ YHFWRU RI ; *LYHQ DQQ[Q PDWUL[ ( DQG IRU D VXIILFLHQWO\ VPDOO H 5 OHW $MHf DQG ;MWf EH GLIIHUHQWLDEOH IXQFWLRQV RI H VXFK WKDW $Mf $ M DQG $Mf ;M 'LIIHUHQWLDWLQJ f ZLWK UHVSHFW WR H DQG WKHQ SXWWLQJ W ZH REWDLQ /
PAGE 26

6XEVWLWXWLQJ f LQWR f ZH REWDLQ $  ;%nf (;  ;%cMf$M <$nf r W O  ;$Ef (;  ;M%nf$ $nM $n2f  (; e ; %A2,$M $c%cf;M$n2f f 6LQFH 97 \ B I IRU L s N N > IRU L N f ZKHUH <" LV WKH ]WK EORFN URZ RI ;aO 3UHPXOWLSO\LQJ f E\ $f +HUH ZH PHQWLRQ WZR ZHOO NQRZQ PHWKRGV WR ILQG %c$f 7R GHULYH WKHVH WZR PHWKRGV ZH QHHG WKH IROORZLQJ r WZR WKHRUHPV 7KHRUHP >+DJ SDJH @ ,I % &S[S DQG & &U[U DUH QRQGHIHFWLYH PDWULFHV WKHQ WKH VROXWLRQ = WR WKH PDWUL[ HTXDWLRQ =%&= 5 LV JLYHQ E\ = 9:8 ZKHUH % 8(8 DQG & 949 DUH GLDJRQDOL]DWLRQV RI % DQG & UHVSHFWLYHO\ DQG : 3URR I RI 7KHRUHP /HW % f§ ^9an58fL Z^Mf LV D U [ S PDWUL[ ZLWK :LM D M f§ X!L 8
PAGE 27

/HW : 9 =8 DQG 9 58 WKHQ WKH ODVW HTXDWLRQ UHGXFHV WR + ( f§ 4: &RPSDULQJ WKH LMf HQWU\ RQ ERWK VLGHV RI WKH SUHFHGLQJ UHODWLRQ ZH KDYH YO58fD 7M f§ /c DQG ZLWK WKLV : = YZX 7KHRUHP $ ,I $ e 5S[S DQG % e 5U[U WKHQ WKH VROXWLRQ = WR WKH PDWUL[ HTXDWLRQ $==% & f LV JLYHQ E\ = 8397 ZKHUH $ O 587 DQG % 9697 DUH UHDO 6FKXU GHFRPSRn VLWLRQV RI $\ DQG % UHVSHFWLYHO\ DQG 3 LV WKH VROXWLRQ WR WKH 6\OYHVWHU HTXDWLRQ 53 f§ 36 87&9 3URRI RI 7KHRUHP /HW $ 8587 DQG % 9697 EH UHDO 6FKXU GHFRPSRVLn WLRQV RI O DQG % UHVSHFWLYHO\ 7KHQ f EHFRPHV 8587= =9697 & 587=9 87=96 87&9 /HW 3 87=9 DQG 87&9 7KHQ WKH ODVW HTXDWLRQ UHGXFHV WR 53 f§ 36 =f ZKLFK LV D 6\OYHVWHU HTXDWLRQ )RU WKH GHWDLOHG VROXWLRQ RI D 6\OYHVWHU HTXDWLRQ UHDGHUV DUH UHIHUUHG WR >*RO SDJH @ ’ 1H[W ZH ZLOO VKRZ KRZ WR XVH WKH DERYH WZR WKHRUHPV WR GHULYH WZR PHWKRGV WR ILQG WKH XQNQRZQ %>^f LQ f 0HWKRG 7KHRUHP f /HW 3 %>-^f DQG 5 <" (;M WKHQ f UHGXFHV WR 3$M $L3 5 f 6ROYLQJ f IRU 3 ZH JHW 3 9:8 ZKHUH $ 8<8 DQG $ GLDJRQDOL]DWLRQV RI $DQG $ UHVSHFWLYHO\ DQG ,) ZXf ZLWK ZP )LOO nL DUH n U? 58fNO Dc f§ XfN

PAGE 28

0HWKRG 7KHRUHP f /HW 3 %>^f DQG &
PAGE 29

$M W
PAGE 30

F M f§ &M ] HQG HQG ;L f§ ;O f§ $ [M?? IRU Mr ^ ;M f§ f f USM AA f f f AUULM LV FROXPQ SDUWLWLRQLQJ < ;a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f§ rf 1H[W VROYH %\ IRU \ LQ WKH IROORZLQJ ZD\ 3XW \Lf DQG WKHQ VROYH >e"  f§ f % L Qf?] f§ % Lf E\ XVLQJ WKH 45 IDFWRUL]DWLRQ WHFKQLTXH WR VROYH DQ RYHUGHWHUPLQHG V\VWHP RI HTXDWLRQV 3XW \O L f§ f ]O L f§ f DQG \L Qf ]L Q f§ f 7KHQ [ 4\ LV WKH HLJHQYHFWRU RI $ FRUUHVSRQGLQJ WR WKH HLJHQYDOXH T 1RZ WKHVH HLJHQYHFWRUV FDQ EH XVHG LQ WKH GLDJRQDOL]DWLRQ PHWKRG WR ILQG WKH HLJHQYDOXHV DQG FRUUHVSRQGLQJ HLJHQYHFWRUV RI WKH SHUWXUEHG PDWUL[ $ H(

PAGE 31

([DPSOH &RQVLGHU WKH PDWUL[ $ :H UHGXFH $ ILUVW WR DQ XSSHU +HVVHQEHUJ IRUP + DQG WKHQ DSSO\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW ZLWK WRO $IWHU LWHUDWLRQV ZH JHW D UHDO 6FKXU GHFRPSRVLWLRQ 67$6 f§ 8 7KH HLJHQYDOXHV RI 8 DUH $ > f§  f§ f§  @ 1H[W ZH XVH WKH WHFKQLTXH GLVFXVVHG LQ WKH ODVW SDUDn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n $?6f4 5 DV ZHOO ([DP@!8 &RQVLGHU WKH PDWUL[

PAGE 32

,I ZH DSSO\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW ZLWK WRO WR =" WKHQ DIWHU LWHUDWLRQ ZH REWDLQ D UHDO 6FKXU IRUP 67%6 e DQG WKH HLJHQYDOXHV RI 8 DUH $ > f§ f§ @ 1H[W ZH XVH WKH WHFKQLTXH GLVFXVVHG SULRU WR ([DPSOH WR ILQG FRUUHVSRQGLQJ HLJHQYHFWRUV +HUH LV WKH PDWUL[ RI HLJHQYHFWRUV n ; 1H[W FRQVLGHU WKH IROORZLQJ PDWUL[ %L ZKLFK ZH JHW E\ SHUWXUELQJ WKH HOHPHQWV RI n n % ,I ZH DSSO\ $OJRULWKP WR %? ZLWK ;T ; DQG WRO WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D EORFN GLDJRQDOL]DWLRQ
PAGE 33

n

PAGE 34

,I ZH DSSO\ WKH 45 DOJRULWKP ZLWK GRXEOH LPSOLFLW VKLIW ZLWK WRO WR WKHQ DIWHU LWHUDWLRQV ZH JHW D UHDO 6FKXU IRUP 67&6 8 DQG WKH HLJHQYDOXHV RI 8 DUH $ >     6r r r r r r r r @ 1H[W ZH XVH WKH WHFKQLTXH GLVFXVVHG SULRU WR ([DPSOH WR ILQG FRUUHVSRQGn LQJ HLJHQYHFWRUV +HUH IRU HDFK SDLU RI FRPSOH[ FRQMXJDWH HLJHQYDOXHV ZH VROYH WKH HTXDWLRQ %[ RQO\ IRU RQH DQG WKHQ WDNH WKH UHDO DQG WKH LPDJLQDU\ SDUWV RI [ DV WZR YHFWRUV FRUUHVSRQGLQJ WR WKH FRPSOH[ FRQMXJDWH SDLU RI HLJHQYDOXHV :H GHQRWH WKH PDWUL[ RI YHFWRUV EY ; 1H[W FRQVLGHU WKH PDWUL[ n n S f§ f§ ,I ZH XVH $OJRULWKP WR WKH SHUWXUEHG PDWUL[ &? & 3 ZLWK ;T ; DQG WRO WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D EORFN GLDJRQDOL]DWLRQ
PAGE 35

ZKHUHDV LI ZH DSSO\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WR 67&?6 WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D UHDO 6FKXU IRUP 4767&?6f4 5 $ 'LIIHUHQWLDO (TXDWLRQ $SSURDFK WR (LJHQFRPSXWDWLRQV ,Q WKH SUHYLRXV VHFWLRQ ZH IRXQG WKDW WKH PDLQ SUREOHP LQ XVLQJ WKH EORFN GL DJRQDOL]DWLRQ PHWKRG LV KRZ WR FKRRVH D PDWUL[ RI DSSUR[LPDWH HLJHQYHFWRUV IRU WKH VWDUWLQJ JXHVV +HUH ZH ZLOO VWXG\ ZKHWKHU WKH (XOHU PHWKRG FDQ EH XVHG WR EORFN GLDJRQDOL]H D QRQGHIHFWLYH PDWUL[ $ ZLWK WKH LGHQWLW\ PDWUL[ DV WKH PDWUL[ RI DSSUR[n LPDWH HLJHQYHFWRUV ,Q QXPHULFDO RUGLQDU\ GLIIHUHQWLDO HTXDWLRQV RQH ZD\ WR REWDLQ WKH (XOHU PHWKRG LV WR GURS WKH QG RUGHU HUURU WHUP LQ WKH 7D\ORU 6HULHV RI WKH JLYHQ IXQFWLRQ ,Q 7KHRUHP GURSSLQJ WKH QG RUGHU HUURU WHUPV ZH JHW WKH H[SUHVVLRQV IRU $Hf DQG ;IMFf ZKLFK FDQ EH XVHG LQ (XOHUnV PHWKRG 1RZ ZH ZLOO GLVFXVV KRZ WR LPSOHPHQW WKH (XOHU PHWKRG WR ILQG WKH HLJHQYDOXHV RI D QRQGHIHFWLYH PDWUL[ $ (XOHUfV PHWKRG IRU WKH GLIIHUHQWLDO HTXDWLRQ ]Wf I ]ff KDV WKH IRUP =Q f§ =Q fWf ZKHUH ]Q LV WKH DSSUR[LPDWLRQ WR ]Q$ef DQG $W LV WKH FRQVWDQW WLPH VWHS +HUH DQG DUH YHFWRU IXQFWLRQV RI W LQ JHQHUDO ,Q WKH EORFN GLDJRQDOL]DWLRQ SURFHGXUH ZH DWWHPSW WR JHQHUDWH D EORFN GLDJRQDO PDWUL[ $ DQG DQ DVVRFLDWHG EORFN PDWUL[ ; VXFK WKDW $; ;$ ,I $M GHQRWHV WKH MWK GLDJRQDO EORFN RI $ WKHQ ZH KDYH $;M ;M$M 7KH GLIIHUHQWLDO HTXDWLRQ JRYHUQLQJ WKH EORFN PDWULFHV KDV WKH IRUP ?Wf f§$ 'LDJ $n $$n2f DQG ;^Wf ;^Wf)^;Wf $Wff ZKHUH 'LDJ$f LV D EORFN GLDJRQDO PDWUL[ IRUPHG IURP WKH GLDJRQDO EORFNV RI WKH EORFN PDWUL[ $ )MM;Wf $ff LV D VTXDUH ]HUR PDWUL[ DQG )W-;Wf $Wff IRU L A M LV WKH VROXWLRQ % WR WKH PDWUL[ HTXDWLRQ %$MWf f§ $WWf%
PAGE 36

< ;Wf 7\SLFDO VWDUWLQJ JXHVV LV $f 'LDJ7f DQG ;f +HQFH WKH (XOHU DSSUR[LPDWLRQ WR WKH GLIIHUHQWLDO HTXDWLRQV FDQ EH H[SUHVVHG DV $ Q $Q $W 'LDJ A;Q $;QAM f§ $Qf DQG ;Q? f§ ;Q $W;Q) ;Q $Qf f :H ZLOO XVH WKH YDOXHV RI $QL DQG ;QL IURP f WR FUHDWH DQ LWHUDWLYH PHWKRG WR FRPSXWH D EORFN GLDJRQDOL]DWLRQ RI DQ Q [ Q UHDO PDWUL[ $ %XW ILUVW ZH ZLOO GLVFXVV KRZ WR FRPSXWH D EORFN GLDJRQDOL]DWLRQ RI DQ XSSHU WULDQJXODU PDn WUL[ )RU DQ\ XSSHU WULDQJXODU PDWUL[ 7 LWV HLJHQYDOXHV DUH WKH GLDJRQDO HOHPHQWV :H ZLOO VKRZ KRZ WR FRPSXWH D EORFN GLDJRQDOL]DWLRQ < O7< RI 7 ZKHUH GLDJ =fQ =f f f f 7f f ZLWK WKH SURSHUW\ WKDW HDFK GLDJRQDO EORFN LV XSSHU WULDQJXODU DQG WKH HLJHQYDOXHV RI '] DQG 'M IRU L M DUH GLVWLQFW /HW < f§ ,Q DQG ?WX f§ WMM_f *LYHQ D WROHUDQFH WRO LI DcM WRO WKHQ GHILQH ] W X W f§ W A / /HW :^M MM EH HTXDO WR WKH LGHQWLW\ PDWUL[ H[FHSW IRU WKH ] Mf HOHPHQW ZKLFK LV 7KHQ WKH SURGXFW :O-7:cM LV DQ XSSHU WULDQJXODU PDWUL[ ZKRVH ] Mf HOHPHQW LV ]HUR 1H[W XSGDWH < E\ < QHZ B \RLG\\ :H XVH WKLV WHFKQLTXH WR ]HUR WcM ZKHQHYHU DcM WRO ] M = $Q HIILFLHQW ZD\ WR ]HUR WcM ZKHQ R L M Q LV WR VWDUW IURP WKH ERWWRP ULJKW FRUQHU RI WKH PDWUL[ =HUR WcM LQ WKH GHFUHDVLQJ RUGHU RI WKH URZ LQGH[ L XQWLO L f§ ,I WRO LV HQFRXQWHUHG WKHQ PRPHQWDULO\ VWRS ]HURLQJ RQ WKH MWK FROXPQ DQG JR WR WKH M f§ fWK FROXPQ &RQWLQXH WKLV SURFHVV DOO WKH ZD\ WR WKH VHFRQG FROXPQ 2QFH WKH VHFRQG FROXPQ LV GRQH WKHQ LQ WKH LQFUHDVLQJ RUGHU RI WKH FROXPQ LQGH[ M JR WR WKRVH FROXPQV ZKHUH VRPH RI WKH WLM DUH QRW ]HURHG LQ WKH ILUVW URXQG HYHQ WKRXJK WKH FRUUHVSRQGLQJ DcM DUH JUHDWHU WKDQ WKH WRO 7KLV WLPH DOVR ]HUR WcM LQ WKH GHFUHDVLQJ RUGHU RI WKH URZ LQGH[ L XQWLO L ZKHQHYHU WKH FRUUHVSRQGLQJ DW! WRO 7KHQ XVH SHUPXWDWLRQV RI URZV DQG FROXPQV WR REWDLQ 'X 'X 8VH WKHVH SHUPXWDWLRQV WR < WR VZLWFK URZV DQG FROXPQV RI <

PAGE 37

,Q IDFW WKH DERYH WHFKQLTXH FDQ EH H[WHQGHG WR D N [ N TXDVLXSSHU WULDQJXODU PDWUL[ 7 e 5UL;Q ,Q WKLV FDVH :^M LV HTXDO WR WKH N [ N EORFN LGHQWLW\ PDWUL[ H[FHSW IRU WKH ] Mf EORFN ZKLFK LV = ZKHUH = LV WKH VROXWLRQ WR WKH PDWUL[ HTXDWLRQ 7LW= f§ =7MM f§7LM :H VXPPDUL]H WKH EORFN GLDJRQDOL]DWLRQ RI 7 LQ WKH IROORZLQJ DOJRULWKP $OJRULWKP *LYHQ D N [ N TXDVLXSSHU WULDQJXODU PDWUL[ 7 e 5W$Q DQG D FRDn OHVFLQJ WROHUDQFH WRO JUHDWHU WKDQ WKH VTXDUH URRW RI WKH XQLW URXQGRII WKH IROORZLQJ DOJRULWKP FRPSXWHV D EORFN GLDJRQDOL]DWLRQ RI 7 6 ,Q ^ 6 LV WKH Q [ Q LGHQWLW\ PDWUL[ ` URZ > @ FRO > @ IRU M N f§ L M f§ PLQLP WRO ZKLOH PLQLP WRO $ L DM PLQ^ _D f§ XM FU e ;7Xf DQG OR e $7MMf LI DW-r WRO URZ > L URZ @ FRO > M FRO @ ^ URZ DQG FRO DUH WZR DUUD\V ZKRVH HOHPHQWV DUH WKH URZ LQGH[ L DQG WKH FROXPQ LQGH[ M UHVSHFWLYHO\ RI DQ HQWU\ 7LM RI 7 ZKLFK ZLOO QRW EH UHSODFHG E\ D ]HUR PDWUL[ ` PLQLP DM HOVH 6ROYH 7D= f§ =7MM f§7LM IRU = DQG WKHQ IRUP ???M 7 :Mn7:LM@ 6 6 :[M HQG

PAGE 38

HQG HQG P OHQJWKURZf IRU O f§ P L URZf f§ M FROf ZKLOH  LI WRO 7 :aM7 :LM 6f 6 :WHQG HQG HQG /HW IF DQG VXSSRVH $7 LV DQ DUUD\ RI $ HOHPHQWV ZKRVH ]WK HOHPHQW LV WKH GLPHQVLRQ RI WKH ]WK EORFN RI 7 DORQJ WKH GLDJRQDO )RU L M GHILQH DV LQ WKH DERYH FRGH ,I DWWRO WKHQ PHUJH EORFNV 7 DQG 7MM WR IRUP D VLQJOH EORFN XVLQJ SHUPXWDWLRQV RI FROXPQV DQG URZV RI 7 8VH WKRVH FROXPQ SHUPXWDWLRQV WR 6 WR PHUJH 6L DQG 6M 8SGDWH NROG DQG 17ROG WR REWDLQ NQHZ DQG 17QHZ 7U\ WKH DERYH PHUJLQJ WHFKQLTXH IRU DOO SRVVLEOH FRPELQDWLRQV RI L M 1RZ ZH DUH UHDG\ WR FRPSXWH D EORFN GLDJRQDOL]DWLRQ RI DQ Q [ Q QRQGHIHFWLYH PDWUL[ $ XVLQJ WKH IROORZLQJ SURFHGXUH $OJRULWKP '\QDPLFDO (LJHQFRPSXWDWLRQVf *LYHQ D QRQGHIHFWLYH PDWUL[ $ ( D WROHUDQFH WRO JUHDWHU WKDQ WKH XQLW URXQGRII D FRDOHVFLQJ WROHUDQFH WROO JUHDWHU WKDQ WKH VTXDUH URRW RI WKH XQLW URXQGRII D FRQVWDQW VWHSVL]H $W VPDOOHU WKDQ DQG D VWDUWLQJ JXHVV GLDJRQDO PDWUL[ $ WKH IROORZLQJ DOJRULWKP XVHV f WR FRPSXWH D EORFN GLDJRQDOL]DWLRQ $ ;$;aO

PAGE 39

'HILQH LRO WROO 1H[W ZH EUHDN WKH VNHWFK RI WKH DOJRULWKP LQWR VHYHUDO VWHSV 6WHS 7DNH ; $ $R U 1% Q WKH QXPEHU RI EORFNV f DQG 16 > @ DQ DUUD\ RI Q HOHPHQWV ZKRVH WK HOHPHQW LV WKH GLPHQVLRQ RI WKH WK GLDJRQDO EORFN RI $ 6WHS /HW P 1% )RU L M GHILQH ',) PLQ^ ?FU f§ XM? D ( $$f DQG XM f $ $f `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f DV IROORZV )LML <$f IRU L 3WIRU L  M ZKHUH 3 3WLV WKH VROXWLRQ WR WKH PDWUL[ HTXDWLRQ 3$M f§ $W3 $P $P" f f f $\YA@ ZKHUH P 1% ZLWK GLP$f IRU L L DQG GLP$f IRU L P OP 1% 'HILQHG P GLDJ ^<$;f W Qf f§ [ ZKHUH W A $U6Lf DQG GLDJ0f Q f LV D YHFWRU IRUPHG IURP W WKURXJK Q HOHPHQWV RI WKH YHFWRU GLDJ0f 2XU DLP LV WR ILQG WKH VPDOOHVW V f f ZLWK WKH SURSHUW\ VGc ILM VGM 7R REWDLQ WKLV ZH GR WKH IROORZLQJ /HW N EH DQ DUUD\ ZLWK WKH SURSHUW\ WKDW rf LLNLf 1H[W IRUP WKH UDWLR U 3NOLOf 8Wf GNf GNW L r}‘

PAGE 40

DQG GHILQH V PLQ^ UW L Q f§ W U ` ,I V $ WKHQ WDNH D WLPH VWHS RI VL]H V LQVWHDG RI WKH QRUPDO WLPH VWHS $W 2WKHUZLVH WDNH WKH QRUPDO WLPH VWHS $W 6WHS 8SGDWH $ $ DQG < DV IROORZV $QHZ $ROG $WA \ROGnM 7 $;ROG B A,GA M B [QHZ ;ROG $W) [ROG$ROGff f\QHZ A AUQHZ f§ 6WHS &RQVLGHU WKH EORFN $ M ZLWK M P ZKHUH P LV GHILQHG DV LQ 6WHS /HW $ M 4847 EH D 6FKXU GHFRPSRVLWLRQ RI $ M 7KHQ 8 1 ZKHUH GLDJ &Q 8 f f f 8X f DQG 1 LV WKH VWULFWO\ XSSHU WULDQJXODU SDUW RI 8 8X LV HLWKHU D [ PDWUL[ RU D [ PDWUL[ /HW 0 EH DQ DUUD\ ZKRVH WK HOHPHQW LV WKH GLPHQVLRQ RI WKH EORFN 8X 8VH $OJRULWKP ZLWK WRO DV WKH FRDOHVFLQJ WROHUDQFH WR UHGXFH 8 WR D EORFN GLDJRQDO PDWUL[ WR XSGDWH WKH DUUD\ 18 DQG WKH QXPEHU RI EORFNV DQG WR JHW WKH LQYHUWLEOH PDWUL[ 6 5HSODFH $ M E\ > ;M E\ ;M46
PAGE 41

B n B n $ ZLWK $T WRO WROO f§ DQG $W f§ WKHQ LW WDNHV LWHUDWLRQV WR WKH GLDJRQDO PDWUL[ $T WR FRQYHUJH WR GLDJ 7KH HLJHQYDOXHV RI DUH $ > @ ([DPSOH ,I ZH DSSO\ $OJRULWKP WR WKH PDWUL[ $ ZLWK $R GLDJ f§ f§ f§ f§f WRO WROO DQG $W WKHQ LW WDNHV LWHUDWLRQV WR WKH GLDJRQDO PDWUL[ $T WR FRQYHUJH WR GLDJ n n f f 7KH HLJHQYDOXHV RI DUH $ > f§ f§ @ 7KLV PHWKRG KDV WKH IROORZLQJ GLVDGYDQWDJH $V WKH VL]H RI WKH JLYHQ PDWUL[ $ JURZV ZH QHHG WR WDNH D VWHSVL]H VPDOOHU WKDQ +RZHYHU ZLWK D VWHSVL]H VPDOOHU WKDQ WRR PDQ\ LWHUDWLRQV DUH UHTXLUHG WR REWDLQ D IHZ GHFLPDO SODFHV RI DFFXUDF\

PAGE 42

$ 'LIIHUHQWLDO (TXDWLRQ $SSURDFK WR (LJHQFRPSXWDWLRQV ZLWK $UPLMRnV 6WHSVL]H ,Q WKH 'LIIHUHQWLDO (TXDWLRQ $SSURDFK WR (LJHQFRPSXWDWLRQV ZH IRXQG WKDW ZLWK D FRQVWDQW WLPH VWHS $ ZKLFK LV XVXDOO\ D VPDOO QXPEHU WRR PDQ\ LWHUDWLRQV DUH UHTXLUHG WR DFKLHYH D IHZ GLJLWV DFFXUDF\ LQ WKH UHVXOWV 6R KHUH ZH ZLOO PRGLI\ WKH 'LIIHUHQWLDO (TXDWLRQ $SSURDFK WR (LJHQFRPSXWDWLRQV E\ YDU\LQJ WKH WLPH VWHS $W LQ HDFK LWHUDWLRQ :H SODQ WR DFKLHYH WKLV E\ XVLQJ $UPLMRfV UXOH IURP RSWLPL]DWLRQ WKHRU\ /HW $ EH DQ Q [ Q UHDO QRQGHIHFWLYH PDWUL[ :H ZLOO ILQG DQ LWHUDWLYH PHWKRG WR FRPSXWH D EORFN GLDJRQDOL]DWLRQ RI WKH IRUP $ ;$; ZKHUH $ GLDJ $L $ $W f LV D EORFN GLDJRQDO PDWUL[ DQG ; > $K ;W @ LV D FRPSDWLEOH EORFN FROXPQ PDWUL[ VXFK WKDW $;M ;M$M $V LQ 6HFWLRQ IRU WKH EORFN HLJHQYDOXH SUREOHP WKH GLIIHUHQWLDO HTXDWLRQV WKDW ZH VROYH DUH $P $Lf 'LDJ ;LW\n$;LWff DQG ;Wf ;Wf);Wf$Wff ZKHUH 'LDJ$f LV D EORFN GLDJRQDO PDWUL[ IRUPHG IURP WKH GLDJRQDO EORFNV RI D EORFN PDWUL[ 0 )MM;Wf $Wff LV D VTXDUH ]HUR PDWUL[ DQG )LM;Wf $Wff IRU L A M LV WKH VROXWLRQ % WR WKH PDWUL[ HTXDWLRQ %$MWf f§ $OWf% f§ <^Wf7$;MWf +HUH
PAGE 43

6R ILf $Q =f ;Q DQG 4$Wf $Q =$Wf $IQL WKH PDWULFHV JHQHUn DWHG E\ D (XOHU VWHS 'HILQH IVf __*Vf__I ZKHUH *Vf $ f§ =Vf4Vf=Vf L +HQFH f __$ ;QL?Q;Q __I DQG $Wf @_$;f$Q$n ) ,I WKH VWDUWLQJ JXHVV EORFN GLDJRQDO PDWUL[ $T DQG WKH VWDUWLQJ JXHVV LQYHUWLEOH nL PDWUL[ ;T DUH JRRG DSSUR[LPDWLRQV RI $ DQG ; LQ WKH IDFWRUL]DWLRQ $ ;$; WKHQ ZH PXVW KDYH I$Wf f +HUH RXU JRDO LV WR ILQG DQ V IRU ZKLFK IVf f KROGV 7R WKLV HQG ZH XVH $UPLMRnV UXOH IURP RSWLPL]DWLRQ WKHRU\ ,Q $UPLMRnV UXOH ZH GHWHUPLQH V LQ WKH IROORZLQJ ZD\ (YDOXDWH IVf DW f f f r r VWRSSLQJ ZKHQ P f 6LPSOLI\LQJ WKH DERYH LQHTXDOLW\ ZH JHW B,Rf ,W WXUQV RXW >+DJ66 SDJH f§ @f WKDW WR XVH WKH DERYH UXOH ZH PXVW KDYH n f &Vf f§f 6R RXU QH[W DLP LV WR GHWHUPLQH nf ZKHQ IVf f§ __*nVf__In DQG $ f§ =Vf4Vf=Vfa 6XSSUHVVLQJ WKH VXEVFULSWV RI $Q DQG ;Q LQ f ZH JHW ILVf $ V A'LDJ >; $;f f§ $f DQG =Vf ; V;) ; $f %HIRUH ZH ILQG nf ZH ZDQW WR SURYH WKH IROORZLQJ IXQGDPHQWDO UHVXOW /HPPD /HW *Vf EH DQQ[Q FRPSOH[ PDWUL[ ZKRVH HOHPHQWV DUH GLIIHUHLLWLDEOH IXQFWLRQV RI V ,I ZH GHILQH IVf __*Vf__If WKHQ G fnf WUDFH *f+f§*Vf?D DV f 3URRI RI /HPPD GHDUO\ IVf O_*rf ) W UDFH *f*ff f

PAGE 44

'LIIHUHQWLDWLQJ f ZLWK UHVSHFW WR V ZH KDYH I^VfIn^Vf WUDFH f§ *Vf+f *^Vf *Vf+A*Vf GV GV + WUDFH G *Vff *Vf *VfKA*^Vf GV GV WUDFH *Vf + G GV *Vf $IWHU VLPSOLILFDWLRQ VHWWLQJ ZH JHW fnf WUDFH *f + G GV *Vf_ R f f 1RZ ZLWK 7OVf $ V 'LDJ ; $;f f§ $f =Vf ; V;) $7 $f *Vf $ f§ =Vf4Vf=Vfa DQG IVf __*Vf__U DQG WKH UHVXOW RI WKH DERYH OHPPD ZH DUH LQ D SRVLWLRQ WR VKRZ WKDW n2f Rf 7KHRUHP /HW $ e 5 Q[Q EH D QRQGHIHFWLYH PDWUL[ 6XSSRVH $ ; $; LV D EORFN GLDJRQDOL]DWLRQ RI $ ZKHUH $ GLDJ $L $ $r f LV D EORFN GLDJRQDO PDWUL[ VXFK WKDW $c DQG $ M IRUL A M KDYH GLVWLQFW HLJHQYDOXHV DQG ; > $L ; ;W @ LV D FRPSDWLEOH EORFN FROXPQ PDWUL[ ,I ZH GHILQH 4Vf $ V 'LDJ; $;f f§ $f =Vf ; V;);$f *^Vf $ f§ =Vf4Vf=Vf DQG IVf ??*Vf??I ZKHUH f LV D EORFN GLDJRQDO PDWUL[ IRUPHG IURP WKH GLDJRQDO EORFNV RI WKH EORFN PDWUL[ 0 )MM ; $f LV D VTXDUH ]HUR PDWUL[ DQG )W; $f IRU L M LV WKH VROXWLRQ 3 WR WKH PDWUL[ HTXDWLRQ 3$M f§ $c3 f§ <"$;M ZKHUH <" GHQRWHV WKH LWK EORFN URZ RI ; WKHQ n2f Rf 3URRI RI 7KHRUHP 7KH UHVXOW f RI /HPPD JLYHV fnf WUDFH *f7 I*Vf?V @ f

PAGE 45

'LIIHUHQWLDWLQJ WKH H[SUHVVLRQ IRU *nVf ZLWK UHVSHFW WR V ZH KDYH G GV *Vf G GV G GV =VffVf=Vf ]X;LQXff ]Xf GV ]VfQXfIW]Vf =Vff>Vf=Vf =Vf G GV PVff]Vf L ]^VfQ^Vf]^Vf GV =Vff=Vf G (YDOXDWLQJ f§ *Vf DW V ZH REWDLQ GV G GV *Vf_V G GV =Vf_ R'f=f ]fIQVf_V R=f GV O =fILf=f L G GV =Vf?L R=f 'LIIHUHQWLDWLQJ 'Vf DQG =Vf ZLWK UHVSHFW WR DQG WKHQ HYDOXDWLQJ WKH GHULYDWLYHV DW V ZH JHW GB GV QVf?V R $ 'LDJ [a0[f DQG G GV =Vf?V ;);?f 8VLQJ WKHVH YDOXHV DQG YDOXHV RI 'f DQG =f LQ f ZH KDYH G GV *mVf_V ;) ; $f $; ; 'LDJ ;n0[f $f ; ;) ; $ff ; U? ; ) ; $f $ $) ; $ff ;aO ;'LDJ ;n0[f ; ;$; 1H[W OHW ); $f$ $); $f 7KHQ LM )$ IRU $ )M IRU L  r K f ZKHUH )M )M $ $f $FFRUGLQJ WR RXU DVVXPSWLRQ IRU L A )WLV WKH VROXWLRQ 9 WR WKH PDWUL[ HTXDWLRQ 3?M f§ DIWHU VLPSOLILFDWLRQ f JLYHV $ L3 <"$;M 6R )$` $L)LM
PAGE 46

8VLQJ f LQWR f ZH KDYH $ ;a[ $; 'LDJ U0Yff ;a[ f§$'LDJ [n$;f ; $$$ $ $$$f 6LQFH *f f§ $ f§ =f8f=f f§ $ $$$ VR f JLYHV f§ *f DQG ZLWK WKLV YDOXH f UHGXFHV WR fnf WUDFH *fW *2fff Rf +HQFH nf f§f SURYLGHG f A ’ 7KH DERYH WKHRUHP LPSOLHV WKDW $UPLMRfV UXOH FDQ EH DSSOLHG WR $OJRULWKP PHQWLRQHG LQ WKH 'LIIHUHQWLDO (TXDWLRQ $SSURDFK WR (LJHQFRPSXWDWLRQV $ PRGLILHG DOJRULWKP WR ILQG WKH HLJHQYDOXHV DQG FRUUHVSRQGLQJ HLJHQYHFWRUV RI D QRQGHIHFWLYH PDWUL[ $ FDQ EH REWDLQHG DV IROORZV $OJRULWKP *LYHQ D QRQGHIHFWLYH PDWUL[ $ 5Q[Q D WROHUDQFH WRO JUHDWHU WKDQ WKH XQLW URXQGRII D FRDOHVFLQJ WROHUDQFH WROO JUHDWHU WKDQ WKH VTXDUH URRW RI WKH XQLW URXQGRII DQ LQYHUWLEOH PDWUL[ $U DQG D EORFN GLDJRQDO PDWUL[ $ WKH IROORZLQJ DOJRULWKP XVHV f WR FRPSXWH D EORFN GLDJRQDOL]DWLRQ $ ;$;aO 'HILQH WRO WROO 1H[W ZH EUHDN WKH VNHWFK RI WKH DOJRULWKP LQWR VHYHUDO VWHSV 6WHS 7DNH ; ; < ;a? $ $ /HW 1' GHQRWH WKH QXPEHU RI GLDJRQDO EORFNV RI $ DQG OHW 16 GHQRWH DQ DUUD\ RI 1% HOHPHQWV ZKRVH ]WK HOHPHQW LV WKH GLPHQVLRQ RI WKH ]WK GLDJRQDO EORFN RI $ 6WHS /HW P f§ 1% )RU L M GHILQH ',) PLQ^ ?D f§ X!? D* $$Mf DQG FF H $$f ` ,I ',) WROO WKHQ PHUJH EORFNV $ DQG $A WR IRUP D VLQJOH EORFN XVLQJ

PAGE 47

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n LQ WKH GHFUHDVLQJ RUGHU RI VL]HV 6WHS &RQVWUXFW )$n $f LQ WKH IROORZLQJ ZD\ 7DNH )MM ; $f ZKLFK LV DQ PM [ PM ]HUR PDWUL[ DQG IRU L A M )cM ; $f FDQ EH GHWHUPLQHG E\ XVLQJ WKH IROORZLQJ ORRS IRU M ;% IRU L f§ ;% LI LrM 6ROYH 3$M $ L3 <" $; M IRU 3 HQG HQG HQG 6WHS /HW IVf __*Vf__I ZKHUH *Vf $ f§ ;Vf$Vf;Vf $nVf ; V)f DQG $Vf $ V 'LDJ ^;aO $;f $f (YDOXDWH I^Vf DW V  ‘ VWRSSLQJ ZKHQ f /HW EH WKH ILUVW YDOXH RI V IRU ZKLFK f LV WUXH

PAGE 48

6WHS 6XSSRVH >$PL $P" f f f AQE@ ZKHUH P 1% ZLWK GLP$f IRU L "U DQG GLP$Wf IRU L P 1% 'HILQH G P GLDJ <$;f W f f§ ZKHUH W A 16Lf DQG GLDJ0fW f LV D YHFWRU IRUPHG W L IURP W WKURXJK Q HOHPHQWV RI WKH YHFWRU GLDJ$If 2XU DLP LV WR ILQG WZR LQGLFHV ] DQG LQ DV IROORZV /HW N EH DQ DUUD\ ZLWK WKH SURSHUW\ WKDW ILNLf /ONLLf 1H[W IRUP WKH UDWLR U B AIHWLf a ANLf ANLf A$UWf DQG OHW UWr PLQ^ L f§ U ` /HW LO W f§ PLQ^IFf  f ` DQG ]X W f§ PD[^ $]f $] f ` 6WHS 8SGDWH $ ; DQG DV IROORZV $ QHZ ; fQHZ PHZ A A QHZ 6WHS &RQVLGHU WKH EORFN $M ZLWK M ZKHUH LV GHILQHG DV LQ 6WHS /HW $ M 4847 EH D 6FKXU GHFRPSRVLWLRQ RI $ M 7KHQ 8 1 ZKHUH f§ GLDJ 8LX 8 f f f 8D f DQG 1 LV WKH VWULFWO\ XSSHU WULDQJXODU SDUW RI 8 8f LV HLWKHU D [ PDWUL[ RU D [ PDWUL[ /HW 18 EH DQ DUUD\ ZKRVH ]WK HOHPHQW LV WKH GLPHQVLRQ RI WKH EORFN 8b 8VH $OJRULWKP ZLWK WRO DV WKH FRDOHVFLQJ WROHUDQFH WR UHGXFH 8 WR D EORFN GLDJRQDO PDWUL[ WR XSGDWH WKH DUUD\ 18 DQG WKH QXPEHU RI EORFNV DQG WR JHW WKH LQYHUWLEOH PDWUL[ 6 5HSODFH $ M E\ e ;M E\ ;M46
PAGE 49

6WHS &RPSXWH f ??$ f§ $n$9n__U *RWR 6WHS XQWLO f WRO ([DPSOH ,I ZH DSSO\ $OJRULWKP WR n f $ B ZLWK $ B B ; WRO K DQG WROO WKHQ DIWHU LWHUDWLRQV WKH GLDJRQDO PDWUL[ $R FRQYHUJHV WR f§ GLDJ 66 7KH HLJHQYDOXHV RI DUH $ > f§ + f§ f§ + ([DPSOH ,I $OJRULWKP LV DSSOLHG WR f§ B ‘ $ ZLWK $ B $nR WKH WROHUDQFH DQG WKH FRDOHVFLQJ WROHUDQFH DUH DV LQ ([DPSOH WKHQ DIWHU LWHUDWLRQV WKH GLDJRQDO PDWUL[ $T EHFRPHV GLDJ f ,Q WKH IROORZLQJ VXFFHVVLYH LWHUDWLRQV FKDQJHV WR f ,f ,f GLDJ ‘ f GLDJ f GLDJ f GLDJ f ,Q WKH DERYH LWHUDWLRQV V f§ LV WKH $UPLMRnV VWHSVL]H )URP WKH GLDJRQDO PDWULFHV f f f LW LV FOHDU WKDW WKH ILUVW DQG WKH WKLUG HQWULHV DORQJ WKH GLDJRQDO DUH VORZO\ DSSURDFKLQJ HDFK RWKHU

PAGE 50

([DPSOH &RQVLGHU WKH PDWUL[ ,I ZH DSSO\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW ZLWK WRO WR $ WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D UHDO 6FKXU IRUP 67$6 ^ DQG WKH HLJHQYDOXHV RI 8 DUH $ >  r rf r r f§ @ 1H[W ZH XVH WKH WHFKQLTXH GLVFXVVHG SULRU WR ([DPSOH WR ILQG FRUUHVSRQGLQJ HLJHQYHFWRUV +HUH IRU HDFK SDLU RI FRPSOH[ FRQMXJDWH HLJHQYDOXHV ZH VROYH WKH HTXDWLRQ %[ f§ RQO\ IRU RQH DQG WKHQ WDNH WKH UHDO DQG WKH LPDJLQDU\ SDUWV RI [ DV WZR YHFWRUV FRUUHVSRQGLQJ WR WKH FRPSOH[ FRQMXJDWH SDLU RI HLJHQYDOXHV :H GHQRWH WKH PDWUL[ RI YHFWRUV E\ ; &RQVLGHU WKH IROORZLQJ SHUWXUEHG PDWUL[ $? ZKLFK ZH REWDLQ E\ SHUWXUELQJ WKH HOHPHQWV RI $ ,I ZH XVH $OJRULWKP WR $? ZLWK ;T f§ ; $R ;aO$; WRO E DQG WROO WKHQ DIWHU LWHUDWLRQV ZH JHW D EORFN GLDJRQDOL]DWLRQ
PAGE 51

UGLDJ ZKHUH < 4GLDJ f§ f§ f§ f47 DQG 4 LV DQ RUWKRJRQDO PDWUL[ ,I ZH DSSO\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW ZLWK WRO WR $ WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D UHDO 6FKXU IRUP 67$6 8 1H[W ZH XVH WKH WHFKQLTXH GLVFXVVHG SULRU WR ([DPSOH WR ILQG FRUUHVSRQGLQJ HLJHQYHFWRUV :H GHQRWH WKH PDWUL[ RI HLJHQYHFWRUV E\ ; &RQVLGHU WKH PDWUL[ ,I ZH XVH $OJRULWKP WR WKH SHUWXUEHG PDWUL[ $? $ 3 ZLWK ;T ; $ ;ar$; WRO DQG WROO WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D EORFN GLDJRQDOL]DWLRQ =a[$?= '? ZKHUHDV LI ZH DSSO\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WR 6 $L6 WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D UHDO 6FKXU IRUP 4I67$6f4 5 $OWKRXJK WKLV PHWKRG FDQ EH DSSOLHG WR ILQG HLJHQYDOXHV RI VRPH PDWULFHV ([DPn SOH f LQ JHQHUDO LW GRHV QRW ZRUN IRU DOO PDWULFHV )RU H[DPSOH LI D PDWUL[ KDV HTXDO HLJHQYDOXHV RU WKH VL]H RI WKH PDWUL[ LV ODUJH WKHQ WKH PHWKRG GRHV QRW ZRUN ([DPSOH f ,Q DOO RI WKHVH FDVHV WZR O[O GLDJRQDO EORFNV DSSURDFK HDFK RWKHU EXW DV WKH LWHUDWLRQV FRQWLQXH WKH UDWH DW ZKLFK WKH\ DSSURDFK HDFK RWKHU JHWV VORZHU DQG VORZHU +RZHYHU LI E\ DQRWKHU PHWKRG D EORFN GLDJRQDOL]DWLRQ $ ;$;aO FDQ EH FRPSXWHG WKHQ ZH FDQ XVH $ DV WKH VWDUWLQJ JXHVV EORFN GLDJRQDO PDWUL[ DQG ; DV WKH VWDUWLQJ JXHVV LQYHUWLEOH PDWUL[ WR FRPSXWH D EORFN GLDJRQDOL]DWLRQ RI WKH

PAGE 52

SHUWXUEHG PDWUL[ $ H( ZKHUH ( LV DQ DUELWUDU\ PDWUL[ DQG H LV D VPDOO VFDODU ([DPSOH DQG f &RQYHUJHQFH RI %ORFN 'LDJRQDOL]DWLRQ RI D 0DWUL[ ,Q 6HFWLRQ ZH GHYHORSHG $OJRULWKP WR EORFN GLDJRQDOL]H D QRQGHIHFWLYH PDWUL[ +HUH ZH ZLOO H[DPLQH WKH VSHHG RI FRQYHUJHQFH RI $OJRULWKP /HW $ EH DQ Q [ Q QRQGHIHFWLYH PDWUL[ DQG OHW $ < $ nL EH D EORFN GLDJn RQDOL]DWLRQ RI $ ZKHUH $ GLDJ $L $ $r f LV D EORFN GLDJRQDO PDWUL[ DQG < > !L
PAGE 53

7K H Q ??'LDJ ;aO$;f $__ __e DQG ; UQHZ \ __e 3URRI RI 7KHRUHP $VVXPH WKDW LI ?ScM? ?TLM? IRU L M Q WKHQ __3__ ,,4,, 1RZ ;aO$; (f
PAGE 54

7KXV 'LDJ$e f§ ($f 'LDJ&f DQG IURP f ZH KDYH 'LDJ$f $;f f§ $ 'LDJ/f +HQFH 0'LDJ 9n$$nf $__ __'LDJ,f__ O_L_O__e f ZKLFK SURYHV WKH ILUVW SDUW RI 7KHRUHP 6LQFH 'LDJA< $;nM f§ GLDJ 'OO 'Nf DQG ) )Mf LV D N [ N EORFN PDWUL[ ZKHUH )Q DQG IRU L A M )LM LV WKH VROXWLRQ % WR WKH PDWUL[ HTXDWLRQ %'M 'L% $0,f VR )' ') ZKHUH *LM IRU W M )LM'M 'W)LM IRU L s M IRU +HQFH )' ') ;aO $ ;f WIRU L M QRQGLDJA$ $
PAGE 55

/HW : 9 )^M8 7KHQ WKH DERYH HTXDWLRQ UHGXFHV WR ,76 f§ ,: 9a[+^M8 9 [/LM8 1RZ VROYLQJ IRU : X!Pf ZH JHW ZcP 3LP TLP ZKHUH SP 9n+LM8f OP D P 0O L 4LP 9n/LM8f OP D P 0O O UD DQG UQ 3 SPf DQG 4 TLPff 7KHQ VLPSOLI\LQJ WKH UHODWLRQ : /HW 3 4 ZH REWDLQ )^ M 938a 948 L 1H[W OHW $M WL(LLM DQG $ 9?eO?9[ EH GLDJRQDOL]DWLRQV RI $M DQG $ L UHVSHFWLYHO\ 8VLQJ f ZH KDYH __$ f§ 'W?? __-6__f ZKHUH W LM 6R WKHUH H[LVW kLkUL DQG 7 VXFK WKDW 8? 8 kL 6L 78 9L L O k DQG  I@ U DQG __e__f DQG __U__ __A__f ZKHUH W 1RZ F f( UfL kf L IV L7M V UfWB ] 2LWf VW 87[8aO *L6W 2LAIf - kLWf 6L L L ZKHUH $L XU[X kL V kUI H WUL 6L L L kUWfLLf kL8aOfa? +HQFH L L?L ,,$+ LLWUW ??4[;8 :2[7L8 ??XPaO XUWXaO 4LAX L L UL__ __W R ,OOfOOf kL Lf 6LPLODUO\ $ O LOO L ZLWK +$,, ??(??f 6LQFH 3LM L 62 9: 9 LM X 9n(06 Q9n(W-8 7 eW-$ $f 8 X f EDNLQJ WKH Pf HQWU\ RQ ERWK VLGHV RI f ZH KDYH 9n=YLf 9bOUW 9nefWfP Y (LM$L r(ff8f OP

PAGE 56

$IWHU VLPSOLI\LQJ WKH ODVW HTXDWLRQ ZH JHW Sc ^9n+D8f P D ZKHUH Vc 9n^(LL$$(LMf8fcP P LL P D P 8O /HW 6 f§ VPf WKHQ 3 (LM8!f 6P 9O(Q8 DQG ZLWK WKLV YDOXH RI 3 f UHGXFHV WR )D 99n(D88 96 4f8a X X (LM 7 5D L X ZKHUH n (LM L 0 p P L A L 7KXV ) 9f§ 6 4f8 6LQFH Vc 9an(O-$$(O-f8f OP P D P 8L DQG TL P OP 62 Y ,,A,.,,A, ,,,r f O O_$_ (D f X PLQ NP f§ 8O OOAO 8L OOLFLO PLQ_ pP /, OLH 2 ( ( 5 ZKHUH 5 5W-f ZLWK ??5? ? __(__f 1H[W ;QHZ ;, )f <, (f,( 5f < (f < (f5 ;QHZ< f§<( < (f 5 +HQFH ; QHZ \ ??(??f 7KLV SURYHV WKH VHFRQG SDUW RI 7KHRUHP ’ 7KH UHVXOW RI WKH DERYH WKHRUHP LPSOLHV WKDW WKH EORFN GLDJRQDOL]DWLRQ RI D QRQn GHIHFWLYH PDWUL[ FRQYHUJHV ORFDOO\ TXDGUDWLFDOO\ %ORFN 6FKXU 'HFRPSRVLWLRQ RI D 0DWUL[ 7KH LWHUDWLYH PHWKRGV ZH GHYHORSHG VR IDU WR ILQG WKH HLJHQYDOXHV DQG FRUUHVSRQGn LQJ HLJHQYHFWRUV RI D PDWUL[ KDYH D OLWWOH SUDFWLFDO LQWHUHVWV XQOHVV ZH KDYH D JRRG

PAGE 57

DSSUR[LPDWLRQ WR WKH PDWUL[ RI HLJHQYHFWRUV DQG LQ VRPH PHWKRGV D JRRG DSSUR[Ln PDWLRQ WR WKH HLJHQYDOXHV DV ZHOO +HUH ZH ZLOO ILQG DQ LWHUDWLYH PHWKRG WR FRPSXWH D EORFN 6FKXU GHFRPSRVLWLRQ RI D PDWUL[ RI $ /HW $ EH DQ Q[Q FRPSOH[ PDWUL[ /HW $ 686+ EH D EORFN 6FKXU GHFRPSRVLWLRQ RI $ ZKHUH 8 LV D N [ N EORFN XSSHU WULDQJXODU PDWUL[ ZKRVH Mf EORFN LV DQ P [ PM PDWUL[ DQG > L 6 f f f r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f DQG 8Hf ZLWK f DQG ef 8 VXFK WKDW $ H(f6Hf 6Hf8Hf 7KHRUHP /HW $ EH DQ Q[Q FRPSOH[ PDWUL[ 6XSSRVH 6 $6 L7 LV D EORFN 6FKXU GHFRPSRVLWLRQ RI $ VXFK WKDW WKH GLDJRQDO EORFNV 8D DQG 8MM IRU L A M KDYH GLVWLQFW HLJHQYDOXHV 7KHQ JLYHQ DQ DUELWUDU\ Q [ Q FRPSOH[ PDWUL[ ( DQG IRU D VXIILFLHQWO\ VPDOO H WKHUH H[LVW FRQWLQXRXVO\ GLIIHUHQWLDEOH IXQFWLRQV Hf DQG 7Hf ZLWK f DQG 8f 8 VXFK WKDW WKH IROORZLQJ PDWUL[ HTXDWLRQ KROGV $ H(f Hf HfHf f 3URRI RI 7KHRUHP )LUVW ZH GLVFXVV WKH HIIHFW RI SHUWXUEDWLRQV LQ WKH HOHPHQWV RI $ RQ WKH FROXPQV RI WKH XQLWDU\ PDWUL[ LQ LWV 6FKXU GHFRPSRVLWLRQ 6+$6 ( 7KH FROXPQV 6M6 VQ RI IRUP D EDVLV IRU &7f *LYHQ DQ DUELWUDU\ Q [ Q PDWUL[ LV DQG IRU D VXIILFLHQWO\ VPDOO H OHW HfBW W(f6Hf Hf EH D WULDQJXODU GHFRPSRVLWLRQ RI H( E\ DQ LQYHUWLEOH PDWUL[ Hf /HW 6+$ HYf

PAGE 58

DQG OHW 4Wf O'4Hf 8Hf EH WKH WULDQJXODU GHFRPSRVLWLRQ RI E\ WKH LQYHUWLEOH PDWUL[ 4^Hf 7KHQ ZH KDYH VHf 6TLHf L Q 7KH IRUP RI WKH PDWUL[ LV LOOXVWUDWHG IRU D [ PDWUL[ ZLWK N f§ VXFK WKDW GLP8Qf GLPLW\f DQG GLPWMMf L XQ XL XQ A A 8O Q O A XE X 8 A A XE A 8 A A A A A A ZKHUH JWM VA(VM (TXDWLQJ WKH MWK FROXPQ RQ ERWK VLGHV RI '4Hf f§ 4Hf8Hf ZH REWDLQ f f KM Q Q FAmLMef eJQHfXLMHf O L f L 6LQFH Mf M VR ZH PXVW KDYH TMf HM ZKLFK LV WKH MWK XQLW YHFWRU 1H[W ZH QRUPDOL]H JHf M Q E\ VHWWLQJ bHf L ; r r r DQG WDNLQJ DV WKH ODUJHVW HOHPHQW )RU VXIILFLHQWO\ VPDOO FODLP WKDW TQHf ZKLFK LV WKH VW FRPSRQHQW RI TLHf ,I QRW OHW TQLHf 7KHQ IURP f ZH JHW Q 8X8f QQ 1RZ OHWWLQJ H f§! ZH ILQG XL L QQ D FRQWUDGLFWLRQ 1H[W ZH ZLOO VKRZ WKDW _cUQLHf_ 0Q?W DV H f§! IRU VRPH 0Q? 6LQFH ?TQFf? VR IURP f ZH REWDLQ If X QQ NQOHf_ < ?QO? /HW _8XHf f§ X QQ _Lf 8 QQ WKHQ _fLHf_ $QLH ZKHUH  0 nLL X QQ

PAGE 59

8VLQJ WKH DERYH DUJXPHQWV WR WKH FRPSRQHQWV TQ L Q f§ LQ WKH GHFUHDVn LQJ RUGHU RI WKH URZ LQGH[ L ZH FDQ VKRZ WKDW TLHf A DQG _LHf_ 0LH ZKHUH 0X $JDLQ IURP f ZH KDYH Q Q XQHf XLLfmLL H < XTQHf Q Q XfHf 8XfTLHf AWLMLHf JLTL?^Wf 7R PDNH WKH DERYH WZR HTXDWLRQV FRQVLVWHQW ZH WDNH "L 8VLQJ VLPLODU DUJXn PHQWV DQG FRQVLGHULQJ WKH HOHPHQWV LQ WKH GHFUHDVLQJ RUGHU RI WKH URZ LQGH[ ZH FDQ VKRZ WKDW ?TLMWf? 0;IRU VRPH M Q f§ L M Q 7R VKRZ ?TWMHf? 0^MH ZH XVH WKH UHVXOWV _Hf_ $H !fff!M DQG _Hf_ 0LMW O ] Q 7R NHHS WKH FRQVLVWHQF\ ZH WDNH Ff /HW 0 P D [ MQ MLQ ;, 7KHQ ?TW-Hf? 0H M Q f§ L M Q 7KXV TcMHf DUH RI RUGHU H IRU M Q f§ L M H[FHSW L8fL DQG Ff ZKLFK DUH HTXDO WR ]HUR 1H[W IURP f ZH KDYH LL Q Q X M M Hf 8LLfTLMHf LFfXnFf f + f &2 m + MHf m r  7KH WHUP RQ WKH ULJKW LV RI RUGHU F VR Q LL LM X -8LL LM < XnfLƒrf
PAGE 60

VDPH [ PDWUL[ ZLWK N f§ +HUH ZH ZLOO XVH WKH UHVXOWV RI SHUWXUEDWLRQV RI WKH FROXPQV RI 48f 3XWWLQJ M r DQG M L LQ f ZH REWDLQ Q Q ;@mLLHf XQHf Q Q AXXTL^Wf JLPLWf mHf Q Q HAJHf $IWHU VLPSOLI\LQJ WKH DERYH HTXDWLRQV ZH JHW XQ Hf Q ALLFf A A??
PAGE 61

+HQFH 7KXV WKH SHUWXUEDWLRQ LQ XM LV RI RUGHU H DV H f§ +HQFH Hf DQG Hf DUH GLIIHUHQWLDEOH IXQFWLRQV RI H DQG WKHLU ILUVW GHULYDWLYHV DUH FRQWLQXRXV 7KLV FRPSOHWHV WKH SURRI RI WKH WKHRUHP ’ 1RZ ZH DUH UHDG\ WR GHULYH WKH QHFHVVDU\ HTXDWLRQV WR ILQG D EORFN 6FKXU GHFRPn SRVLWLRQ $ 686 *LYHQ DQ Q [ Q FRPSOH[ PDWUL[ er DQG IRU D VXIILFLHQWO\ VPDOO F OHW Hf DQG 8Ff EH FRQWLQXRXVO\ GLIIHUHQWLDEOH IXQFWLRQV RI W VXFK WKDW f DQG 8f 8 'LIIHUHQWLDWLQJ f ZLWK UHVSHFW WR H DQG WKHQ SXWWLQJ H ZH REWDLQ $6n f (6 6?f8 68n^ f 6AO6nL2f 6K(6 6+6f f8 8nf f 6LQFH 6+ $ 86+ VR f JLYHV 86+6n^ f 6(6 6+6n^f8 8n^ f f 7DNLQJ WKH MWK FROXPQ EORFN RQ ERWK VLGHV RI f ZH KDYH 86+6nM f 6Q(6M 6+6?f8M 8nML f 6f 1H[W FRQVLGHU WKH IROORZLQJ H[SUHVVLRQV IRU 6MHf L: (6L%Hf f ZKHUH LV DQ PW [ PM PDWUL[ :H QRUPDOL]H f E\ WDNLQJ %MMWf DQG %LMHf IRU L M 'LIIHUHQWLDWLQJ f ZLWK UHVSHFW WR W DQG WKHQ VHWWLQJ H ZH JHW 6nf  6%n-2f W LO f

PAGE 62

6LQFH 6 LV XQLWDU\ VR 6f 6M IRU L A M IRU L M f 8VLQJ f DQG f ZH JHW 6+6nf %nf 6R f UHGXFHV WR 8%@^ f 6+(6M %nf8 &M f ZKHUH %Mf LV WKH MWK FROXPQ EORFN RI %nf +HQFH IRU HDFK M ZH KDYH WKH IROORZLQJ WZR PDWUL[ HTXDWLRQV L < 8D%82f 6(6 < %n08OL 8ncf IRU LM M N $ A PA A ,9 O N e bbf (6 e f8X IRU r M M N O L 7KH ODVW WZR HTXDWLRQV FDQ EH UHZULWWHQ DV N L 8nLML f 6"(6M < 8X%c$fn(%nLOf8M IRU L M M f LO M %nLMLI\8MM a 8X%ALf e 8X%^Mf IRU r M M N rf f 6ROXWLRQ RI f LV GHSHQGHQW RQ WKH VROXWLRQ RI f 6R ZH IRFXV RQ KRZ WR ILQG WKH VROXWLRQ RI f )URP WKH ULJKW KDQG VLGH RI f LW LV FOHDU WKDW WR ILQG %>f ZH QHHG HQWULHV EHORZ %>^f RQ WKH MWK EORFN FROXPQ QDPHO\ "-OMf I"MAf DQG HQWULHV RQ WKH OHIW RI %>^ f RQ WKH WK EORFN URZ QDPHO\ Af -Bf 3LFWRULDOO\ WKHVH FDQ EH GHVFULEHG ZLWK ]HUR VXSSUHVVHG LQ %>^f DV

PAGE 63

7KDW LV WR GHWHUPLQH %> ZH QHHG HQWULHV MXVW DERYH WKH DUURZKHDGV GLUHFWLQJ WRZDUGV WKH OHIW DQG MXVW RQ WKH ULJKW RI WKH GRZQZDUGV DUURZKHDGV +HUH ZH ZLOO JLYH D VNHWFK WR ILQG %nW/HW & 6M (6M ,I L N WKHQ & N M & 8X%> DQG LI M WKHQ & &f§A %nX8LM 1H[W ZH QHHG WR VROYH WKH PDWUL[ rL L HTXDWLRQ %>`8M f§ Lf3>‘ & IRU %>\ 7R VROYH WKLV HTXDWLRQ OHW 3 %IW) 8MM DQG 8X 7KHQ LW EHFRPHV *3 f§ 33 ZKHUH 5 f§& 6LQFH ) DQG DUH XSSHU WULDQJXODU PDWULFHV VR XVLQJ WKH IROORZLQJ PHWKRG GHWDLO LV LQ >*RO SDJH @f ZH FDQ VROYH WKH PDWUL[ HTXDWLRQ *3 f§ 3) 5 IRU 3 /HW 5 >T n DQG 3 >SLS" f f L3PM @ EH FROXPQ SDUWLWLRQLQJV WKHQ VROYH LL ILOOf 3O 7 A P3P P O IRU S LM 2QFH ZH REWDLQ 3 WKHQ %> 3 IRU  M M $Q $OJRULWKP IRU %ORFN 6FKXU 'HFRPSRVLWLRQ RI D 0DWUL[ ,Q 6HFWLRQ ZH GHULYHG WKH QHFHVVDU\ HTXDWLRQV WR ILQG D EORFN 6FKXU GHFRPn SRVLWLRQ RI D PDWUL[ +HUH ZH SODQ WR GHYHORS DQ DOJRULWKP XVLQJ $UPLMRfV UXOH IURP RSWLPL]DWLRQ WKHRU\ WR ILQG D EORFN 6FKXU GHFRPSRVLWLRQ RI D PDWUL[ /HW $ EH DQ Q [ Q FRPSOH[ PDWUL[ 6XSSRVH $ 686 LV D EORFN 6FKXU GHn FRPSRVLWLRQ RI $ ZKHUH 8 LV D N [ N EORFN XSSHU WULDQJXODU PDWUL[ VXFK WKDW 3Q DQG 8MM IRU L M KDYH GLVWLQFW HLJHQYDOXHV DQG 6 > 6L 6r @ LV D FRPSDWLEOH M EORFN FROXPQ XQLWDU\ PDWUL[ VXFK WKDW $6M 6M8MM *LYHQ DQ DUELWUDU\ FRPSOH[ L L PDWUL[ ) DQG IRU D VXIILFLHQWO\ VPDOO F OHW 6Hf DQG )Hf EH DQDO\WLF IXQFWLRQV RI H VXFK WKDW 6f 6 8f ) DQG $ F(f6Hf 6Hf8Hf f $ ILUVW RUGHU 7D\ORU ([SDQVLRQ JLYHV Ff m 6f H6nf )Hf } 8^f H8nf

PAGE 64

7R GHYHORS DQ DOJRULWKP VXSSRVH 686+ LV DQ DSSUR[LPDWH EORFN 6FKXU GHFRPSRn VLWLRQ RI % :H LGHQWLI\ $ DQG ( LQ f ZLWK 686+ DQG % f§ 686+ UHVSHFWLYHO\ /HW nf 6*6 ^f ZKHUH *cM6 8f LV DQ [ UULM ]HUR PDWUL[ IRU L M DQG IRU L M *LM6 Wf LV WKH VROXWLRQ 3 WR WKH PDWUL[ HTXDWLRQ 38M f§ I]3 & ZKHUH & 6 E 686+f 6£  8X*LM &f  &f& O )URP f ZH KDYH 6f& Inf 1H[W VXEVWLWXWLQJ WKH YDOXHV RI er DQG nf LQ WKH DERYH HTXDWLRQ DQG WKHQ VLPSOLn I\LQJ ZH REWDLQ 8n^ f 8*^6 8f 6K%6 8 *6 8f8 1RZ VXEVWLWXWLQJ WKH YDOXHV RI IIf DQG 6nf LQWR f DQG VXSSUHVVLQJ ]HURV LQ 6f DQG 8f ZH JHW 6Hf } 6^, H*68ff 8Wf m > HW*&f L%L*Wf&f 7DNLQJ H LQ f ZH REWDLQ QHZ QHZ ,I WKH VWDUWLQJ JXHVV XQLWDU\ PDWUL[ 6R LV QRW D JRRG DSSUR[LPDWLRQ RI 4 ZKHUH % 454,^ LV D 6FKXU GHFRPSRVLWLRQ RI /" WKHQ LQ WKH XSGDWH 6QHZ 6ROG, 6ROG 8ROGff WKH LQFUHPHQW 6ROG* 6ROG 8ROGf PD\ EH WRR ODUJH +RZHYHU ZLWK D ODUJH LQFUHPHQW ZH PD\ OLNHO\ KDYH LQVWDELOLWL\ LQ WKH XSGDWLQJ SURFHVV DQG WKH DOJRULWKP PD\ GLYHUJH 7R UHVWRUH WKH FRQYHUJHQFH RI WKH DOJRULWKP DQG WR KDYH D

PAGE 65

VWHDG\ FKDQJH LQ WKH YDOXHV RI QHX DQG 8QHZ ZH QHHG D VPDOO LQFUHPHQW LQ HDFK LWHUDWLRQ 7R DFKLHYH WKLV ZH UHGHILQH HDFK LWHUDWH DQG LQWURGXFH D VPDOO SRVLWLYH SDUDPHWHU /HW W EH D SRVLWLYH SDUDPHWHU GHILQH 6Wf 6ROG W* R0 eR£ff DQG 8ROGW XROG* 6ROG8ROGf 6ROGf %6ROG 8ROG 6ROG8ROGf 8ROGf 7KHQ f 6ROG f QHZ f 8ROG DQG f QHZ 'HILQH IWf ??=Wf??) ZKHUH =Wf % 6Wf8Wf6Wfn 7KHQ f ??% fff + ) DQG f __OfLOfOf U :KHQ WKH VWDUWLQJ JXHVV R DQG 8R DUH JRRG DSSUR[LPDWLRQV RI 4 DQG 5 LQ WKH IDFWRUL]DWLRQ % 454 WKHQ ZH PXVW KDYH f f $V HDUOLHU KHUH RXU JRDO LV WR ILQG D W IRU ZKLFK IWf 2f KROGV 7R WKLV HQG ZH XVH $UPLMRnV UXOH IURP RSWLPL]DWLRQ WKHRU\ ,Q $UPLMRnV UXOH ZH GHWHUPLQH W LQ WKH IROORZLQJ ZD\ (YDOXDWH IWf DW W W VWRSSLQJ ZKHQ P L fm}f $V LQ 6HFWLRQ WR XVH WKH DERYH UXOH ZH PXVW KDYH n2f f 6R RXU QH[W DLP LV WR GHWHUPLQH n2f ZKHQ IWf __=f__U DQG =Wf % 6Wf8Wf6Wf? 6XSSUHVVLQJ WKH VXSHUVFULSWV RI 6ROG DQG 8ROG LQ WKH GHILQLWLRQV RI Wf DQG 8Wf ZH JHW Lf W*68ff W >8* f f f f : LWK WKH DERYH GHILQLWLRQ RI f ZH ZLOO VKRZ LQ WKH QH[W WKHRUHP WKDW nf f WRUH LQ I /HW $ ( &L;Q DQG VXSSRVH $ 6/U6 LV D EORFN 6FKXU GHFRPSRVLWLRQ RI $ ZKHUH 8 LV D N [ N EORFN XSSHU WULDQJXODU PDWUL[ VXFK WKDW 8D DQG 8MM IRU L A M KDYH GLVWLQFW HLJHQYDOXHV DQG 6 > 6? A D@ LV D FRPSDWLEOH EORFN FROXPQ XQLWDU\ PDWUL[ ,I ZH GHILQH  f 6Wf DV LQ =^Wf $ f§ 6Wf Wf6Wf O DQ G P ??=Wf??) WKHQ Inf Rf

PAGE 66

3URRI RI 7KHRUHP 7KH UHVXOW f RI /HPPD JLYHV IfaP? R WUDFH ]f+ M=Wf?,0 f 'LIIHUHQWLDWLQJ WKH H[SUHVVLRQ IRU =Wf ZLWK UHVSHFW WR W DQG WKHQ HYDOXDWLQJ WKH GHULYDWLYH DW W ZH KDYH G GW =Wf?W G GW 6Wf?W R8f6f 6f_LLf8Rff 6fef6f GB GW 6Wf?W R6f f 1H[W GLIIHUHQWLDWLQJ 8Wf DQG 6Wf ZLWK UHVSHFW WR DQG WKHQ HYDOXDWLQJ WKH GHULYDn WLYHV DW W ZH REWDLQ GB GW 8Wf?W 8*6 8f 6K$6 8 *6 8f8 DQG M6Wf?W 6*68f 8VLQJ WKHVH YDOXHV DQG YDOXHV RI f DQG 6n2f LQ f ZH JHW G GW =Wf R 6*^6 8f86+ 6 8*6 8f 6K$6 8 *^6 8f8f 6 + 686K6*68f6 + +HQFH f§=Wf?W GW 686 + =^ f =f DQG ZLWK WKLV YDOXH f UHGXFHV WR G fLf_ f WUDFH =f f§=fff f Vr IWP?W R f SURYLGHG f s ’ 7KXV $UPLMRfV UXOH FDQ EH DSSOLHG WR ILQG D EORFN 6FKXU GHFRPSRVLWLRQ RI D PDWUL[ $ DQG ZH ZLOO FRPSXWH D EORFN 6FKXU GHFRPSRVLWLRQ RI $ DV IROORZV

PAGE 67

$OJRULWKP %ORFN 6FKXU 'HFRPSRVLWLRQf *LYHQ $ & Q [ Q D WROHUDQFH WRO JUHDWHU WKDQ WKH XQLW URXQGRII D FRDOHVFLQJ WROHUDQFH WROO JUHDWHU WKDQ WKH VTXDUH URRW RI WKH XQLW URXQGRII DQ XSSHU WULDQJXODU PDWUL[ 8T DQG D XQLWDU\ PDWUL[ 6R WKH IROORZLQJ DOJRULWKP XVHV f WR FRPSXWH D EORFN 6FKXU GHFRPSRVLWLRQ $ 686+ 'HILQH WRO WROO :H EUHDN WKH VNHWFK RI WKH DOJRULWKP LQWR VHYHUDO VWHSV 6WHS 7DNH 6R 8 8R /HW 1% GHQRWH WKH QXPEHU RI GLDJRQDO EORFNV RI 8 DQG OHW 16 GHQRWH DQ DUUD\ RI 1% HOHPHQWV ZKRVH WK HOHPHQW LV WKH GLPHQVLRQ RI WKH WK GLDJRQDO EORFN RI 8 6WHS /HW P 1% )RU L M GHILQH D PLQ ^ ?D f§ XM? D $f DQG X $8MMf ` ,I RWLM WROO WKHQ PHUJH EORFNV 8D DQG 8MM WR IRUP D VLQJOH EORFN XVLQJ D XQLWDU\ WUDQVIRUPDWLRQ WR 8 3RVWPXOWLSO\ 6 E\ WKH VDPH XQLWDU\ WUDQVIRUPDWLRQ 8SGDWH 1%ROG DQG 16ROG WR JHW 1%QHZ DQG 16QH[Y 7U\ WKH DERYH FRXSOLQJ SURFHGXUH IRU DOO SRVVLEOH FRPELQDWLRQV RI L M P 8VLQJ D XQLWDU\ WUDQVIRUPDWLRQ DUUDQJH GLDJRQDO EORFNV RI 8 LQ WKH GHFUHDVLQJ RUGHU RI VL]HV 3RVWPXOWLSO\ 6 E\ WKH VDPH XQLWDU\ WUDQVIRUPDWLRQ 6WHS &RQVWUXFW *68f DV IROORZV )RU L M WDNH *WM6 8f f§ ZKLFK LV DQ UULL;PM ]HUR PDWUL[ DQG IRU L M *W-6 8f FDQ EH GHWHUPLQHG E\ XVLQJ WKH IROORZLQJ ORRS % $6 68 IRU M 1% f§ IRU L 1% f§ M & 6 %M LI L 1% 1% & & e 8LN*NM^68f Nf§L O HQG

PAGE 68

LI LL F F<-*LN^V XfXNM N HQG 6ROYH 38`@ 83 & IRU 3 *LM6 8f f§ 3 HQG HQG 6WHS /HW I^Wf __=f__) ZKHUH =Wf $ 6Wf8Wf6^Wfn 6^Wf 6^, W*6 eff DQG 8Wf 8 W 8*^6 8f 6+$6 8 *6 efef (YDOXDWH IWf DW L fffVfrrnf VWrSSLQJ ZKHQ rf O _f f f /HW S EH WKH ILUVW YDOXH RI W IRU ZKLFK f LV WUXH 6WHS 6XSSRVH > WPLPL 8QEQE @7 ZKHUH P 1% ZLWK GLPWWf IRU L f§ DQG GLP>f IRU L P $e/ 'HILQH G P GLDJ 8*68f 8 *68f8f W Qf ZKHUH  < 16Lf DQG }n GLDJ0f  Qf LV D YHFWRU IRUPHG IURP W WKURXJK Q HOHPHQWV RI WKH YHFWRU GLDJ0f 2XU DLP LV WR ILQG WZR LQGLFHV DQG LX DV IROORZV /HW N EH DQ DUUD\ ZLWK WKH SURSHUW\ WKDW ?ANLf ?ILNLLf? 1H[W IRUP WKH UDWLR 3NWLOf 9NLf GNLf G IFWOf DQG OHW ULR PLQ^ _Uc L Q f§ 5H Uf t _UW_ ` /HW LO W PLQ^ $f"nf NL f ` DQG LX f§ W f§ PD[^ t]f IFm f ` 6WHS ,-SGDWH 8 DQG DV IROORZV QHZ

PAGE 69

QHZ 6ROG 3* 6rOG8ROGff /HW = 6 QHZ O UQHZ &QHZ U 6WHS &RQVLGHU WKH EORFN 8MM ZLWK M P ZKHUH P LV GHILQHG DV LQ 6WHS /HW 8MM 454+ EH D 6FKXU GHFRPSRVLWLRQ RI 8MM )LQG D XQLWDU\ PDWUL[ 3 VXFK WKDW 3853 9 1 ZKHUH GLDJ 9Q 9 f 9NN f $U LV WKH VWULFWO\ XSSHU WULDQJXODU SDUW RI 8 N PM DQG PLQ^ ?D f§ X!? D DQG OM $9f ` QHX WRO 5HSODFH 8MM E\ & DQG 6M E\ 6M43 8SGDWH 1%ROG DQG 16ROG WR JHW 1L DQG 16QHZ 7U\ WKH DERYH GHFRXSOLQJ SURFHGXUH IRU DOO M VXFK WKDW M P 6WHS &UHDWH D [ GLDJRQDO EORFN E\ PHUJLQJ 8X`X DQG 8^XMX E\ D XQLWDU\ WUDQVIRUPDWLRQ 3RVWPXOWLSO\ E\ WKH VDPH XQLWDU\ WUDQVIRUPDWLRQ 6WHS /HW 6 ROG 45 EH D 45IDFWRUL]DWLRQ RI 6ROG 7KHQ WDNH 6 QHZ 4 DQG 8 QHZ 4+=4 ZKHUH = LV WKH PDWUL[ IURP 6WHS 6WHS &RPSXWH f __$ f§ =??S DQG JRWR 6WHS XQWLO f WRO ([DPSOH ,I ZH DSSO\ WKH UHDO YHUVLRQ RI $OJRULWKP WR WKH PDWUL[ $ LQ ([DPSOH ZLWK 8T GLDJ f§ f 6R WRO DQG WROO a ? WKHQ DIWHU LWHUDWLRQV WKH XSSHU WULDQJXODU PDWUL[ 8T FRQYHUJHV WR 7KH HLJHQYDOXHV RI 8 DUH $ > f§ f§ f§ @ ,I ZH XVH WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WR $ ZLWK WKH VDPH WROHUDQFH WKHQ DIWHU LWHUDWLRQV ZH JHW D UHDO 6FKXU GHFRPSRVLWLRQ $ 4547 ZKHUH f n 5 ,2 DQG 4

PAGE 70

RI $ $? L[ $[ ZKLFK ZH REWDLQ ,I ZH DSSO\ WKH UHDO YHUVLRQ RI $OJRULWKP WR $? ZLWK 6R 4 8 5 WKH WROHUDQFH DQG WKH FRDOHVFLQJ WROHUDQFH DUH WKH VDPH WKHQ DIWHU LWHUDWLRQ ZH JHW D EORFN 6FKXU GHFRPSRVLWLRQ 37$?3 f§ IL ZKHUHDV LI ZH XVH WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WR 47$?4 WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D UHDO 6FKXU IRUP 9747$;4f9 U ([DPYOH ,I WKH UHDO YHUVLRQ RI $OJRULWKP LV DSSOLHG WR WKH PDWUL[ $ LQ ([DPSOH ZLWK 8R GLDJ f§ f§ f 6R WKH WROHUDQFH DQG WKH FRDOHVFLQJ WROHUDQFH DUH DV LQ ([DPSOH WKHQ DIWHU LWHUDWLRQV WKH XSSHU WULDQJXODU PDWUL[ 8T FRQYHUJHV WR ,I ZH XVH WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WR $ ZLWK WKH VDPH WROHUDQFH WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D UHDO 6FKXU IRUP ([DPSOH &RQVLGHU WKH IROORZLQJ PDWULFHV $ DQG 3 n n n L f L L $ 3 L DQD U f§ L

PAGE 71

,I ZH DSSO\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WR $ ZLWK WRO fE WKHQ DIWHU LWHUDWLRQV ZH JHW D UHDO 6FKXU IRUP 6 $6 8 7KH HLJHQYDOXHV RI 8 DUH $ >f§ f§ @ 1RZ LI WKH UHDO YHUVLRQ RI $OJRULWKP LV DSSOLHG WR WKH SHUWXUEHG PDWUL[ $? $ 3 ZLWK 8R 6R WROO DQG WKH WROHUDQFH LV DV DERYH WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D EORFN 6FKXU GHFRPSRVLWLRQ 47$?4 IW ZKHUHDV LI ZH DSSO\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WR 6U$?6 WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D UHDO 6FKXU IRUP 9767$?6f9 ,? $OWKRXJK WKH UHDO YHUVLRQ RI $OJRULWKP ZRUNV IRU DOO PDWULFHV $ ZLWK D GLDJn RQDO PDWUL[ IRUPHG IURP WKH GLDJRQDO HOHPHQWV RI $ DV WKH VWDUWLQJ XSSHU WULDQJXODU PDWUL[ 8R DQG 6R LW KDV VRPH GLVDGYDQWDJHV )RU H[DPSOH LI WKH VL]H RI WKH PDWUL[ LV ODUJH WKHQ ZH QHHG WR WDNH D YHU\ ODUJH FRDOHVFLQJ WROHUDQFH FRPSDUHG WR WROO f WKURXJK VHYHUDO LWHUDWLRQV DW WKH EHJLQQLQJ +RZHYHU ZLWK D ODUJH FRDn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f6 E\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW ZKHUH 6 $6 8 LV D UHDO 6FOQLU IRUP RI WKH PDWUL[ $ +HUH WKH RULJLQDO PDWULFHV $ DMf DUH UDQGRP PDWULFHV ZKHUH DWDUH LQWHJHUV DQG DM WKH PDWULFHV ( Hf DUH UDQGRP PDWULFHV ZKHUH WW@ DUH UHDOV DQG HM WKH WROHUDQFH WRO DQG WKH FRDOHVFLQJ WROHUDQFH WROO

PAGE 72

VL]H QXPEHU RI LWHUDWLRQV IRU RXU DOJRULWKP QXPEHU RI LWHUDWLRQV IRU WKH 45 DOJRULWKP ZLWK GRXEOH VKLIW H $ W( $ H( 67^$ H(f6 a f ([DPSOH ƒ ,I $OJRULWKP LV DSSOLHG WR WKH PDWUL[ $ ZLWK 8 GLDJ rn rn r rf ZKHUH WKH HOHPHQWV RI 8T DUH WKH XQLIRUPO\ GLVWULEXWHG SRLQWV RQ D FLUFOH ZKLFK LQFOXGHV DOO *HUVFKJRULQ GLVNV IRU $ 6R WRO DQG WROO WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D EORFN 6FKXU GHFRPSRVLWLRQ 6 $6 8 ZKHUH 8 1 GLDJ rn rn r  f DQG 1 LV WKH VWULFWO\ XSSHU WULn DQJXODU SDUW RI 8 )LJXUH VKRZV KRZ WKH LQLWLDO HLJHQYDOXHV RI 8R FRQYHUJHG WR WKH HLJHQYDOXHV RI $ ,I ZH XVH WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WR $ ZLWK WKH DERYH WROHUDQFH WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D UHDO 6FKXU IRUP 47$4 5 1H[W FRQVLGHU WKH IROORZLQJ PDWUL[ $? ZKLFK ZH JHW E\ SHUWXUELQJ WKH HOHPHQWV RI f

PAGE 73

)LJXUH 7KH SDWK RI FRQYHUJHQFH RI WKH HLJHQYDOXHV RI WKH PDWUL[ $ ,I ZH XVH $OJRULWKP WR $? ZLWK 8T  6R 6 WKH WROHUDQFH DQG WKH FRDOHVFLQJ WROHUDQFH DUH DV DERYH WKHQ DIWHU LWHUDWLRQV ZH JHW D EORFN 6FKXU GHFRPSRVLWLRQ 3$?3 + ZKHUHDV LI ZH DSSO\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WR 47$?4 ZLWK WKH VDPH WROHUDQFH WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D UHDO 6FKXU IRUP 97 47 $;4f9 7 ([DPSOH ,I $OJRULWKP LV DSSOLHG WR WKH PDWULFHV n ‘ n F n B B f DQG (

PAGE 74

ZLWK GLDJRQDO PDWULFHV $ GLDJ f§ f§ f GLDJ " f ( GLDJ "n f DQG f GLDJ f§ f§ f§ f§ f§ "f DV e ZKHUH WKH HOHPHQWV RI $ ( DQG  DUH WKH XQLIRUPO\ GLVWULEXWHG SRLQWV RQ FLUFOHV ZKLFK LQFOXGH DOO *HUVFKJRULQ GLVNV IRU L" & DQG ( UHVSHFWLYHO\ 6R WRO DQG WROO a WKHQ DIWHU DQG LWHUDWLRQV ZH REWDLQ EORFN 6FKXU GHFRPSRVLWLRQV RI % & DQG ( UHVSHFWLYHO\ ,I ZH XVH WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WKHQ DIWHU DQG LWHUDWLRQV ZH REWDLQ UHDO 6FKXU IRUPV RI L" & DQG UHVSHFWLYHO\ ZKHUHDV ( GLYHUJHV ([DPSOH ,I ZH DSSO\ $OJRULWKP WR $ 9GLDJ f;a? ZKHUH ; 3GLDJ f4 3 DQG 4 DUH RUWKRJRQDO PDWULFHV ZLWK 8T GLDJ "  "n "n " "n "n "n " "n "n f§ f ZKHUH WKH HOHPHQWV RI 8T DUH WKH XQLIRUPO\ GLVWULEXWHG SRLQWV RQ D FLUFOH ZKLFK LQFOXGHV DOO *HUVFKJRULQ GLVNV IRU $ 6T WRO DQG WROO WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D EORFN 6FKXU GHFRPSRVLWLRQ 6+$6 8 ZKHUH 8 1 GLDJ f§ f DQG 1 LV WKH VWULFWO\ XSSHU WULDQJXODU SDUW RI 8 ,I ZH XVH WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WR $ WKHQ DIWHU LWHUDWLRQV ZH JHW D UHDO 6FKXU IRUP TWDT 5 1H[W FRQVLGHU WKH SHUWXUEHG PDWUL[ $? $ 3 ZKHUH 3 LV WKH PDWUL[ IURP ([DPSOH ,I ZH DSSO\ $OJRULWKP WR L ZLWK 8T 8 6R 6

PAGE 75

cV WKH WROHUDQFH DQG WKH FRDOHVFLQJ WROHUDQFH DUH WKH VDPH WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D EORFN 6FKXU IRUP 3+$?3 IL ZKHUHDV LI ZH DSSO\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WR 47$?4 WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D UHDO 6FKXU IRUP YWTWDTfY U ([DPSOH ,I $OJRULWKP LV DSSOLHG WR WKH PDWUL[ & LQ ([DPSOH ZLWK 8 GLDJ r W rn } Wrn r rn rn rn rn r rn f§ r rn f§ r f§ r f§ r f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 ZKHUHDV LI ZH DSSO\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WR 47&?4 WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D UHDO 6FKXU IRUP 9747&?4f9 U ,Q WKH IROORZLQJ WDEOH IRU GLIIHUHQW YDOXHV RI Q ZH JLYH WKH QXPEHU RI LWHUDWLRQV UHTXLUHG WR REWDLQ EORFN 6FKXU GHFRPSRVLWLRQV RI $ E\ $OJRULWKP ZLWK GLDJRQDO PDWULFHV DV 8R ZKHUH WKH HOHPHQWV RI 8R DUH WKH XQLIRUPO\ GLVWULEXWHG SRLQWV RQ FLUFOHV ZKLFK LQFOXGH DOO *HUVFKJRULQ GLVNV IRU $ 6R DQG UHDO 6FKXU IRUPV RI $ E\ WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW +HUH $ DW-f DUH UDQGRP PDWULFHV

PAGE 76

)LJXUH 7KH SDWK RI FRQYHUJHQFH RI WKH HLJHQYDOXHV RI & ZKHUH DcM DUH UHDO QXPEHUV DQG D[M WKH WROHUDQFH WRO a DQG WKH FRDOHVFLQJ WROHUDQFH WROO f VL]H QXPEHU RI LWHUDWLRQV IRU RXU DOJRULWKP QXPEHU RI LWHUDWLRQV IRU WKH 45 DOJRULWKP ZLWK GRXEOH VKLIW $OJRULWKP FDQ EH DSSOLHG WR DOO PDWULFHV $ ZLWK 6R f§ 7 DQG GLDJRQDO PDn WULFHV DV 8 ZKHUH WKH HOHPHQWV RI 8R DUH WKH XQLIRUPO\ GLVWULEXWHG SRLQWV RQ FLUFOHV ZKLFK LQFOXGH DOO *HUVFKJRULQ GLVNV IRU $ 8VXDOO\ WKH VWHSVL]H LQ WKH ILUVW LWHUDWLRQ EHFRPHV ,I ZH XVH WKLV VWHSVL]H WKHQ DIWHU RQH LWHUDWLRQ DOO GLDJRQDO HOHPHQWV RI WKH UHVXOWLQJ XSSHU WULDQJXODU PDWUL[ EHFRPH UHDO DQG WKLV FUHDWHV SUREOHP LQ WKH

PAGE 77

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f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f LV XVHG LQ HDFK LWHUDWLRQ WR REWDLQ D UHDO 6FKXU

PAGE 78

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f WR GHQRWH WKH ]WK SURFHVVRU ,Q 6WHS ZLWK S SURFHVVRUV ZH FDQ DFFHOHUDWH WKH PHUJLQJ RI GLDJRQDO EORFNV LQ WKH IROORZLQJ ZD\ )RU D IL[HG ] L 1% DQG 1% f§ L S ZH FDQ ILQG D PLQ^ ?FU f§ XM? D ( $ eff DQG X $8MMf ` M L O] 9 LQ VHSDUDWH SURFHVVRUV ,I DM WROO WKHQ ZH FDQ PHUJH 8MM ZLWK 8X LQ WKH LQFUHDVLQJ RUGHU RI M ,I 1% f§ L S WKHQ ZH KDYH WR XVH VRPH SURFHVVRUV PRUH WKDQ RQFH 3URFef ZLOO FRPSXWH T ZKHUH e e ] S 1% 3DUDOOHO SURFHVVRUV FDQ DOVR EH XVHG WR DUUDQJH GLDJRQDO EORFNV RI 8 LQ WKH GHFUHDVLQJ RUGHU RI VL]HV /HW >[@ GHQRWH WKH VPDOOHVW LQWHJHU VXFK WKDW [ >[@ [ ,I 8 KDV N EORFNV DORQJ WKH GLDJRQDO WKHQ QRW PRUH WKDQ f§f§f§f§ VZDSV DPRQJ WKH GLDJRQDO EORFNV DUH QHFHVVDU\ WR DUUDQJH WKHP LQ WKH GHFUHDVLQJ RUGHU RI VL]HV ,Q D VLQJOH SURFHVVRU ZH KDYH WR H[HFXWH f§ A NN f§ f WLPHV WR ILQLVK VZDSV %XW LQ D SDUDOOHO FRPSXWHU ZLWK S SURFHVVRUV ZH FDQ XVH SURFHVVRUV ZKHUH P S LI S N VZDSV E\ H[HFXWLQJ DW PRVW NNOf P DQG P N LI N S WR ILQLVK NN f§ f VZDSV LQ HDFK SURFHVVRU )RU H[DPSOH LI WKH GLPHVnf LV ILYH VD\ 16 > L? D D @7 ZLWK FL? D D D D f DQO

PAGE 79

WKHUH DUH WZR SURFHVVRUV WKHQ ZH FDQ VZDS WKH HOHPHQWV RI 16 LQ WKH IROORZLQJ ZD\ mL D m F m D D[ D D DV D D D? m m D D m D? &O D DV D D DL D D m D DL +HUH ZH XVH RQH SURFHVVRU WR VZDS EORFNV HQFORVHG E\ D EUDFH RQ WKH ULJKW DQG WKH RWKHU SURFHVVRU WR VZDS EORFNV HQFORVHG E\ D SDUHQWKHVLV RQ WKH ULJKW (DFK SURFHVVRU GRHV FRPSDULVRQV DQG VZDSV ,Q 6WHS WR FRPSXWH WKH VWULFWO\ ORZHU WULDQJXODU PDWUL[ 8f ZH FDQ XVH SDUDOOHO SURFHVVRUV 6XSSRVH $ >$[ $S@ 6 >6L 3@ DQG 8 >8? 8S? DUH FROXPQ SDUWLWLRQLQJV RI $ DQG 8 UHVSHFWLYHO\ ZKHUH HDFK EORFN FROXPQ KDV ZLGWK U 7R FRPSXWH $6 f§ 68 ILUVW ZH ILQG I DQG WKHQ ILQG % $6 f§ N 6LQFH 8M 8LM f f f 8IML XWL H &U[U VR 'N X 68L \VXOW N OS 7KXV HDFK =fr FDQ EH FRPSXWHG LQ D VHSDUDWH SURFHVVRU 6LQFH 8 LV XSSHU WULDQJXODU VR 'S LQYROYHV PXFK PRUH ZRUN WKDQ '? 7R ORDG EDODQFH ZH DVVLJQ 3URFef WKH FRPSXWDWLRQ RI 'e S Qf 68= S Qf  6r S S Qe S Qf L DV VXJJHVWHG LQ >*RO SDJH @ 1H[W ZLWK 6M L Lf f 67 SM F U ; U % $ 6O f§ Mr 'L $ L L 6R S SURFHVVRUV FDQ EH XVHG WR FRPSXWH % ZKHUH FDQ EH RYHUZULWWHQ E\ % $V ZH PHQWLRQHG LQ 6HFWLRQ WR FRPSXWH ZH QHHG HQWULHV EHORZ *;RQ WKH MWK EORFN FROXPQ QDPHO\ *[?M *MY%ML DQG HQWULHV RQ WKH OHIW RI *‘ M RQ WKH WK EORFN URZ QDPHO\ *L 7R ORDG EDODQFH ZH ZLOO XVH WKH IROORZLQJ WZR VWUDWHJLHV WR FRPSXWH WKH HOHPHQWV RI 8f? RQH IRU S .% DQG WKH RWKHU IRU S $L

PAGE 80

)LJXUH 7KH GLUHFWLRQ LQ ZKLFK WKH HOHPHQWV RI *68f DUH FRPSXWHG ZKHQ WKH QXPEHU RI SURFHVVRUV DUH JUHDWHU WKDQ KDOI RI WKH WRWDO EORFNV 6XSSRVH S 1% aa 7KHQ ZH XVH 3URFOf WR FRPSXWH *MYEM M 1% f§ LQ WKH LQFUHDVLQJ RUGHU RI WKH FROXPQ LQGH[ M DQG WKH RWKHU S f§ SURFHVVRUV WR FRPSXWH WKH UHPDLQLQJ HOHPHQWV RI *68f LQ WKH GHFUHDVLQJ RUGHU RI WKH URZ LQGH[  DV ZH H[SODLQ EHORZ :H XVH 3URFf WR FRPSXWH *WL L 1% M 3URFf WR FRPSXWH *c L 1% f§ &RQWLQXLQJ WKH DVVLJQPHQW LQ WKLV ZD\ ZH XVH 3URFSf WR FRPSXWH *SB L 1% S ,I 1% f§ S WKHQ ZH ZLOO FRQWLQXH WKLV DVFHQGLQJ RUGHU RI DVVLJQPHQWV WR WKRVH SURFHVVRUV H[FHSW 3URFOf 7KDW PHDQV 3URFf WR FRPSXWH *S L 1% S 3URFf WR FRPSXWH *LS_L L 1% OS DQG LQ WKLV RUGHU LI QHFHVVDU\ 3URFSf WR FRPSXWH *LS L 1% f§ S f§ 3LFWRULDOO\ LW FDQ EH SUHVHQWHG E\ DUURZ KHDGV ZKHUH WKH HOHPHQWV DUH FRPSXWHG WRZDUGV WKH DUURZ KHDGV DV LQ )LJXUH

PAGE 81

)LJXUH 7KH GLUHFWLRQ LQ ZKLFK WKH HOHPHQWV RI *68f DUH FRPSXWHG ZKHQ WKH QXPEHU RI SURFHVVRUV DUH OHVV WKDQ KDOI RI WKH WRWDO EORFNV 1H[W VXSSRVH S 1% aaRa 7KLV WLPH ZH ZLOO DVVLJQ HDFK SURFHVVRU WR FRPSXWH WKH HOHPHQWV LQ WKH GHFUHDVLQJ RUGHU RI WKH URZ LQGH[ L 3URFOf WR FRPSXWH *L L $7" 3URFf WR FRPSXWH L $7" $VVLJQLQJ WKH FROXPQV LQ WKLV RUGHU ZH XVH 3URFSf WR FRPSXWH *S L $7"S 1H[W ZH ZLOO UHSHDW WKH RUGHU RI DVVLJQPHQW DV LQ WKH ILUVW VWUDWHJ\ 3URFOf WR FRPSXWH *SL L 1% S 3URFf WR FRPSXWH *S L 1% S DQG LQ WKLV RUGHU 3URFSf WR FRPSXWH *3 L 1% S ,I QHFHVVDU\ ZH ZLOO UHSHDW LW 3LFWRULDOO\ LW FDQ EH VKRZQ E\ DUURZ KHDGV ZKHUH WKH HOHPHQWV DUH FRPSXWHG WRZDUGV WKH DUURZ KHDGV DV LQ )LJXUH ,Q ERWK VWUDWHJLHV ZH FDQ QRW VWDUW DOO SURFHVVRUV VLPXOWDQHRXVO\ ,Q WKH ILUVW VWUDWHJ\ EHIRUH 3URFef e FRPSXWHV DQ HOHPHQW *\ ZH QHHG WR PDNH VXUH WKDW !fff DOUHDG\ FRPSXWHG DQG LI L 1% WKHQ PDNH VXUH WKDW 3URF f DOUHDG\ FRPSXWHG *MVM%Mf 7KH QXPEHU RI IORSV UHTXLUHG E\ WKH VHULDO

PAGE 82

DOJRULWKP WR FRPSXWH *68f LV D FRPSOLFDWHG IXQFWLRQ RI WKH EORFN VL]HV RI 8 ,I DOO EORFNV DUH O[O PDWULFHV WKHQ WKH DOJRULWKP UHTXLUHV f§Q IORSV ,Q 6WHS WR HYDOXDWH IWf IRU GLIIHUHQW YDOXHV RI L ZH FDQ XVH SDUDOOHO SURFHVVRUV 7R FRPSXWH WKH ORZHU WULDQJXODU PDWUL[ / ZKHUH /cM W*LM IRU M L $-% ZH FDQ XVH RQH RI WKH VWUDWHJLHV GLVFXVVHG HDUOLHU WR FRPSXWH *6 ef GHSHQGLQJ RQ WKH QXPEHU RI SURFHVVRUV S $VVLJQ 3URFef I M WR WDNH / MM ZKHUH SD e D DQG M 1% :H FDQ XVH WKH LGHDV GLVFXVVHG LQ WKH DQDO\VLV RI 6WHS WR ILQG PDWUL[ SURGXFWV 1H[W ZH ZLOO JLYH D VNHWFK WR ILQG WKH LQYHUVH RI D PDWUL[ LQ D GLVWULEXWHG PHPRU\ PXOWLSURFHVVRU ZKHUH SURFHVVRUV DUH VHW XS LQ D ULQJ $Q HIILFLHQW ZD\ WR ILQG WKH LQYHUVH RI D PDWUL[ $ LV WR ILQG WKH /8 IDFWRUL]DWLRQ RI $ ZLWK SDUWLDO SLYRWLQJ ILUVW /HW SLY Q f§ f EH D ]HUR YHFWRU &RQVLGHU WKH IROORZLQJ QRGH SURJUDP ZKLFK LV GXH WR + *ROXE DQG & ) 9DQ /RDQ ZKHUH e LV WKH LGHQWLW\ RI WKH eWK SURFHVVRU % $ Qe S Qf SLYRW SLYe S Q f§ f L FRO e S Q? / OHQJWKFRf ZKLOH T / LI M FROTf $ M Q ^ )LQG WKH SHUPXWDWLRQ LQGH[ SLYMf DQG *DXVV YHFWRU $M QMf ` 'HWHUPLQH N ZLWK M N VR ?%NTf ? __%M QTf__ SLYRWTf f§ N %M /f %N /f %M QTf %^M Q Tf%M Tf 6HQG SLYRWTf %M Tf WR SURFHVVRUV RQ WKH ULJKW ^ 8SGDWH ORFDO FROXPQV ` LI T / %M T /f %M Q T /f %^M Q Tf%M T /f 22

PAGE 83

HQG M M L HOVH 5HFHLYH SLYMf DQG $M QMf IURP WKH OHIW DQG LI WKH SURFHVVRU RQ WKH ULJKW LV QRW WKH SURFHVVRU ZKLFK FRPSXWHG $M Q Mf DQG M LV OHVV WKDQ WKH ODVW FROXPQ LQGH[ RI WKH SURFHVVRU RQ WKH ULJKW WKHQ VHQG SLYMf DQG $M Q Mf WR WKH ULJKW ^ 8SGDWH ORFDO FROXPQV ` %M ef %SLYMf /f %^M QT /f %^M QT /f $M QMf%MT /f M M HQG HQG $IWHU HDFK SURFHVVRU LQ D SSURFHVVRU ULQJ ILQLVKHV WKH H[HFXWLRQ RI WKH DERYH FRGH ZH ZLOO JHW WKH /8 IDFWRUL]DWLRQ RI $ 3URFef ZLOO KRXVH $ Qe S Qf LQ D ORFDO DUUD\ % DQG SLYe S Q f§ f LQ D ORFDO YHFWRU SLYRW 7KH DERYH /8 IDFWRUL]DWLRQ RI $ LV RI WKH IRUP 3$ /5 $IWHU VLPSOLI\LQJ ZH JHW $aO 5aO /aO3 DQG WKHUHIRUH WR REWDLQ $ ZH QHHG WR GHWHUPLQH /aO "BO DQG 5a@ /aO )LUVW ZH ZLOO GLVFXVV KRZ WR ILQG WKH LQYHUVH % RI D ORZHU WULDQJXODU PDWUL[ ZKHUH / LV RYHUZULWWHQ E\ % &RQVLGHU WKH IROORZLQJ QRGH SURJUDP >/N Q IFf@ N e S Q M FRO e S Q $U OHQJWKFRf ZKLOH M Q LI M FrL>Tf LI M O 6HQG 'M UDf WR SURFHVVRUV RQ WKH OHIW

PAGE 84

HQG 8SGDWH ORFDO FROXPQV ` LI T LI M 'M Q T f§ f 'M Q T f§ f 'M QTf'M T f HQG HQG '^KTf 'MTf LI M Q e!M QTf 'M QTf'MTf HQG f P A M M HOVH LI M FROf 5HFHLYH /M QMf IURP WKH ULJKW DQG LI WKH SURFHVVRU RQ WKH OHIW LV QRW WKH SURFHVVRU ZKLFK KRXVHV /M QMf DQG M LV JUHDWHU WKDQ WKH ILUVW FROXPQ LQGH[ RI WKH SURFHVVRU RQ WKH OHIW WKHQ VHQG /M QMf WR WKH OHIW ^ 8SGDWH ORFDO FROXPQV ` 'MOTf 'MOTf/M-f LI M Q 'M Q Tf 'M Q Tf /M QMf'M Tf HQG f f

PAGE 85

LI FRO^Tf S M HQG HOVH f f M M HQG HQG HQG ,I HDFK QRGH LQ D SSURFHVVRU ULQJ H[HFXWHV WKH DERYH FRGH WKHQ ZH ZLOO JHW WKH LQYHUVH RI = DQG XSRQ FRPSOHWLRQ 3URFef ZLOO KRXVH /N Q Nf IRU N e S Q LQ D ORFDO DUUD\ 2XU QH[W PRYH LV WR ILQG WKH LQYHUVH % RI DQ XSSHU WULDQJXODU PDWUL[ ZKHUH 5 LV RYHUZULWWHQ ZLWK % $ QRGH SURJUDP FDQ EH VWUXFWXUHG DV IROORZV >5^ N IFf@ N e S Q M Q FRO e S Q 1 OHQJWKFR=f T 1 ZKLOH M LI M FROTf LI M Q 6HQG M Tf WR SURFHVVRUV RQ WKH ULJKW HQG ^ 8SGDWH ORFDO FROXPQV ` LI T 1 'MT 1f 'MT 1f'MTf LI '^ MOTO 1f 'O M ?T 1f 'OMOTf'MWTO1f HQG

PAGE 86

HQG 'MTf O'MTf LI M MOTf '^ M Tf'^MTf HQG HOVH LI M FR9f 5HFHLYH L"O M M f IURP WKH OHIW DQG LI WKH SURFHVVRU RQ WKH ULJKW LV QRW WKH SURFHVVRU ZKLFK KRXVHV 5^ M Mf DQG M LV OHVV WKDQ WKH ODVW FROXPQ LQGH[ RI WKH SURFHVVRU RQ WKH ULJKW WKHQ VHQG 5 M Mf WR WKH ULJKW ^ 8SGDWH ORFDO FROXPQV ` 'MT1f 'MT1f5M-f LI M 'OMOT1f 'OMOT1fa 5 M Mf'MT 1f HQG LI FROTf f§ S T T L HQG HOVH HQG HQG HQG

PAGE 87

,I HDFK SURFHVVRU LQ D SSURFHVVRU ULQJ H[HFXWHV WKH DERYH QRGH SURJUDP WKHQ XSRQ FRPSOHWLRQ 3URFef ZLOO KRXVH 5 IF Nf IRU N e S Q LQ D ORFDO DUUD\ ,Q WKH /8 IDFWRUL]DWLRQ RI $ 5 FDQ EH VWRUHG LQ WKH XSSHU WULDQJXODU SDUW RI DQG WKH XQLW ORZHU WULDQJXODU PDWUL[ / FDQ EH VWRUHG LQ WKH VWULFWO\ ORZHU WULDQJXODU SDUW RI $ 7KHQ ZH FDQ XVH WKH DERYH FRGH WR ILQG WKH LQYHUVH RI WKH XSSHU WULDQJXODU SDUW RI O DQG DIWHU D VOLJKW PRGLILFDWLRQ RI WKH QRGH SURJUDP ZKLFK ILQGV WKH LQYHUVH RI D ORZHU WULDQJXODU PDWUL[ ZH FDQ XVH LW WR ILQG WKH LQYHUVH RI WKH VWULFWO\ ORZHU WULDQJXODU SDUW RI $ 6LQFH $aO 5aO/aO3 VR RXU QH[W JRDO LV WR FUHDWH D QRGH SURJUDP WR ILQG WKH SURGXFW 5aO/aO &RQVLGHU WKH IROORZLQJ QRGH SURJUDP % $? Qe S Qf M FRO e S Q / OHQJWKFRf T ZKLOH M Q LI M FROTf LI M 6HQG %^ MTf WR SURFHVVRUV RQ WKH OHIW HQG ^ 8SGDWH ORFDO FROXPQV ` LI M f % MTf%MO T f§ f ^ LV D T f§ GLPHQVLRQDO URZ ]HUR YHFWRU ` HQG f f A HOVH LI M FROf 5HFHLYH $ ? MMf IURP WKH ULJKW DQG LI WKH SURFHVVRU RQ WKH OHIW LV QRW WKH SURFHVVRU ZKLFK KRXVHV M Mf DQG M LV JUHDWHU WKDQ WKH ILUVW FROXPQ LQGH[ RI WKH SURFHVVRU RQ WKH OHIW WKHQ VHQG MMf WR

PAGE 88

WKH OHIW ^ 8SGDWH ORFDO FROXPQV ` %^ M Tf LI FR"f S M %^ M "f J HQG $ MMf%^M Tf HOVH M M HQG HQG HQG $IWHU HDFK SURFHVVRU LQ D SSURFHVVRU ULQJ H[HFXWHV WKH DERYH FRGH ZH ZLOO REWDLQ WKH SURGXFW 5aO /aO ZKLFK RYHUZULWHV $ DQG XSRQ FRPSOHWLRQ 3URFef ZLOO KRXVH YO S Qf LQ D ORFDO DUUD\ % 6LQFH $ LV RYHUZULWWHQ E\ DQG WKH SHUPXWDWLRQ PDWUL[ 3 LV HQFRGHG LQ WKH Q f§ f GLPHQVLRQDO YHFWRU SLY VR $aO 5aO/aOf3 FDQ EH FRPSXWHG LQ WKH IROORZLQJ ZD\ IRU N Q f§ f§ LI N SLYNf $ U $Wf $ QSLY$ff HQG HQG 1H[W LI ZH VWRUH WKH XSSHU WULDQJXODU PDWUL[ 8 LQ WKH XSSHU WULDQJXODU SDUW RI D Q [ Q PDWUL[ DQG WKH VWULFWO\ ORZHU WULDQJXODU PDWUL[ *6 8f LQ WKH VWULFWO\ ORZHU WULDQJXODU SDUW RI WKHQ WKH FRGH ZKLFK LV XVHG WR FRPSXWH WKH SURGXFW 5aO /aO FDQ EH DSSOLHG WR % WR ILQG WKH SURGXFW 8*6 8f 6LPLODUO\ LW LV SRVVLEOH WR ILQG WKH

PAGE 89

SURGXFW *6O7f8 LQ D PRUH HIILFLHQW ZD\ $IWHU FRPSXWLQJ 8*68f 6 6 0 DQG *f 8f8 ZH FDQ WDNH DGYDQWDJH RI SDUDOOHO SURFHVVRUV WR FRPSXWH 8Wf 8 W 8*68f 6aO$6 f§ 8 f§ *68f8f 1H[W DIWHU =>Lf $ 6Wf8^Wf6WfaO LV FDOFXODWHG ZH FDQ ILQG IWf \-WUDFH=Wf+=Wff LQ WKH IROORZLQJ ZD\ /HW [O Q U U Sf EH D ]HUR YHFWRU /HW = =Wf DQG U f§ 7R FRPSXWH =n = ZH XVH WKH 3 IROORZLQJ SURFHGXUH % =^ Q e fU eUf \ [If IRU L U \ > Qf@IO Qf HQG +HUH HDFK SURFHVVRU FDQ H[HFXWH WKH DERYH FRGH VLPXOWDQHRXVO\ DQG XSRQ FRPn SOHWLRQ 3URFef KRXVHV [ef LQ D ORFDO YDULDEOH \ 7KHQ IWf \M[? [S ,Q 6WHS LI 1% f§ P S WKHQ ZH FDQ QRW XVH DOO SURFHVVRUV WR ILQG LO DQG LX 6XSSRVH 1% f§ S 8VLQJ D VLPLODU WHFKQLTXH DV GHVFULEHG LQ WKH DQDO\VLV RI 6WHS ZH FDQ JHW DQ DUUD\ N VXFK WKDW ?S!NLf? _SrL f KROGV /HW rrm f a 3NLf GNLf f§ GNLf 7R GHWHUPLQH ,U r PLQ^ U L 1% LQ FDQ XVH S SURFHVVRUV DV IROORZV /HW V 1% P 3 f§ 5H Urf t _U_ ` ZH 'HILQH ILO E f f 3N mf 3IFVOVf f f F F f f r IOf f f f f %S f 3NSf§fV?Sf§ff 3 NSf§?fV?1% f§ Pf A IW f DrSfOSOfDOf SO fO$UePf > S f§ f @U XO Sf DQG Y Sf 1H[W FRQVLGHU WKH IROORZLQJ SURFHGXUH %U J 4 Xef 

PAGE 90

Y ef / OHQJWKf UO / f IRU L / f§ WL 6L Ub LKL HQG IRU = f§ LI 5HUf $ HQG HQG LI  HOVH YM HQG ,I HDFK SURFHVVRU H[HFXWHV WKH DERYH FRGH WKHQ XSRQ FRPSOHWLRQ 3URFef KRXVHV Xef DQG Xef 1H[W WKH GHVLUHG PLQLPXP LV Xef UQLQ^ ef e S ` DQG LI WKH FRUUHVSRQGLQJ Xef A WKHQ GHILQH LO W f§ PLQ^IFXeff N^Y=f f` DQG LX W PD[^OXARffRf f`f ,Q 6WHS ZH FDQ XVH SDUDOOHO SURFHVVRUV DV GHVFULEHG LQ WKH DQDO\VLV RI 6WHS DQG 6WHS WR REWDLQ 8QHZ 6QHZ DQG = 6QHZ8QHZ 6nff ,Q 6WHS LI P S WKHQ ZH FDQ DVVLJQ D VHSDUDWH SURFHVVRU WR LPSOHPHQW WKH FRGH WR HDFK GLDJRQDO EORFN 8MM M P ,I P S WKHQ DVVLJQ 3URFef WR LPSOHPHQW WKH FRGH WR GLDJRQDO EORFNV 8MM ZKHUH M e e S P 7KXV ZH KDYH FRO e S "U / OHQJWKFRf IRU T /

PAGE 91

D FRO^Tf? ,PSOHPHQW 6WHS WR 844 HQG ,Q 6WHS ZH FDQ XVH SDUDOOHO SURFHVVRUV WR ILQG D 45 IDFWRUL]DWLRQ RI 6 +HUH ZH ZLOO GLVFXVV KRZ WR ILQG D 45 IDFWRUL]DWLRQ RI D PDWUL[ $ LQ D GLVWULEXWHG PHPRU\ PXOWLSURFHVVRU ZKHUH SURFHVVRUV DUH LQWHUFRQQHFWHG LQ D ULQJ /HW Q f§ f EH D ]HUR YHFWRU &RQVLGHU WKH IROORZLQJ SURFHGXUH ZKLFK LV GXH WR &/ ,, *ROXE DQG & ) 9DQ /RDQ % $ S Qf F e S Q f§ f M T FRO e S Q / OHQJWKFRf ZKLOH T / LI M FROTf $ M Q ^ )LQG +RXVHKROGHU YHFWRU $M Q Mf ` D __%M QTf__ LI D %M QTf TXLW HOVH D%MTf?%^MTf? HOVH D HQG %^M Tf %M Q Tf %M "L f" "M Tf HQG L QTf F^Tf 6HQG FUf DQG %M QTf WR SURFHVVRUV RQ WKH ULJKW R ^ 8SGDWH ORFDO FROXPQV `

PAGE 92

LI T / %M T /f %M Q T /f FTfYY+ %M Q T /f HQG M M HOVH 5HFHLYH EMf DQG $M QMf IURP WKH OHIW DQG LI WKH SURFHVVRU RQ WKH ULJKW LV QRW WKH SURFHVVRU ZKLFK IRUPHG $M QMf DQG M LV OHVV WKDQ WKH ODVW FROXPQ LQGH[ RI WKH SURFHVVRU RQ WKH ULJKW WKHQ VHQG Mf DQG $M Q Mf WR WKH ULJKW ^ 8SGDWH ORFDO FROXPQV ` B 9 a $M QMf %M QT /f %^M QT /f EMfYY+%M QT /f M M HQG HQG $IWHU HDFK SURFHVVRU LQ D SSURFHVVRU ULQJ H[HFXWHV WKH DERYH FRGH ZH ZLOO JHW D 45 IDFWRUL]DWLRQ RI $ DQG 3URFef ZLOO KRXVH $ Qe S Qf LQ D ORFDO DUUD\ DQG A S Q f§ f LQ D ORFDO YHFWRU F 7KH XSSHU WULDQJXODU SDUW RI $ LV RYHUZULWWHQ E\ WKH XSSHU WULDQJXODU SDUW RI DQG FRPSRQHQWV M Q RI WKH MWK +RXVHKROGHU YHFWRU DUH VWRUHG LQ $M QMf M Q +HUH ZH QHHG DQ H[SOLFLW IRUP RI WKH XQLWDU\ PDWUL[ 4 :H ILQG 4 XVLQJ WKH IROORZLQJ SURFHGXUH ZKLFK XVHV +RXVHKROGHU YHFWRUV $M Q Mf ZKHUH M Q DQG WKH YHFWRU E WR IRUP 4 H[SOLFLWO\ DQG ]HURV VXEGLDJRQDO HOHPHQWV RI $ /HW 4 f§ ,Q % $O Qe S Qf? 4 Qe S Qf F e S Q f§ f M f§ Q f§ FRO e S Q / f§ OHQJWKFRf T f§ / ZKLOH M LI M FROTf

PAGE 93

6HQG Ff DQG %M QTf WR SURFHVRUV RQ WKH OHIW ^ 8SGDWH ORFDO FROXPQV ` Y %M QTf %M QTf OM QT /f n QT /f FTfYY+ ,M QT /f M M HOVH 5HFHLYH Mf DQG $M Q Mf IURP WKH ULJKW DQG LI WKH SURFHVVRU RQ WKH OHIW LV QRW WKH SURFHVVRU ZKLFK KRXVHV $M Q Mf DQG M LV JUHDWHU WKDQ WKH ILUVW FROXPQ LQGH[ RI WKH SURFHVVRU RQ WKH OHIW WKHQ VHQG Mf DQG $M Q Mf WR WKH OHIW ^ 8SGDWH ORFDO FROXPQV ` 9 a $M Q-f ,M QT /f ,M QT /f EMfYY+,M QT /f? M M LI FRO^Tf S M T T HQG HQG HQG ,I HDFK SURFHVVRU LQ D SSURFHVVRU ULQJ H[HFXWHV WKH DERYH FRGH WKHQ XSRQ FRPn SOHWLRQ 3URFef ZLOO KRXVH NNf IRU N e S Q LQ D ORFDO DUUD\ %b DQG 4 Qe S Qf LQ D ORFDO DUUD\ 1H[W ZH FDQ XVH SDUDOOHO SURFHVVRUV DV GHVFULEHG LQ WKH DQDO\VLV RI 6WHS WR ILQG WKH SURGXFW 8QUZ 4+=4 )LQDOO\ LQ 6WHS ZH FDQ XVH PXOWLSURFHVVRUV DV GHVFULEHG LQ WKH DQDO\VLV RI 6WHS WR HYDOXDWH f $= __U

PAGE 94

&+$37(5 (,*(19$/8(6 2) 6<00(75,& 0$75,&(6 'LDJRQDOL]DWLRQ RI D 6\PPHWULF 0DWUL[ XVLQJ $UPLMRnV 6WHSVL]H /HW $ EH DQ Q [ Q V\PPHWULF PDWUL[ 6XSSRVH WKH HLJHQYDOXHV RI $ DUH DOO GLVWLQFW ,Q WKLV VHFWLRQ ZH ZLOO ILQG DQ LWHUDWLYH PHWKRG WR FRPSXWH D GLDJRQDOL]DWLRQ $ 4$4W ZKHUH $ GLDJ $L$$Q f DQG 4 LV DQ RUWKRJRQDO PDWUL[ VXFK WKDW WKH GLDJRQDO HOHPHQWV RI $ DUH WKH HLJHQYDOXHV RI $ DQG WKH FROXPQV RI 4 DUH HLJHQYHFWRUV RI $ &RQVLGHU WKH IROORZLQJ DOJRULWKP WR FRPSXWH WKH HLJHQYDOXHV DQG FRUUHVSRQGLQJ HLJHQYHFWRUV RI D PDWUL[ $ 'HWDLO LV JLYHQ LQ >,ODJ66 SDJH f§ @ RU LQ &KDSWHU \LLHZ ; QHZ QHZ ; \If7 $[I I s 9 m OrQHZ ? \QHZ $[IG UOG ; r AQHZ f IRU M IRU M WR Q WR Q $OJRULWKP LQ f FDQ EH UHZULWWHQ LQ WKH IROORZLQJ ZD\ $ QHZ ; UQHZ 'LDJ
PAGE 95

,Q f LI ZH SXW 9 QHZ ) ;rOG$QHZf DQG $ROG ;ROGf n $;ROG WKHQ LW UHGXFHV WR WKH IRUP $ ; QHZ 'LDJURGf $ROG< n2OG QHZ UROG ?$QHZ [ ? +HQFH LQ HDFK LWHUDWLRQ ZH FDQ XSGDWH WKH HLJHQYDOXHV DQG FRUUHVSRQGLQJ HLJHQYHFn WRUV LQ WKH IROORZLQJ ZD\ $ $ QHZ QHZ PHZ $ AQHZ ROG? r AROGAnROG 'LDJ ) $
PAGE 96

WKH LQFUHPHQW ) $QHZf PD\ EH WRR ODUJH +RZHYHU ZLWK D ODUJH LQFUHPHQW ZH PD\ OLNHO\ KDYH LQVWDELOLWL\ LQ WKH XSGDWLQJ SURFHVV DQG WKH DOJRULWKP PD\ GLn YHUJH 7R UHVWRUH WKH FRQYHUJHQFH RI WKH DOJRULWKP DQG WR KDYH D VWHDG\ FKDQJH LQ WKH YDOXH RI
PAGE 97

EWDWDWEW %f$$%f $%%$ +HQFH $% f§ %$ LV D V\PPHWULF PDWUL[ ’ :LWK WKH DERYH GHILQLWLRQ RI I^Vf QRZ ZH DUH UHDG\ WR ILQG nf DQG ZLOO VKRZ WKDW nf Rf 7KHRUHP /HW $ f 5 Q[Q EH D V\PPHWULF PDWUL[ 6XSSRVH LWV HLJHQYDOXHV DUH DOO GLVWLQFW 'HILQH =Vf V)$f *Vf ORWU>>=^Vff $=VfVM DQG IVf __*Vf__I ZKHUH Ir$f G?M &/-G L L IRU L M IRU L A M ORWL?%f LV D VWULFWO\ ORZHU WULDQJXODU PDWUL[ IRUPHG IURP WKH VWULFWO\ ORZHU WULDQJXODU SDUW RI % 7KHQ nf f 3URRI RI 7KHRUHP 8VLQJ WKH UHVXOW f RI /HPPD ZH KDYH fnf WUDFH *f7A*Vf?V f G f 'LIIHUHQWLDWLQJ *Vf ZLWK UHVSHFW WR DQG WKHQ HYDOXDWLQJ WKH GHULYDWLYH DW ZH REWDLQ G :8R ?RW7?f§=faG=^Vf?D R=faO$=f =faO$a=Vf R_f 'LIIHUHQWLDWLQJ =Vf ZLWK UHVSHFW WR V ZH KDYH f§ =Vf GV ) +HQFH f§=Vf? GV ) 6XEVWLWXWLQJ WKLV YDOXH DQG WKH YDOXH RI =f LQ f DQG WKHQ VLPSOLI\LQJ ZH JHW G GV *Vf?D ORWU $) f§ )$f f

PAGE 98

6LQFH *f ORWU=f $=ff ORLU ,$f ORWU$f VR f EHFRPHV ff WUDFH ORWU]Off7 ORWU $) f§ )$fAM f %HFDXVH $ LV V\PPHWULF VR ORWU $ff XSWU $f ZKHUH XSWUf LV D VWULFWO\ XSSHU WULDQJXODU PDWUL[ IRUPHG IURP WKH VWULFWO\ XSSHU WULDQJXODU SDUW RI $ +HQFH f UHGXFHV WR fnf WUDFH XSWU Of ORWU $) f§ )$ff f 1H[W IRU L s M IW@ D cM &OMM OLL DQG ML -O FO D f§ FL D 6LQFH $ LV V\PPHWULF VR I X IML 7KDW LV ) LV D VNHZ V\PPHWULF PDWUL[ +HQFH E\ /HPPD $) f§ )$ LV V\PPHWULF 1RZ $) )$ Q Q L D LL &/ L &/ L Q ( &/ L L &/ ^ FO D ‘B A FL D Q DL&LL? Q DQ &/ L L Q &OL&OL? Q m LD mA &/ ^^ ]D L L L FO DQ Wr Q DAQL &/ f Q U VL QB DQLDL? GQL&/W A \\ e D QLD c f§A DQ L ?DQQ DQ [f§?D DQ L ?DQQ DQ f§ &/ L ^&OLQ Q e DQD LQ Lf§?&/QQ DQ DQ Q f§ D L D ^Q Q e &OLDLQ Lf§?&LQQ DQ FO FLD f f mr D LQ L ?DQQ &/ L L 7KHUHIRUH f JLYHV fnf WUDFH Q Q emX &/ N L &/ ^ Q D N L D IF M \ A ]f§ FL } L DNN DQ [rN ; ; ; Q Q ; e mr e &ONL&OL Q e FLNLDW N Lf§L D FLD D NN D D mr ; ;

PAGE 99

; ; Q AQOQ mQL m}Qf§ Qf§ L L mQ f§ ,Q f§ LAQ f§ m ] mQLmLQf§ L f§ O 4QQ D ,, ; ; ; ; ? ZKHUH ; DUH QXPEHUV 6LPSOLI\LQJ WKH ULJKW KDQG VLGH RI WKH DERYH H[SUHVVLRQ ZH JHW fnf Q Q ( DnN mLEmmO Q ( m N D M Q Q Nf§ L f Dmm L DNN f§ DLL ( Dr N DND L m f§ 4LL fr Q ( m $r L m} Q mQ Q } L m $ $r mLW LA$ mQ}mLQf§ Qf§ L O m Q f§ Q f§ WAQ f§ O m} ( mQLmLQ f§ L O IOQQ mL DG? mm m m f§ m m f m mQQmQO mQO m m,Q m m QQ D QQ m mm mm r K m f§ m D f§ D QQQ DQr Q O K m f m QQ D QQ m mQQmQQ f§ mQQf§OmQf§ ,Qf§ r r n L mQ f§,Q , fWamm m mQf§,Qf§ m QQ D QQ m Q f§ Q f§ m f§ m m f m m a m m a m m a m m f§ m ffff m Q f§ Q f§ m Q f§ Q m Q f§ ,Q mQf§Qf§ m L QQ m Q f§ Q f§ m QQ mQf§,Qf§ mQf§Qf§ mQQ mQf§Qf§ mQf§Qf§ mQf§,Qf§ D QQ mQf§,Qf§ e m r!Q f f f 6R nf f SURYLGHG f A ’ +HQFH 7KHRUHP LPSOLHV WKDW $UPLMRfV UXOH FDQ EH DSSOLHG WR WKH DOJRULWKP LQ f $ GHWDLOHG VNHWFK RI WKH DOJRULWKP WR ILQG WKH HLJHQYDOXHV RI D V\PPHWULF PDWUL[ $ ZKRVH HLJHQYDOXHV DUH DOO GLVWLQFW LV DV IROORZV

PAGE 100

$OJRULWKP 6\PPHWULF 'LDJRQDOL]DWLRQf *LYHQ D V\PPHWULF PDWUL[ $ ( 5Q;Q ZKRVH HLJHQYDOXHV DUH DOO GLVWLQFW DQ RUWKRJRQDO PDWUL[ ; DQG D WROHUDQFH WRO JUHDWHU WKDQ WKH XQLW URXQGRII WKH IROORZLQJ DOJRULWKP FRPSXWHV D GLDJRQDOL]DWLRQ ;7$; f§ $ DQG $ LV RYHUZULWWHQ ZLWK WKH GLDJRQDO PDWUL[ $ 6WHS 7DNH ; f§ ; &RPSXWH $QHZ ;7$ROG; DQG f 6WHS &RQVWUXFW ) f ZKHUH ORWU $f ??I IRU L ID M &O LM D MM &O L L IRU L M 6WHS /HW IVf __*f__I ZKHUH *Vf ORWU \, V)f $ V)fM (YDOXn DWH IVf DW V n f VWr33QJ ZKHQ IVf ,f f f /HW W EH WKH ILUVW YDOXH RI V IRU ZKLFK f KROGV DQG OHW < W) /HW < 45 EH WKH 45 IDFWRUL]DWLRQ RI < 6WHS 8SGDWH $ DQG ; DV IROORZV $QHZ 47$ROG4 DQG ;QHZ ;ROG4 6WHS (YDOXDWH f __ORWU$f ? ?MU *R WR 6WHS XQWLO f WRO ZKHUH WRO LV WKH WROHUDQFH IRU WKH GHVLUHG DFFXUDF\ RI WKH HLJHQYDOXHV ([DPSOH ,I $OJRULWKP LV DSSOLHG WR WKH PDWUL[ ZLWK ;T DQG WRO f WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D GLDJRQDOL]DWLRQ ; $; GLDJ f§ f ,I ZH XVH WKH 45 PHWKRG IRU D V\PPHWULF PDWUL[ ZLWK VLQJOH LPSOLFLW VKLIW WR $ WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D GLDJRQDOL]Dn WLRQ 47$4 GLDJ f§ f 1H[W FRQVLGHU WKH IROORZLQJ PDWUL[ $?

PAGE 101

ZKLFK ZH REWDLQ E\ SHUWXUELQJ WKH HOHPHQWV RI $ $ L ,I ZH XVH $OJRULWKP WR $? ZLWK ; ZLWK $nR ; DQG WRO WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D GLDJRQDOL]DWLRQ
PAGE 102

,I ZH XVH $OJRULWKP WR &? ZLWK ; $$ DQG R WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D GLDJRQDOL]DWLRQ <7$^< f§ IL ZKHUHDV LI ZH DSSO\ WKH 45 PHWKRG IRU D V\PPHWULF PDWUL[ ZLWK VLQJOH LPSOLFLW VKLIW WR ;7&?; WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D GLDJRQDOL]DWLRQ 67;7&?;f6 f§ 7 %ORFN 'LDJRQDOL]DWLRQ RI D 6\PPHWULF 0DWUL[ XVLQJ $UPLMRA 6WHSVL]H /HW $ EH DQ Q [ Q V\PPHWULF PDWUL[ 2XU JRDO LV WR ILQG DQ LWHUDWLYH PHWKRG WR FRPSXWH D EORFN GLDJRQDOL]DWLRQ $ 4$47 ZKHUH $ GLDJ $L $ $ f LV D EORFN GLDJRQDO PDWUL[ DQG 4 > 4L f f f 4N ` LV D FRPSDWLEOH EORFN FROXPQ RUWKRJRQDO PDWUL[ VXFK WKDW $4M 4M$M +HUH $M GLDJ $\ $M $M f LV DQ M [ M VFDODU PDWUL[ ,W FDQ DOZD\V EH DUUDQJHG VR WKDW WKH HLJHQYDOXHV RI $ DQG $M IRU L M DUH GLVWLQFW )URP WKH EORFN GLDJRQDOL]DWLRQ RI D PDWUL[ $ LQ 6HFWLRQ ZH KDYH WKH IROORZLQJ ORFDOO\ TXDGUDWLFDOO\ FRQYHUJHQW DOJRULWKP $
PAGE 103

LV DQ UULM [ PM ]HUR PDWUL[ DQG )D ;ROG $ QHZ? ;RGf $; ROG f f ?\Z IRU L  M ZKHUH ;RIf f $;ROG6M LV WKH ^LMf EORFN RI [ rfG? $;RO,Q f LI ZH SXW \ QHZ ) ;ROG ?QHZf DQG $rG ROGn $;ROG WKHQ WKH DOJRULWKP UHGXFHV WR $ ; QHZ 'LDJ QHZ ;ROG< UROG AROG\ROG QHZ +HQFH WKH LWHUDWLYH PHWKRG WR ILQG WKH HLJHQYDOXHV DQG FRUUHVSRQGLQJ HLJHQYHFWRUV RI D PDWUL[ $ FDQ EH UHZULWWHQ DV IROORZV $ $ \ [ QHZ QHZ nQHZ QHZ QHZ \ROGA 'LDJ $ ) $QHZf =VfRGA $ROG=VfROGDQG IVf __*Vf__I ZKHUH QRQGLDJ%f LV D EORFN PDWUL[ IRUPHG IURP WKH $ [ K EORFN PDWUL[ % E\ UHSODFLQJ WKH GLDJRQDO EORFNV E\ ]HUR PDWULFHV 6XSSUHVVLQJ WKH VXSHUVFULSWV LQ

PAGE 104

=VfQHZ DQG *Vf ZH KDYH =Vf V)$f DQG *Vf QRQGLDJ =VfaO $=Vff +HQFH f __QRQGLDJ $f __I DQG f __QRQGLDJ ^Vf f KROGV 7R WKLV HQG ZH XVH $UPLMRfV UXOH IURP RSWLPL]DWLRQ WKHRU\ ,Q $UPLMRfV UXOH ZH GHWHUPLQH V LQ WKH IROORZLQJ ZD\ (YDOXDWH IVf DW V VWRSSLQJ ZKHQ IVf O If f 7R XVH WKH DERYH LQHTXDOLW\ ZH PXVW PDNH VXUH WKDW IVf VDWLVILHV n2f P +HQFH RXU QH[W DLP LV WR GHWHUPLQH nf ZKHQ IVf __*Vf__I DQG *Vf QRQGLDJ =Vff $=Vf!M 7KHRUHP /HW $ 5 Q[Q EH D N [ $ EORFN V\PPHWULF PDWUL[ VXFK WKDW $MM GLDJ DMDM D f  DQ "" [ "" VFDODU PDWUL[ ,I ZH GHILQH =Vf V)$f *Vf QRQGLDJ =Vff $=VfA DQG IVf f§ __*Vf__I ZKHUH )MM$f LV DQ UULM [ QLM ]HUR PDWUL[ )LM$f $ 9 M f§ D IRU L ‘ ‘ M DQG QRQGLDJ %f LV IRUPHG IURP WKH N [ $ EORFN PDWUL[ % E\ UHSODFLQJ WKH GLDJRQDO EORFNV E\ ]HUR PDWULFHV WKHQ nf f 3URRI RI 7KHRUHP 7KH UHVXOW f RI /HPPD JLYHV I^4fIIVf?V R WUDFH A*f7A*n£f_V RM f 'LIIHUHQWLDWLQJ *Vf DQG =Vf ZLWK UHVSHFW WR DQG WKHQ HYDOXDWLQJ WKH GHULYDWLYHV DW V ZH JHW IIVf_V QRQGLDJA=fA=f_V R=f$=f =f$A=6f_V RI]Vf _V )

PAGE 105

6LQFH =f VR DIWHU VLPSOLILFDWLRQ f JLYHV G GV *mVf_D R QRQGLDJ $) f§ )$f *f 1H[W *nf QRQGLDJ =f 0=ff QRQGLDJ ,$,f QRQGLDJ$f +HQFH IURP f ZH REWDLQ GV WUDFH QRQGLDJff QRQGLDJ $) f§ )$ff f %XW $ LV V\PPHWULF VR QRQGLDJff 7 QRQGLDJ]Of +HQFH f UHGXFHV WR WUDFH QRQGLDJLfQRQGLDJY) f§ )$ff f 1H[W IRU L A M )^M $ &OM f§ O^ DQG )ML $ f f Mr Dc f§ D 6LQFH $ LV V\PPHWULF VR )cM f§)M[ 7KDW LV ) LV D VNHZ V\PPHWULF PDWUL[ DQG WKHUHIRUH E\ /HPPD $)f§ )$ LV D V\PPHWULF PDWUL[ 1H[W $) )$ \ $X$J N f r D[D[ N UO ( AA L D }r D $?L$^ \ O ?f§ \$L$W L m ‘ r D $ $NL$Q A $NL$Q $NL$L $N$ L m r ( Ln O AN L L A f r Dr mL 7KHUHIRUH IURP f ZH KDYH \\ $ A WnME Dr D L L r r $$N ( A$A ( L O A $ AL L L A mL ‘r f f U f§ VrNW rWN 7 Dr a Dm ff8f

PAGE 106

W UDFH X N $ $ VL-O6, a D[D ( N $ML$Q O ; ; ; W O DM DL ngN ; e $OL e D D N $ML$L $M$ \ ‘ r ; ; ngM f f ; ; 9 $NX I $$LWn < $M $LNf§ W O AN f§ t L LANO LAM W O t N M L -< -< L $MM$MN ( r $$LN OgM O ? ZKHUH < DUH UHFWDQJXODU PDWULFHV $IWHU VLPSOLI\LQJ WKH ULJKW KDQG VLGH RI WKH DERYH H[SUHVVLRQ ZH REWDLQ fe6f DV r r WUDFH  $[M >   W O DL ngM N WUDFH ( N $D$LR $LW$ M -r Q MLAL } A W O DM r rgM M $M]L$LNf§ Y $A$Af§L WUDFH $NQ A ( m LMWN tNf§ WUDFH $ $$ A $L$Q mL f§ D m f§ m WUDFH $X WUDFH $ $NN$NL A $N?$Q DL f§ DN DN f§ DL $$ $$ m f§ A m f§ D WUDFH $W $NN$N $IF$ t a DN DN f§ r!

PAGE 107

6LQFH $LM D L $NN$NN $NNL$NLNL WUDFH $N?N AA $$ I UDFH $ M K ;/? f§ &/ f§ &/ M I AA A A WUDFH W E DN f§ 2N f f WUDF AA AA O &/ f§ &/ f§ D &, f§ &/? &/ f m $NLN$NN $Nf§Q$NNL WUDFH $NNL K 2N a 2N 2N f§ 2N WUDFH $N N $NNL$NLN $NNL$NLN 2N f§ 2N &/ N &O N f§ $NN$NN? $NN$NN? WUDFH 28X K DO f§GN GN&/N $7 $MM DQG $Q DUH VFDODU PDWULFHV VR $MM$-W M rf GrMM f FL M $M r $M L r f§ FL $ }f DQG +HQFH R A f f L f f Ar ,n0 O D L D &OM &OL D &/M$ML &/ ? $M^ &OL D &OL f§ &/ W$ ML WUDFH $ $ $ X D L f§ DM *M f§ &=M WUDFH U f f AMmf f 1H[W ZH NQRZ WKDW IRU DQ\ PDWULFHV & DQG WUDFH &f WUDFH &f DQG WUDFH &'f WUDFH '&f SURYLGHG WKH PDWUL[ PXOWLSOLFDWLRQV H[LVW /HW WUDFH $LM $ ML $Kf 7KHQ FOHDUO\ WUDFH mM $ML$Xf7f WUDFH WUDFH $Q$?M$M^f $OVR WUDFH $M $ML$Xff WUDFH $ML$Qf $^$ +HQFH WUDFH ,WWUDFH $ML $ML$X FLL FX $Q $ X $LM DM &OL $LM $ MM $W M $MM &OL f§ &/ &OL f§ &O /L DL DM f§ DL D f§ D M D L D D f§ D M aM rr Dc L D 8VLQJ WKH UHVXOWV IURP f DQG f LQ f ZH KDYH rf:8R WUDFH ,f f f f WUDFH A$A$NLAM WUDFH LA$A$A

PAGE 108

%XW WUDFH $ WUDFH >ƒUNNBO$NNLf f f WUDFH *fU*ff WUDFH QRQGLDJ $fQRQGLDJ $ff WUDFH 7 A WUDFH $A$IFLf WUDFH D$f WUDFH $A$Af + E WUDFH $AA$AL G +HQFH f \LHOGV 2ff§IVf?V GV 2f 6R nf f SURYLGHG f A ’ 7KH UHVXOW RI WKH DERYH WKHRUHP LPSOLHV WKDW $UPLMRfV UXOH FDQ EH DSSOLHG WR WKH DOJRULWKP LQ f ,Q 7KHRUHP nf f§f KROGV RQO\ ZKHQ $ LV V\PPHWULF 6R ZH QHHG WR UHVWRUH WKH V\PPHWU\ RI $ LQ WKH XSGDWH $QHZ A
PAGE 109

6WHS IDNH ; ; &RPSXWH $ QHZ ;7$ROG; DQG f __ QRQGLDJ $f??S ZKHUH QRQGLDJ f LV GHILQHG DV HDUOLHU 6WHS /HW 1% Q WKH QXPEHU RI GLDJRQDO EORFNV RI $ f 16 > @ DQ DUUD\ RI Q HOHPHQWV ZKRVH LWK HOHPHQW LV WKH GLPHQVLRQ RI WKH ]f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ff 6WHS &RQVWUXFW EORFN PDWUL[ ) )^Mf DV IROORZV 7DNH )MM ZKLFK LV DQ PM [ PM ]HUR PDWUL[ DQG IRU L A M FDQ EH GHWHUPLQHG E\ XVLQJ WKH IROORZLQJ ORRS IRU M 1% IRU 1% LI LrM 6ROYH =$ f f MM ) fM = HQG $= $WIRU = HQG HQG

PAGE 110

6WHS /HW IVf __*Vf__I! ZKHUH *Vf QRQGLDJ V)f O$, V) (YDOXDWH IVf DW V AA VWRSSLQJ ZKHQ IVf O ,f f f /HW W EH WKH ILUVW YDOXH RI V IRU ZKLFK f KROGV /HW = W) DQG OHW = 45 EH D 45 IDFWRUL]DWLRQ RI = 6WHS 8SGDWH $ DQG ; DV IROORZV $QHZ 47$ROG4 DQG ;QHZ ;ROG4 ZKHUH 4 LV WKH RUWKRJRQDO PDWUL[ IURP 6WHS 6WHS &RPSXWH f __QRQGLDJ $f __I DQG JRWR 6WHS XQWLO f WRO ([DPSOH ,I $OJRULWKP LV DSSOLHG WR WKH PDWUL[ ZLWK $ WRO f§ DQG WROO B WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D GLDJRQDOL]DWLRQ ;7$; GLDJ f ,I ZH XVH WKH 45 PHWKRG IRU D V\PPHWULF PDWUL[ ZLWK VLQJOH LPSOLFLW VKLIW WR $ WKHQ DIWHU LWHUDWLRQV ZH JHW D GLDJRQDOL]DWLRQ 6U$6 GLDJ f 1H[W FRQVLGHU WKH IROORZLQJ PDWUL[ $L ZKLFK ZH REWDLQ E\ SHUWXUELQJ WKH HOHPHQWV RI $ $? ,I ZH XVH $OJRULWKP WR $? ZLWK $R LWHUDWLRQV ZH REWDLQ D GLDJRQDOL]DWLRQ n7 ; WRO f 0< $ ,I DQG WROO WKHQ DIWHU $ ,I ZH DSSO\ WKH 45 PHWKRG IRU D V\PPHWULF PDWUL[ ZLWK VLQJOH LPSOLFLW VKLIW WR ;7$?; WKHQ DOVR DIWHU LWHUDWLRQV ZH REWDLQ D GLDJRQDOL]DWLRQ 47;7$?;f4 ([DPSOH ,I $OJRULWKP LV DSSOLHG WR WKH PDWUL[

PAGE 111

% 4GLDJf4U ZKHUH 4 LV DQ RUWKRJRQDO PDWUL[ ZLWK ;T WRO DQG WROO WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D GLDJRQDOL]DWLRQ ;7%; GLDJf ZKHUHDV LI ZH XVH WKH 45 PHWKRG IRU D V\PPHWULF PDWUL[ ZLWK VLQJOH LPSOLFLW VKLIW WR % WKHQ DIWHU LWHUDWLRQV ZH JHW D GLDJRQDOL]DWLRQ 67%6 GLDJ f ([DPSOH ,I $OJRULWKP LV DSSOLHG WR 5RVVHU PDWUL[ 5 ZLWK 9 WRO n n DQG WROO f§ WKHQ DIWHU LWHUDWLRQV ZH DJRQDOL]DWLRQ ;75; GLDJ 6 f ,I ZH DSSO\ WKH 45 PHWKRG IRU D V\PPHWULF PDWUL[ ZLO LPSOLFLW VKLIW WR 5 WKHQ DIWHU LWHUDWLRQV ZH JHW D GLDJRQDOL]DWLRQ 6U56 GLDJ f 1H[W FRQVLGHU WKH PDWUL[ U ,I ZH DSSO\ $OJRULWKP WR WKH SHUWXUEHG PDWUL[ "> I! ZLWK 9R $n WRO DQG WROO WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D GLDJRQDOL]DWLRQ

PAGE 112


PAGE 113

([DPSOH ,I $OJRULWKP LV DSSOLHG WR :LONLQVRQfV [ WHVW PDWUL[ ZLWK ;T WRO ^? DQG WROO WKHQ DIWHU LWHUDWLRQV ZH REWDLQ D GLDJRQDOL]DWLRQ ;7:; GLDJ f ZKHUHDV LI ZH XVH WKH 45 PHWKRG IRU D V\PPHWULF PDWUL[ ZLWK VLQJOH LPSOLFLW VKLIW WR : WKHQ DIWHU LWHUDWLRQV ZH JHW D GLDJRQDOL]DWLRQ 67?96 GLDJ f§ f ([DPSOH ,I $OJRULWKP LV DSSOLHG WR :LONLQVRQnV [ WHVW PDWUL[ 1RWH WKDW LI DW OHDVW WZR GLDJRQDO HOHPHQWV RI WKH JLYHQ PDWUL[ $ DUH HTXDO DQG WKH RII GLDJRQDO HOHPHQWV DUH ODUJH FRPSDUHG WR WKH GLDJRQDO HOHPHQWV WKHQ $OJRn ULWKP FDQ QRW EH DSSOLHG GLUHFWO\ WR $ ,Q WKLV FDVH ILUVW ZH ILQG D +HVVHQEHUJ IRUP + RI $ DQG WKHQ DSSO\ $OJRULWKP WR WKH +HVVHQEHUJ IRUP ,I ZH DSSO\ $OJRULWKP WR DQ Q [ Q V\PPHWULF PDWUL[ $ ZLWK $fR D WROHUDQFH R DQG D FRDOHVFLQJ WROHUDQFH R WKHQ ZH REWDLQ D GLDJRQDOL]DWLRQ ; $; $ 1H[W ZH FDQ XVH ; DV WKH VWDUWLQJ JXHVV RUWKRJRQDO PDWUL[ WR REWDLQ D GLDJRQDOL]DWLRQ RI D VOLJKWO\ SHUWXUEHG V\PPHWULF PDWUL[ $ F( ZLWK YHU\ UDSLG FRQYHUJHQFH ZKHUH ( LV DQ DUELWUDU\ V\PPHWULF PDWUL[ DQG H LV D VPDOO VFDODU ,Q WKH IROORZLQJ WDEOH IRU GLIIHUHQW YDOXHV RI Q ZH JLYH WKH QXPEHU RI LWHUDWLRQV UHTXLUHG WR REWDLQ GLDJRQDOL]DWLRQV RI $ W( E\ $OJRULWKP ZLWK $R ; DQG GLDJR QDOL]DWLRQV RI ; $ H(f; E\ WKH 45 PHWKRG IRU D V\PPHWULF PDWUL[ ZLWK VLQJOH LPSOLFLW VKLIW +HUH WKH RULJLQDO PDWULFHV $ DMf DUH UDQGRP V\PPHWULF PDWULFHV

PAGE 114

ZKHUH DcM DUH LQWHJHUV DQG DcM DQG WKH PDWULFHV ( HMf DUH UDQGRP V\PPHWULF PDWULFHV ZKHUH H=DUH UHDO QXPEHUV DQG HW7KH WROHUDQFH LV WRO DQG WKH FRDOHVFLQJ WROHUDQFH LV WROO VL]H QXPEHU RI LWHUDWLRQV IRU RXU DOJRULWKP QXPEHU RI LWHUDWLRQV IRU WKH 45 DOJRULWKP ZLWK VLQJOH VKLIW $ H( $ H( 67^$ H(f6

PAGE 115

&+$37(5 &21&/86,21 ,Q WKH LQWURGXFWLRQ ZH UHYLHZHG WKH DSSOLFDWLRQ RI HLJHQYDOXH VHQVLWLYLW\ DQG HLJHQYHFWRU VHQVLWLYLW\ WR GHULYH DQ LWHUDWLYH PHWKRG WR ILQG DOO HLJHQYDOXHV DQG FRUn UHVSRQGLQJ HLJHQYHFWRUV RI D PDWUL[ $ ZKRVH HLJHQYDOXHV DUH DOO GLVWLQFW 7R XVH WKLV PHWKRG ZH QHHG D PDWUL[ RI DSSUR[LPDWH HLJHQYHFWRUV ;R ,I $fR LV QRW D JRRG DSSUR[LPDWLRQ RI ; LQ WKH IDFWRUL]DWLRQ $ $ $$ ZKHUH $ LV D GLDJRQDO PDWUL[ WKHQ WKH PHWKRG GLYHUJHV $V GLVFXVVHG LQ 6HFWLRQ WKH DSSOLFDWLRQ RI HLJHQ YDO XHVHQVLWLYLW\ DQG HLJHQYHFWRU VHQVLWLYLW\ FDQ EH WKHRUHWLFDOO\ H[WHQGHG WR GHULYH DQ LWHUDWLYH PHWKRG WR ILQG DOO HLJHQYDOXHV DQG FRUUHVSRQGLQJ HLJHQYHFWRUV RI D QRQGHIHFn WLYH PDWUL[ $ VRPH RI ZKRVH HLJHQYDOXHV PD\ EH LGHQWLFDO 7KH REYLRXV GLIILFXOW\ RQH HQFRXQWHUV LQ XVLQJ WKLV PHWKRG LV KRZ WR FKRRVH D PDWUL[ RI DSSUR[LPDWH HLJHQYHFn WRUV IRU WKH VWDUWLQJ JXHVV ,Q 6HFWLRQ ZH VKRZHG WKDW WKH EORFN GLDJRQDOL]DWLRQ PHWKRG FRQYHUJHV ORFDOO\ TXDGUDWLFDOO\ ,I E\ DQRWKHU PHWKRG D PDWUL[ RI HLJHQn YHFWRUV ; RI D PDWUL[ $ FDQ EH GHWHUPLQHG LQ 6HFWLRQ ZH DSSOLHG WKH 45 PHWKRG ZLWK GRXEOH LPSOLFLW VKLIW WR REWDLQ WKH HLJHQYDOXHV DQG WKHQ VROYH %[ IRU [ ZKHUH % $ f§ $ WR REWDLQ D FRUUHVSRQGLQJ HLJHQYHFWRU WR WKH HLJHQYDOXH $ L Q 7R VROYH %[ IRU [ ZH DVVXPH [" f PRYH WKH LWK FROXPQ RI % WR WKH ULJKW KDQG VLGH RI WKH HTXDO VLJQ DQG WKHQ VROYH DQ RYHUGHWHUPLQHG V\Vn WHP RI HTXDWLRQV E\ WKH 45 IDFWRUL]DWLRQ WHFKQLTXH f WKHQ WKH EORFN GLDJRQDOL]DWLRQ PHWKRG FDQ EH XVHG WR ILQG WKH HLJHQYDOXHV RI D VOLJKWO\ SHUWXUEHG PDWUL[ $ H( ZLWK $n DV WKH VWDUWLQJ JXHVV PDWUL[ RI HLJHQYHFWRUV ZLWK UDSLG FRQYHUJHQFH ZKHUH ( LV DQ DUELWUDU\ PDWUL[ DQG H LV D VPDOO VFDODU

PAGE 116

,Q WKH 'LIIHUHQWLDO (TXDWLRQ $SSRDFK WR (LJHQFRPSXWDWLRQV ZH XVH WKH (XOHU PHWKRG 7KH GLIIHUHQWLDO HTXDWLRQV WKDW ZH VROYH IRU WKH HLJHQSUREOHP DUH $Wf f§$Wf 'LDJ OLfn0OLff DQG ;Wf ;^Wf) $Wff ZKHUH 'LDJ0f LV D EORFN GLDJRQDO PDWUL[ IRUPHG IURP WKH GLDJRQDO EORFNV RI WKH EORFN PDWUL[ $ )MM ;Wf $ff LV D VTXDUH ]HUR PDWUL[ DQG )cM ^;Wf $Wff IRU L A M LV WKH VROXWLRQ % WR WKH PDWUL[ HTXDWLRQ %$MWf f§ $cWf%
PAGE 117

ZLWK GRXEOH LPSOLFLW VKLIW WR $ ZH FDQ ILQG D UHDO 6FKXU IRUP $ 4547 7KHQ DSSO\LQJ $OJRULWKP WR ZH FDQ ILQG D EORFN GLDJRQDOL]DWLRQ 5 3$3 +HQFH $ ;$;a@ LV WKH EORFN GLDJRQDOL]DWLRQ RI $ ZKHUH ; 43 f WKHQ ; FDQ EH XVHG DV WKH VWDUWLQJ JXHVV LQYHUWLEOH PDWUL[ DQG $ FDQ EH XVHG DV WKH VWDUWLQJ JXHVV EORFN GLDJRQDO PDWUL[ WR REWDLQ D EORFN GLDJRQDOL]DWLRQ RI D SHUWXUEHG PDWUL[ $ H( ZLWK YHU\ UDSLG FRQYHUJHQFH ZKHUH ( LV DQ DUELWUDU\ PDWUL[ DQG F LV D VPDOO QXPEHU 6LQFH DOO PDWULFHV $ DUH QRW GLDJRQDOL]DEOH VR LQ 6HFWLRQ ZH GLVFXVV WKH IDFn WRUV VHQVLWLYLW\ LQ WKH 6FKXU GHFRPSRVLWLRQ UHODWLYH WR SHUWXUEDWLRQV LQ WKH FRHIILFLHQW PDWUL[ WR FRPSXWH D EORFN 6FKXU GHFRPSRVLWLRQ RI D PDWUL[ $ ,Q 6HFWLRQ ZH SUHVHQW DQ DOJRULWKP WR FRPSXWH D EORFN 6FKXU GHFRPSRVLWLRQ RI D PDWUL[ $ 7KLV DOJRULWKP LV EDVHG RQ VHQVLWLYLW\ UHVXOWV IRU WKH IDFWRUV LQ WKH 6FKXU GHFRPSRVLWLRQ UHODWLYH WR SHUWXUEDWLRQV LQ WKH FRHIILFLHQW PDWUL[ DQG $UPLMRfV UXOH IURP RSWLPL]Dn WLRQ WKHRU\ +HUH RQH QHHGV D VWDUWLQJ JXHVV XQLWDU\ PDWUL[ 6R DQG D VWDUWLQJ JXHVV XSSHU WULDQJXODU PDWUL[ 8R ZKRVH GLDJRQDO HQWULHV DSSUR[LPDWH WKH HLJHQYDOXHV RI WKH PDWUL[ $ ,I ZH WDNH 6R DQG D GLDJRQDO PDWUL[ ZKRVH GLDJRQDO HOHPHQWV DUH XQLIRUPO\ GLVWULEXWHG SRLQWV RQ D FLUFOH LQ WKH FRPSOH[ SODQH ZKLFK LQFOXGHV DOO *HUVFKJRULQ GLVNV IRU $ DV 8R WKHQ WKH DOJRULWKP ZRUNV IRU DOO PDWULFHV 2QFH D EORFN 6FKXU GHFRPSRVLWLRQ $ 4848 LV FRPSXWHG WKHQ WKLV DOJRULWKP FDQ EH XVHG WR ILQG D EORFN 6FKXU GHFRPSRVLWLRQ RI D VOLJKWO\ SHUWXUEHG PDWUL[ $ W( ZLWK DV WKH VWDUWLQJ JXHVV XSSHU WULDQJXODU PDWUL[ DQG 4 DV WKH VWDUWLQJ JXHVV XQLWDU\ PDWUL[ ZLWK YHU\ UDSLG FRQYHUJHQFH ZKHUH ( LV DQ DUELWUDU\ PDWUL[ DQG H LV D VPDOO QXPEHU ,Q 6HFWLRQ ZH SUHVHQW VRPH QRGH FRGHV ZKLFK DORQJ ZLWK VRPH PRGLILn FDWLRQV WR $OJRULWKP FDQ EH XVHG LQ D SDUDOOHO FRPSXWHU ZKHUH SURFHVVRUV DUH VHW XS LQ D ULQJ

PAGE 118

,OO ,Q &KDSWHU ZH GLVFXVVHG KRZ WR H[SORLW WKH V\PPHWU\ RI D PDWUL[ LQ WKH GL DJRQDOL]DWLRQ SURFHVV ,Q 6HFWLRQ ZH VKRZHG KRZ WR PRGLI\ WKH GLDJRQDOL]DWLRQ PHWKRG GLVFXVVHG LQ &KDSWHU :H LQWURGXFHG $UPLMRfV VWHSVL]H LQ WKH PRGLILHG PHWKRG WR PDNH LW PRUH SUDFWLFDO 7KLV PHWKRG ZRUNV IRU DOO V\PPHWULF PDWULFHV ZLWK GLVWLQFW HLJHQYDOXHV ,Q 6HFWLRQ ZH DOVR PRGLILHG WKH EORFN GLDJRQDOL]DWLRQ PHWKRG ZKLFK LV GLVFXVVHG LQ 6HFWLRQ $JDLQ $UPLMRfV VWHSVL]H LV LQWURGXFHG WR PDNH WKH PHWKRG SUDFWLFDO ,I DW OHDVW WZR GLDJRQDO HOHPHQWV RI D V\PPHWULF PDWUL[ $ DUH HTXDO DQG WKH RII GLDJRQDO HOHPHQWV DUH ODUJH FRPSDUHG WR WKH GLDJRQDO HOHn PHQWV WKHQ ZH FDQ QRW DSSO\ WKLV PHWKRG GLUHFWO\ WR $ ,Q WKLV FDVH ILUVW ZH UHGXFH $ WR D V\PPHWULF WULGLDJRQDO PDWUL[ + XVLQJ WKH +HVVHQEHUJ UHGXFWLRQ WR $ DQG WKHQ DSSO\ WKH DOJRULWKP WR +

PAGE 119

5()(5(1&(6 >$WN@ >*RO@ >+DJ@ >+DJ@ >+HO@ >+HQ@ >,,RX@ >.DO62D@ >.DO62K@ >.DO6O@ >/DQ@ ( $WNLQVRQ $Q ,QWURGXFWLRQ WR 1XPHULFDO $QDO\VLV QG (GLWLRQ -RKQ :LOH\ 6RQV 1HZ 1RO!@ % 1REOH $SSOLHG /LQHDU $OJHEUD 3UHQWLFH,,DOO )=QJOHZRRG &OLI7V 1>5XG@ : 5XGLQ 3HDO DQG &RPSOH[ $QDO\VLV UG (GLWLRQ 0F*UDZ+LOO ,QF 1HZ 5XK@ $ 5XKH f$Q $OJRULWKP IRU 1XPHULFDO 'HWHUPLQDWLRQ RI WKH 6WUXFWXUH RI D *HQHUDO 0DWUL[ +, 7

PAGE 120

>6WH@ : 6WHZDUW ,QWURGXFWLRQ WR 0DWUL[ &RPSXWDWLRQV $FDGHPLF 3UHVV 1HZ 6WH@ : 6WHZDUW f$OJRULWKP +45 DQG (;&+1* )RUWUDQ 6XEURXn WLQHV IRU &DOFXODWLQJ DQG RUGHULQJ WKH (LJHQYDOXHV RI D 5HDO 8SSHU +HVVHQ EHUJ 0DWUL[f $&0 7UDQV 0DWK 6RIW 9RO 1R >6WH@ : 6WHZDUW DQG -LJXDQJ 6XQ 0DWUL[ 3HUWXUEDWLRQ 7KHRU\ $FDGHPLF 3UHVV 6DQ 'LHJR &$ >:LOO + :LONLQVRQ 7KH $OJHEUDLF (LJHQYDOXH 3UREOHP &ODUHQGRQ 3UHVV 2[IRUG

PAGE 121

%,2*5$3+,&$/ 6.(7&+ 6LQFH P\ VFKRRO GD\V KDYH EHHQ IDVFLQDWHG E\ PDWKHPDWLFV DQG VFLHQFH FRPn SOHWHG P\ KLJK VFKRRO FHUWLILFDWH LQ ILUVW GLYLVLRQ IURP 6DUDEDUL +LJK 6FKRRO 'DUUDQJ ,QGLD 7KHQ VHOHFWHG WKH PDWKHPDWLFV DQG VFLHQFH VWUHDP IRU P\ SUHXQLYHUVLW\ GHn JUHH DQG FRPSOHWHG LQ WKH VHFRQG GLYLVLRQ IURP 0DQJDOGDL &ROOHJH 0DQJDOGDL ,QGLD ZDV QRW VXUH ZKDW FDUHHU WR FKRRVH ZKHQ ZDV D KLJK VFKRRO DQG D SUHXQLYHUVLW\ VWXGHQW %XW DIWHU ILQLVKHG D SUHXQLYHUVLW\ GHJUHH GHFLGHG WR SXUVXH DQ DFDGHPLF FDUHHU 6XEVHTXHQWO\ SXUVXHG WKH XQGHUJUDGXDWH SURJUDP LQ PDWKHPDWLFV ZLWK KRQRUV LQ $U\D 9LG\DSHHWK &ROOHJH *XZDKDWL ,QGLD DQG FRPSOHWHG P\ EDFKHORUnV GHJUHH LQ WKH VHFRQG FODVV 7KHUHDIWHU LQ DQ HQGHDYRU WR PRYH FORVHU WR P\ DPELWLRQ HQUROOHG LQ WKH JUDGXn DWH SURJUDP LQ PDWKHPDWLFV DW *DXKDWL 8QLYHUVLW\ *XZDKDWL ,QGLD FRPSOHWHG P\ PDVWHUnV GHJUHH ZLWK WKH ILUVW UDQN LQ WKH GHSDUWPHQW $IWHU EHLQJ H[SRVHG WR VFLHQn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

PAGE 122

:LOOLDP +DJHU $IWHU KH DJUHHG WR VXSHUYLVH P\ UHVHDUFK ZH GLG IXUWKHU UHVHDUFK RQ RQH RI KLV SDSHUV ZKLFK H[DPLQHG WKH DSSOLFDWLRQ RI WKH HLJHQYDOXHV DQG HLJHQYHFWRUV VHQVLWLYLW\ LQ WKH HLJHQFRPSXWDWLRQV :H VXFFHGHG LQ FUHDWLQJ VRPH DOJRULWKPV ZKLFK XVH WKH DERYH LGHDV WR FRPSXWH WKH HLJHQYDOXHV DQG HLJHQYHFWRUV RI D PDWUL[

PAGE 123

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f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nK $Gnf§ 'RQDOG : +HDUQ 3URIHVVRU RI ,QGXVWULDO DQG 6\VWHPV (QJLQHHULQJ 7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH 'HSDUWPHQW RI 0DWKHPDWLFV LQ WKH &ROOHJH RI /LEHUDO $UWV DQG 6FLHQFHV DQG WR WKH *UDGXDWH 6FKRRO DQG SWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI WKH 'RFWRU RI ZDV 3KLORVRSK\ 'HFHPEHU 'HDQ *UDGXDWH 6FKRRO

PAGE 124

81,9(56,7< 2) )/25,'$ ,,, LOO PX L}n f§


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 E868ZUNUH_VAGVYP INGEST_TIME 2017-07-12T21:18:57Z PACKAGE AA00003258_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES