Sparse Matrices

SPARSE MATRICES REGINALD P. TE WARSON Department of Applied Mathematics and Statistics State University of N e w York Stony Brook, N e w York

A C A D E M I C P R E S S New York and London

1973

0 1973, OF

OR F O R M OR

OR

OR

York,

111

York 10003

United Kingdom Edition published by 24/28

OF

(MOS) 1970 OF

To HEIDI, ANITA, A N D MONIQUE

Preface

book

book

xi

xii

Preface

1962-1964, I ;

on. on on

1968, on

1969

1970. on

1969,

1970 book on on 1970 book book

book.

book.

xiii

Preface

1,

2,

3,

do 2.

4. ;

2

3

5,

by

6.

7 7

xiv

Preface

8,

EFI, on

8.

Acknowledgments

L.

I book book

I.

K.

xu

CHAPTER

I

Preliminary Considerations

1.I.

Introduction

A

I .2. Sparse Matrices A a

nx n

I

2

1 Preliminary Considerations

n n.

i

;

on

1.3. Packed Form of Storage

pucked form;

1.3.

3

Packed Form of Storage

(for

by

for ;

n

A z E,

E

IuLf)] > E

DEFINITION n -k + 1 by

(2.5.5)

i 2 k, 2 k,

B, A") by

ujyk-

by (2.5.6)

G,

=

(i, j )

BkBk'B,,

B,' B, Proof (i

bti

A(,), + k - 1, q + k + k - 1, q + k + k - 1, + k + k - 1, q + k A'k")

Bk,

q)

g:;)

24

2 me Gaussian Elimination

C C b{t)(l - bfJ)bf’, p # i, q # j = C C ei‘B,eq(l - eplBkeq)ep‘Bkej,

(2.5.7)

=

p

=

P

4

P

4

1 - b!: M

b$) = 0 b$ = 0.

i

1 - bfJ = 0

=

1 - e,’B,e, = e i M e , - e,‘B,e,

(2.5.8)

=

-

e,’B,e,

=

0,

p # i, q # j , q= (n - k

=

e,’(M - B,)eq

+

-

eq’B,’ep, e,’B,e,

(2.5.8), P

4

eqeq’Bk’C e,e,’B,ej

= e;B,

P

4

= ei’BkBLBkej,

1,eqeq‘ = 1,ePe,,’ = I , -, +

by

2.5.5. (2.5.9)

COROLLARY s=

u:!)

p (2.5.20)

+k-1 g$

cc,p

=

lc$+!k-l,j+k.-ll

i,j

(E

ct

+ k - 1, t =

by

> E,

25

2.5. Minimizing the Total Number of Nonzero Elements in EFI

by

(ik)s

U

n-k A(k)

')

tjJk)s

((k)s,

((k)s,

rfk)s

so by

2.5.9,

sub( ( k ) ~

a:!)

no

A("+1)= L,Pn...L,P,LlP,AQlQ2...Q,,

(2.5.22)

L = L,P,. . . L,P2LlPl,

(2.5.12)

Q1Q2...Qn

(2.5.13)

A("+')=

=Q

A-' =

Q

Pks

on

THEOREM

0,

Qfr-lE.

n'),

(2.5.24)

v(~)s.

a~~k-,,j+k-l

n

26

2 The Gaussian Elimination

by

Gk, (2.5.25)

Gk

= (Bk

-

I n - k + l)M(Bk

- In-k+

l)?

M

2:;’

Proof

2.5.5,

2;:)

(2.5.16)

1bi:)b$),

= P

=

p #

q # j,

4

(ei’BkC.‘,- ei’C.‘,)(Vk’Bkej - b‘ej), n-k

Vk 8 j

=

ei’(Bk

- In-,+

l)GC.‘,’(Bk

= ei’(Bk - zn-k+l)‘(Bk

- In-k+

- In-,+

+ 1.

I k j

l k j ?

VkVk’ =

(2.5.10)]

EL’$

(2.5.1

~ u ~ ~ k ~ l ,>j E,+ k - l ~

= i.j

u

a$)

Bk = Bk’

+k - 1 =s

B+k -1=

(2.5.17)

27

2.5. Minimizing the Total Number of Nonzero Elements in EFI

(2.5.16),

k::)

-

= i

c1

(ei’Bk&

-

ei’Bkl/k,

ei’BkV,2 1. (2.5.18)

i

ei’Bkh = e,’Bkh,

/‘Lkik-

I.a+k-

11

’

E,

+k -

ei’B,V, A‘k’.

do,

A

A

v(,)s

(2.5.29)

THEOREM

A

k

=

1,2,. . . ,n - 1,

n -k A‘”’) (2.5.4), qlk), i > k (2.5.20)

qjk) = -a::+

by =

> k.

28

2 The Gaussian Elimination

Proof

u $ + ' ) = u$")

> k,

= a$), i, j uij) =

by

As

(2.5.3),Qk = pk' (2.5.4), (2.2.3),

=

a$), i , j > k.

( k + 1)

=

(k + 1) =

'ji

-

&k)a(k)/$p, rk k J

-

a(k)a(k)/*(k), Jk k i ukk

JI

' i (j k + l )

= a$+1),

a!k) =

a(?),

JJ

?Ik)

= d:!)/&)

=

=

a:!),

i, j > k,

i, j 2 k.

JI

(2.5.2),(2.2.3), l)

(2.5.2),

j > k,

'ij

a:!+

i,,j > k on k

i > k, (2.2.11)

(2.5.4),

- d$)/li$), ;

(2.5.10)

(i,j ) ,

by by (2.5.16)

(2.5.15), (2.5.21)

G k

= BkMkBk

&!k-

1, j + k -

ei'Gkej is

2.5.14 (2.5.2), (2.5.3), l/uiyk- l , j + k - 1 , arik-

(2.5.4) &'Bkej - 1 1, j + k -

ek'BkVk- 1 q(k) eLA(k+'). # 0, p # i, (e:Bk& - l)(&'&ej - 1)

29

2.5. Minimizing the Total Number of Nonzero Elements in EFI

(2.5.22)

;:2

ei’G,ej

=

la$!k-

l,j+k-

> E,

1.J

k

Gk

G,

G,

B,(M - 6Bk‘)Bk 6 1-6 6 on

by Lks

ck,

O

< 6 < 1,

Bk

by

n-k A(,+’)

A^(,)

30

2

The Gaussian Elimination

2.6. Storage and Use of the Elimination Form of Inverse q(k)s

i?$) # 0,

>k

0, > k, ali") qik' # 0, > k ti$) # 0, i > q(kk) no a;\+ ') = 1 U Pk Qk (2.5.3) k a$+

(2.5.4), a;:+

l),

t

=

Pk

2n (2.2.12), (2.2.10), ((k)s by U

QkS

Pks

i?@ =

1.

Qk, (2.5.12).

(2.2.11) uks

by

((k)~

U.

A(k)

q(k)

((k)

k

1.3.

A(k'

by (2.1.1)

(2.2.1),

b 1969 ;

2.5,

A

2.7.

31

Bibliography a d Comments

2.7. Bibliography and Comments

by

al. by

on

by do no

3

CHAPTER

3

Additional Methods for Minimizing the Storage for EFI

3.1. Introduction

do ;

2. ;

33

34

3 Additional Methods for Minimizing the Storage for EFI

no

3.2. Methods Based on A Priori Column Permutations Q, P

(3.2.1)

PAQ

=

A,

A^ P

Q.

by

;

on

P,

Q A

Q

A. A

so DEFINITION B,,

rik’

cy)

2.5.

B,

(3.2.2) n-k

n -k + 1.

+ 1,

ei

35

3.2. Methods Based on A Priori Column Permutations

i;;)= (

(3.2.3)

p-

-

by

j,

(3.2.4) y y )

=

i

2:;)

=

( c y )-

i

(rlk)bl;) = 1,

i =

bL5) = 1.

(cik)- l)(rLk)-

2.5.14,

(j

A'k),

+k (a + k

-

A A

y$')s.

k n, on A

yj')

by

A - ',

on

5; p

ylip's,

A

is

j,

cy)

is

(3.2.3), by (3.2.5)

for =

(3.2.6)

dy) =

(d'jk' -

1i I l k ) ,

i

b$) = 1,

-

i

=

1.

36

3 Additional Methods for Mininlizing the Storage for EFI

A

A$%, A cjl)s.

yy's

AT)

$)

:

2.5.14,

(3.2.7)

ei)Gkej,

= i

ei'B,ej

i

=

1.

1 ej'B,'eiei) = ej'B,' C eiei) = ej)Bk'ln-,+ i i

=

ei'B,ej = 1,

(3.2.8)

=

X i e;,

(ej'B,'G,ej)/b'B,ej,

e,'B,ej

i

=

1, e j B, .

A by

,

I

c(jl)s, y y ' s

2,

A = AQ.

(3.2.9) A Ae, e,,

2

=

=

-

Ae,

2, =

AQe,,

I,

Qe,,

Q.

A

I,

Q. 2 by

cJ1)s, yJ1)s,

AJ1)s, by

37

3.2. Methods Based on A Priori Column Permutations

A(') = 2

(2.5.3), A^(k)

by = p A(k)

A(k)

(3.2.20)

k

3

1, by

Pk a

Fik)=

(3.2.12)

(a

Flk),

i

+k -

by

Idlyk,- 1,kl > E ,

i

Flk)

(3.2.2).

rlk)

(2.5.10).

E

F!k)

by

rik)

1966). n-k

(3.2.12)

A^(k) (i

+k -

+1

pik)

A^(k),

> 1, 7{k_+,'),

by

A(k+l ) (3.2.13)

by (2.5.2),(3.2.10),

A(k+')

Proof

A^(k) by 8,

8,

n -k by

(i

+1 8,

Bk+ (i -

6$),

(2.5.4).

Bk+

+ k - 1, + k - 1) A^(k) &. a!"!,- = 0, 6:;) = 0 a(k) A('+') (3.2.13) = Pi? ', rl!''') = I,k

i, j ,

38

3 Additional Methods for Mihh&z&e the Storage for EFI

1.k

&$!k-

hik) = 1

,Z o

= 1,

T(o)

b$) = 0. o

Bk A(k’ = 1

(3.2.25)

b!k) = 13

0) =

= 0)

= P(k)

- 1

plk’ - 1

3).

=

n -k

Bk

of

(Pik)-

-

hik! = hi:) = 1.

k)

(Pik)

Q)

-

-

k),

Fll)

=

8,

f‘k’ - 1 4) ( n - k ) n ( - lk ) ( 1 - n - k

’

Plk)

al7kk)I.k = 0,

on

r:”

ril),

=

B,

ei’B, Vl

A‘”

=

A’

k, Plk) Pi.4) = p

jz(k). k i

(3.2.26) n

-k

1

Pik)

+1 rik)

k. k

Bk.


p2 ( p l , q l ) , ( p 2 ,q l ) , ( p 2 ,q2)

as hij = 0

B,

i 2p,

triangular corner

Fig. 3.7.1. Triangular corner.

i, - i,

B

i, > i , ,

A,,,

i,

A,,,

Proof

pi, by Si1

pi, I

Bi1

(3.7.24)

=

Pi2

i,

Biz,

i,

- ill i, > i, ,

- (i2

< Biz.

Bit

Biz = pi, + (iz pi, - pi, > i, (3.7.25)

- i,,

c Biz.

Biz

A,,, ,

i,

(3.7.14)

(3.7.19, A,,, A,,,

i,

/3

B i, > i , ,

pi, - pi, < i,

i , c i, (3.7.26)

p 12. - p . 11= i

2

- 4,

-

i,.

66

3 Additional Methods for Minimizing the Storage for EFI

i,

< i < i,, pi, - pi = i, p,

p,

- i.

i, ,

i,

p,

+1

q, + 1, ( p , , q,)

q1 ( p l , q,), ( p , ,q l )

B p, p,

=

q,,

B

3.7.2.

Fig. 3.7.2.

Block triangular form.

B

3.7.13,

3.8.

The Band Form A

on

=

q,

67

3.8. The Band Form

2.5.19, on A

a (3.3.2)

A.

(3.8.2)

PBP

B.

=

qi < i,

pi = i - q i ,

(3.8.2)

biqi (3.8.3)

i

p

p

B,

= P

i

pi. B

Band Form no

p.

pi,

a

FIRST METHOD

A by 3.8.1.

fi

0

B

B

68

3 Additional Methods for Minimizing the Storage for EFI B

B X X X . .

X X . X . X . X . X

...

. X . X X X

X

Fig. 3.8.1.

8,

by 4

1,2, . . . , n

1.

p

2.

Vertex List B

n

pi = /jp,

p - p,

(Pi+ ( p -

go

x 's,

8

< p,

i < p),

7.

2. If

go

7.

3.

go 6.

4. go 6.

3

:

4

so 1 5. =

1,2,. . . , n,evL(j)

A^

= PAP'.

69

3.8. The Band Form

6.

4

3

go 5.

7. go

1.

SECOND METHOD

1969).

1.

i pi,

R pi

i,

pi, =

pi

3.8.2 1.

2.

2*m 2

B

7*

3*

5*

8*

Fig. 3.8.2. Example of a vertex renumbering scheme.

1*

70

3 Additio~lMethods for Minimizing the Storage for EFI

3.8.2

2*, 4*,

2, 3,

9*

4,

3.

2, 3*,

on.

3,

2 5;

3.8.2 4*)

3 6*

6*, 5*, 5* 8,

6, 7,

5,6,7,

8

1

4.

R

n

R

as 2, 3 , 4 5.

B

B

A^).

3.8.3, 3.8.2. 8

5.

B

B I * X X x 2 * x x x x x x 3* x x x 4* x x x x x 5* x x 6* x 7* x 8* 9* x x x x x x x x 10*

\

X X

x x x x x x x x x x

X

x x x x x x x x

I x x x x 2 x x x x 3 x x x x x x 4 x x x x x 5 x x x x 6 x x x x x 7 x x x 0 x x X X X X 9 x X X

.\

10

Fig. 3.8.3. Matrix B and its permuted form 8.

X

X

X

)

)

) )

71

3.8. The Band Form

by Pmin

pmin pi,

i,

d pi,

Pmin

1

+ tpmax,

p,,,

B

B.

THIRD METHOD An

Pi

on

(3.8.2) 1968).

1.

B

B

B B. Row Interchange

RI

1,2,. . . ,n,

2.

by n-1 1 (1,2), (n,n - l), (2,3), ( n - 1, n - 2), . . . , up

72

3 Additional Methods for Minimizing the Storage for EFI

no

3.

no

B

P,

RI(j)

+

j = 1,2,. . . ,n,e,,(,)

FOURTH METHOD FOR REDUCING THE BANDWIDTH

length of their intersection i

vi

p vi

3.8.4

3.8.4.

1 < i,

i’s.

< fl +

i, i, ; i, , 2p

i,

+1

i, n

- 2p.

i,

i, i,.

1

28 + 1

vis

i i

n

- 2p. ui

73

Band Form

3.8.

n - 28

28

+1 +1

n.

@

B (i, j )

i

vi

B

by

(3.8.4)

vi = e i ' i W ,

V

by @

1. (3.8.4).

vi

2.

p,,, :

z

n)/2n (p,,, - 1)/2 on

(t-

8

on

no

3.8.4

vis

3.

28

3.8.5). vp ep)@ei # 0 ei'vej# 0

vp = p.

vis by by

vir

p j

no q $ NW

e;Weq = 0,

NW

74

3 Additional Methods for M-

the Storage for EFI

NW

SE

Fig. 3.8.5.

a

a

vis on on 3.8.5. 4.

3.8.5) :

ei'V = 1,

V

ie

0, p

j~

=

NW.

: E

3.8.5. 5.

3 4

P

B=

75

3.9. Other Desirable Forms

3.5, 3.6, 3.7,

3.8

3.9. Other Desirable Forms 3.9.1 1971). Singly Bordered Block Diagonal Form Doubly Bordered Block Diagonal Form Bordered Block Triangular Band Triangular Forms Singly Bordered Band Form Bordered Band Form 3.9.1 SBBDF

DBBDF

BBTF

BBNTF

SBBF

DBBF

Bordered Doubly

Fig. 3.9.1. Some simple desirable forms.

3.9.2

B

1966, 1968).

by

76

3 Additional Methods for Minimizing the Storage for EFI

Fig. 3.9.2. Two other desirable forms.

on ;

attachment set

1965). 1971, p. 125). B.

vi

vi

(3.8.4),

vi B).

vis et al., 1970).

3.9.3). 3.6 3.9.3.

3.7

77

3.9. Other Desirable Forms

I

I

Fig. 3.9.3. Modified forms of BTF and BNTF.

3.9.3. by

3.7.13 3.9.3

no

3.5

3.6) 1971b),

(1969)

3.6). by

[l,4,3,6,5,1].

3.9.4. no

parallel 3.9.4,

no

b, c,

d.

order

78

3 Additional Methods for Minimizing the Stornge.for EFI

1

I

2

3

X

5

6

X X

2

X

x

x

X

x

x

X

3

x

4

5

4

x

X

x x

6

x

Fig. 3.9.4.

;

c

2 - 1 = 1.

3.9.5,

Order

I

4

3

6

!

5

0

B-2-E

I

B-E

i

I

E

IB-c

3

E

a

~

b

B

d

Fig. 3.9.5.

3.9.4

E [6,5]

d

3.9.4, 5

1

79

3.10. Inverses of BTF and BBTF

on. 3.9.6. 5

I Fig. 3.9.6.

1

X/Torn

4

3

element

2

6

m x El

The matrix associated with the relabeled digraph.

3.9.6

3.10. Inverses of BTF and BBTF A,,, i

by (3.3.1). 1972; 1.

=

1,2,. . . ,p

1972).

i = p - 1, p - 2,. . . , 2 , 1

A,,

by

elimination

A,,

Uii APi Gaussian A,,.

U,, A,,

< i.

Ajps,

< i.

80

3 Additional Metbods for MinimMog tbe Storage for EFI

2. Up, Aj$,j # p

I

Up,

no ji,

j

i # p.

i, i = p, p - 1,. . . ,2,1, :

Uii.

A,,

Uii

I

Aji

0

< i. A j i , j # i.

no

U,,,

3.11. Bibliography and Comments

1,4,

3.7

Q, 3.8 to

et

good 3.9.2

by

3.11.

81

Bibliography a d Comments

1961;

CHAPTER

4

Direct Triangular Decomposition

4.1.

Introduction

z

A U, (4.2.2)

A

=

ZU.

1968 ; (4.2.2)

A - 1 = U-1z-1

U-'

t-' 83

84

4

Direct Triangular Decomposition

2 =

x

b

(4.1.3)

Ly

=

(4.1.4)

ux

=y

y

b

no by

x,

2. 2

Ag)

4.2.

The Crout Method

2

(i,]]

by lij

U

U

k = 1,2,. .. ,n. L U

k

k- 1

=

1,

+ p = 1 lipUpkr

i

> k,

> k,

k- 1

(4.2.1)

Iik = aik

-

1l i p U p k ,

p= 1

lkp = 0

p > k, k- 1

ukk =

-1

> i, upk =

lip = 0

uij,

il k.

1,

ukk = 1,

4.2.

85

The Crout Method

U

k-1

k-1

U U

E

E

?I

by

(4.2.3)

i = 1,2,... , n ;

1.11 = a.11,

U

(4.1.1)

U

(4.1.1)

Z ll1el’, (4.2.4)

ulj = alj/lll,

j > 1.

k = 1, (. . .) = 0, (4.2.1) 4.2.2),

E

L2>

(4.2.3) In

(4.2.4)

U

Okk

7

Fig. 4.2.1.

4

2

Storage for the Crout method.

4.2.1

k

- 1

el.

86

4

+ U, , U,,

U

k

U,]

-

Direct Triangular Decomposition

z

1

I,,

U, (4.2.1)

a,,

E,

[?] [

(4.2.5)

A,

lkk = a,,

li+,,,

A, "k]1 -

=

1

[;:]., by (4.2.2)

= ei'Az1.

U, (4.2.6)

A12 =

(A12 - L Z u 3 ) / a k k ,

1

ukJ+,= Al,ej.

U

z,

:

E, (4.2.7)

U, (4.1.3)

(4.1.4) lkk,

yk =

(4.2.8)

X,

f

= y, -

U, U

ukpxp?

k

y

k

=

:

x

1,2,. . . ,n

= n, n -

1,... , I ,

p=k+ 1 0

n

p= 1

p=n+ 1

1 (..-) (4.2.7)

1

(

-

0

.

)

y

(4.2.2) by (Alb)

(4.2.2) =

b,

up,"+ = y,,.

by

1965). (4.2.1),

(4.2.9)

llskl

=

11ik1 3 I

i 2 k,

li,

87

4.3. Minimiziig the Fill-in for the Crout Method

by

i

E

>k

of A

P =

O (4.2.ZO)

lkk = 0 lsk

2, =

4.3.

U

L-

Minimizing the Fill-infor the Crout Method

n-k Bk

+1

88

4 Direct Triallgulu Decomposition

4.2.1 (i,j]

by b!:). b$), i < k, 2

Bk b$),i 2 k,
F:”) o

@.

=

on 4.3.5

I,,

by up,).

by by

by (4.5.3)

4.6.

93

Desirable Form for Triangular Decomposition

no

(4.5.2) (4.5.3)

no

A^

on 2.

4.6.

P =

Desirable Forms for Triangular Decomposition 3,

t=

’

4.2,

E

by (2.2.6),

.. . (2.2.3) L,- . .

-

- = I,,

(2.2.4) e,

1

94

4

Direct Triangular Decomposition

=

4.7. Bibliography and Comments

by

(1963), (1969),

(1969), (1971). 1967;

1972).

al. (1969).

k,

ukk

=1

Ikk

(1963), (1969),

=

1.

al., 1970).

(1963, 1968), (1971),

CHAPTER

5

The Gauss-Jordan Elimination

5.1.

Introduction

Gauss-Jordan elimination.

U

A-' Product Form

Inverse

95

96

5 The GnussJordan Elimination

5.2.

The Basic Method

b = b,

1963;

x

1965).

of A(') = A(k)

A(,)

k

= 1,2,. . . ,n,

A(k).

(i,j)

A("+') =

k-1 e, by

u$)

A(k) A @ +1 ) = T A ( k )

(5.2.1)

k

(5.2.2)

= 1,

3

+ ( [ ( k ) - ek)e;, by

((k)

(5.2.3)

[lk) = -&)/ui?,

[ik)= l/ui\).

i#k A(') =

(5.2.1)

A("+')= I , ,

T, - * . TZTlA = I , ,

of A

Product Form oflnuerse A-' = T, * * * TZT1.

(5.2.4)

((k) ')

by

1963 ; 1968 ;

1954; 1963 ; et ul., 1969 ;

1962; 1966,1967a; et ul., 1969).

97

5.3. The Relationship between the PFI and the EFI

5.4. =b

no

& A(k) (5.2.1). A(k)

A(k+')

(5.2.1),

PFI.

2.4

5.3. The Relationship between the PFI and the EFI 2.2

U

by

by no

U

by

U U-

U

U -'

by

k

k,

k

U

U

U -' U-' U -')

U -' by

98

5 The G a d o r d a n Elimination

U :

U k + '= ) Ok Uk), k

(5.3.2)

~ ( 1= )

(5.3.2)

u, U ( n + l )= I n

8, = I n

(5.3.3)

= 1,2,. . . , n,

+ t(k)eL by

f(k)

(5.3.4)

t!k)= 0,

[ui:)

(i,j)

i2k

-@,

i k,

-

a$

=

2

0,

kpk

113

6.3. Minimizing the Nonzeros in the RGS Method

2,

ei

(6.3.2Z)

n -k

rn

V,

$)

+ 1,

c y ) = V’BkZj.

= ei’B,T/,

A

B,

(6.3.22) A L G O R I T H M

A.

R,

n k 1.

cy)

=

1.

c:,) =

b$ by

2. (t + k k = n, go

2.

cy),

> 1, go

j =

0

b:) = 1

R,. 3,

k

k

go

G,

n -k

+1

R, G,

,

R,, . 3.

A R, +

,

,

R,, ,

A.

Remarks

1 A

6.3.

2,

A

1. on A ,

RGS

,

In-,+ ,

I,

+ 1.

114

6 Orthogonalization Methods

A^ A^

3.

6.4.

(6.4.2)

=

:

The Householder Triangularization Method n A(k+' )= HkA(k),

(6.4.2)

Hk=lm-uk

k -1

=

1,2,. . . , n

9( k )9W',

qck)

by

m

(6.4.4)

fikZ

=

1

c(k

= f i k 2 f fik&)

i=k

A(') =

flk

&).

n n

by rn 2 n). rn = n. (6.4.5)

Householder Triangularization A("+') m -n n-1

H,H,,-,**.H1

=

n

H H

by I?,

6.4.

I15

The Householder Triangularhation Method

(6.4.1)

H A ( ” + 1= ) H A =.A =

fiv

A D - ‘ = 8’.

(6.4.6)

8‘

O-’ (6.4.6),

1965). (6.4.5);

H

qck)s

aks

1968a). :

(6.4.7)

by (6.4.1) (6.4.4),

k a,+’)

Proof

=

- 1

T

Ack+’)

A(k)

a!:+” = 0, i > k.

Bk

(6.4.2)

k

(6.4.3),

Hk I, ; (6.4.3)

q(k)’aik) = (aii) f Sk)ai7+ =

a$+ l ) = 0, > k.

& Bk&)

+

(6.4.1), (6.4.4),

rn

1

i=k+ 1

Bk2 =

ak,

(6.4.1)

(6.4.2),

- .ik) - qCk)

aj;kk+

1)

=

- 1

-+ Bk

116

6 Orthogonalizatiot~Methods

n A(k).

n

-

k

k

-

+1

+1

by

n-k B, A ( k )by bjk) = BkZj Z;BkZj = bj?)

In-k+

Zj

bikl = 1,

(6.4.8)

by (6.3.5).

Gk

(6.4.1), (6.4.2), (6.4.3),

Proof 1)

4

=,

(6.4.4),

4 > k,

- a; I ( #k)"af))Q(k),

m

C a$)a$) ~f:flkaiy,

9W' a,, (k) =

i=k

ag) # 0,

bikI = 1 i=k

=4 -k

A(k),

bik)by

A(k)

no

+ 1.

by)' * by) = 1, (bik)'* by))6y)'bik) hik)

bikY* by) = 1

q(k)af)# 0, as

6ik)'bik)= 0, by n-k+

1

(6.3.6)

c

=

1.

117

6.5. ”he Fill-infor the RCS wrsus the HT Metbod

u& = 0, u~k”u= ~ k0, )

fiCk)‘uik) # 0

u,

by

u

u$) # 0.

A(k)

(6.4.9)

Bk

Gk Bk

on G k . Proof In-k+

Pk by

Qk Bk by

PkBkQk,

(6.3.5) (QklBk’pk’

* PkBkQk)

= Qk’(Bk’

Qk‘gk(Pk‘PkBkQk

= Pi*Pk

=Q

(6.4.8)

k Q i

QklGkQk,

= Qk* Q i = PLPk =

In-k+,,

(6.4.9),

s 2 k, +k -

3 @d+k-

* Bk)B,’BkQk

#

A(‘+ ‘). B, do

on

6.5.

on

The Fill-in for the RGS versus the HT Method

fi’

no

118

6 Orthogonaliiation Methods

fi'

by H , ' H , ' . . . H,'

A

fi'

n

rfk)s

fi' fi'

uks

no

fi' by 6.3.4, 6.4.8,

RGS

6.4.9,

RGS

by

Gk

RGS

(n - k

+

Bk

(n - k

x

RGS

+

m x (n - k

+1

n-k A(k)

ffk)

RGS

A(k)

fi' RGS

by

6.6. The Jacobi Method

by

A("+'),

n-1

A(k) k-1 a$) (i, j ) a$) # 0, i > k u$) #

,i > k

$2

a{;), i

A(k) A(k),

by

> k. k

+

119

6.6. The Jacobi Method

A‘,’, (6.6.2)

R,,

= 1,

:

+ (t - l)(f?kek’+ e,e,‘) + w(eke;

(6.6.2)

Z = a&)/(&)z

0=

R,, p ) , (p,

+ abkk)Z)1/2

a$)/(ag

+

a$q1/2.

k),

by by 5, w , - w, r, A‘,’ R,,A(,)

(p,

RpkA(,) p,

i # k

- e,,ek’),

e / e j = 0, i # j ,

(6.6.1)

(6.6.3)

ei’RpkA(,)= ei’A(,).

+ (Z - 1)ek’ + We,’)A‘k’ = 7ek’A(,)+ oe,’A(k!

(6.6.4)

ek‘R,kA‘k’ = (ek’

(6.6.5)

f?p’RpkA(k) = (ed

+ (Z - l)ep’- wf?k’)A‘k’

- ze ‘A(,) - oe,’A(k).

A(k’.

(6.6.6)

(6.6.5)

(6.6.2)

f?,‘R,kA‘k’C?k = 7 4 2 - W U i i ) = 0.

(6.6.4)

(6.6.5), A(,).

R,kA(k’ q > p,

a$)

k q.

a$ = 0.

q.

# 0,

a$) = 0, a:

second order interaction third higher order interactions

k

p

120

6 OrthogoaalizatioaMethods

As

n -k by

Bk

+1

A , by

(s

+ k - 1, + k -

A(,)

(6.6.4)

on

i

(6.6.5),

on bl:)

bj:) = 1,

Bk

=

s B, e,'Bke,.

bf) = 1, bf)

=

1

by (3.2.6),

rp,

=

bf)

i

=

1

i

1bl:'ei'Bk&,

=

(3.2.2)

i

(6.6.7)

dik) = e,'BLBk&.

djk)=

(6.6.8)

j

:

e), bi:)

=

rjk)s (6.6.9)

rjk),

=

i

i

s

(s,j)

by

Bk f?,'Bk'* Bkej = 1,

bl:) = 1.

1

6.7.

I21

Bibliography and Comments

s (6.6.10)

yit) = =

1e,’Bk’* Bkej - rtk) et’(Bk’* Bk)& - es’Bk&, s

t

:

(6.6.11)

6.7. Bibliography and Comments

(1965). (1966)

(1967).

by

(1962).

by

CHAPTER

7

Eigenvalues and Eigenvectors

7.1.

Introduction

:

Method

Householder Method 6. 1965 ;

I23

124

7 Eigenvalues and Eigenvectors

aij = 0, i > j

+1 (Fox,

7.4, by

no

by by

The Givens Method

7.2.

by k-1

n -k-1 k

. . . ,n 6.6,

(7.2.2)

=In

+ (T -

lei+ 1

+

-

epe;+

1)

+

a$

k.

(7.2.2)

+

= pk

pk*

+ 2, k + 3,

7.2.

125

The Givens Method

ei‘Rpk= ei’ i # k

+

+ 1, A\k)

e;+ lRpkA‘k’= (ei+ 1

(7.2.3)

=

re; +

A(k’

+ (z - l)ei+ + oe,’)A(k’ A(k’+ we,’Atk), 1

eP ‘ Rp k A(k)= 7e ’A(k)- we;+ lA(k).

(7.2.4)

k

+1

p A(k’.

RpkA(k’ RpkA(k’R>k,

p

k+1 RpkA(k’.

e>Ay’ek = e,’RpkA‘k’R>kek =

( T ~ , ’ A( ~we;+ ) lA‘k’)ek

- 7 4 2 - wa(kk!1 . k = 0

7 = a (kk+) l , k . / [ ( a $ ) ) z

0

(k,

= a$)/[(a$))’

Aik)

-k

(aik!l . k ) z l l ’ z

+ (aiki

1,k)2]1’z-

e,’A‘:)e, = 0. by k + 2, k + 3,. . . ,n

(7.2.2),

A(k)

by A(k+l )

aiyk,t+k- # 0, s # t - 1,

+ 1,

by do 6.6

6.6,

k

126

k

7 Eigenvalues and Eigenvectors

k

+1

+1 A(k),A\k)

n

n -k

-

k

+1

Bk

A(k) by

by

(k + 1, k)

A(k)

# t - 1 by by

by

by

r

X X

Fig. 7.2.1.

Rotations for a band matrix.

X

127

7.3. The Householder Method

1

1

1

n = 10, 1 = 3,

7.2.1,

udl

on (6.6.1),

w

t

7.2.1, (7,3)

u41

R34

(10,6)

R9,10

(10,6) by

(3,l)

(4,l)

The Householder Method

7.3.

by 1965).

k-1

A(k)

k

. . . ,n

(7.3.1)

n

6.4.

[ A ( ' )=

A("-') A(k+1) = H k A(k)Hk ,

k

H - I - cck-

(7.3.2)

k -

n

=

1,2,..., n - 2,

lq(k)q(k)r,

q(k) fj!k) = 0, i

(7.3.3)

vk+ A(k)

=

a (kk +) l,k

by


k

+1

+ 2, k + 3,

128

7 Eigenvalues and Eigenvectors

n

8,'

(7.3.4)

1

=

uk

(&))'3

= 8,'

f 8k'ikl

1, k

i=k+l

ukklI , , . n -k

n -k

Bk

A',) by

+1

by 6.4.8,

(7.3.5)

THEOREM

1

=

-

AW =

(7.3.6)

H k A(,)r

by

Hk

by Gk

Proof A(,)

by

Bk

n

-

k

n -k 6.4.8.

+1

6.4.8,

G,,

6.4.9

B, s# 3

3

(s

+ k, 3 + k A(,+') 3

-1 A(,)

+ 1,

= 1

A(,) A(,)

H,dk)

+ 1,

(s

by

+ k, 3 + k -

by H , , by H , . A")

7.4.

129

Reduction to the Hessenberg Form

A

S

Bk

es'(Bk'* Bk)v =

(7.3.7)

i

ej'(B,'

* Bk)v.

(6.3.7).

by

7.4. Reduction to the Hessenberg Form k

A(k) i >j

a$) = 0

Lk

k

-

(7.4.1)

-

1 j < k.

2.2)

+ 2, k + 3,. . . ,n

. . . ,n

+1

k = 1,2,

2

A("- ')

A('+') = L k + l A'k'L-' k+l,

=

1 ? 2 , . * . , n - 2,

(7.4.2)

9"' (7.4.3)

(7.4.2) (7.4.4)

(7.4.3),

Lk;ll = I ,

-

q(k+l)ei+l

by

130

7 Eigenvalues and Eigenvectors

A',) by

A(,+')

ui:) # 0, i > k

(k +

+ 1, a$) # 0 , j

n-k -1

>k

+ 1, (k +

n -k-1

A',).

(k +

:

by

Lk+l,

k

by

(k +

j

(k +

+1 I,;:,.

n

-

Lk+l.

n-k

n -k

k-1 Bk

+1

A',) by

by (7.4.5)

THEOREM

(s

(k + 1, k)

+ k, t + k -

A',)

by

# t - 1,

by

by L,+ G,

(s,t )

by # t -1

Proof

(k + 1, k)

A(k)

by (i

+ k,j + k -

E,

(i, j ) A',',

2.5.5,

no

(k +

Lk;ll

A',',

7.4.5

(k +

Lk;ll,

(k + by Lk;ll. n -k +1

A'k)

by

by

B",

N,

A','

7.4.

I31

Reduction to the Hessenberg Form

n -k

+1

I@'

N,. II -

by

k

+1

by

(7.4.6)

THEOREM

(p

+ k - 1,q + k -

(k + 1, k)

A',)

by

(k

Lk;ll,

by

+

by (7.4.7)

yb",' = e,'N,(N,

* Z'4)Bk)e,.

Proof A',)

k

6);:

=

e,'mk'(N,

Bk,

+k-1

I*

Bk

(i,q)

N, p

+k-1 k + 1,

q

* I'4'B&,,

7.4.5

7.4.6, (i + 1,j) aL2k,t+k-

Bk (7.4.8)

gi:)

+ y$

=

+ yly

i,j

(i,j)

Bk,

# j

1, j ) ,

+ 1,

G',).

A

on s

t.

s

7.4.5,

V,

(3.2.2)

t

:

B, n -k

132

7 Eigenvalues and Eigenvectors

(3.2.2)

(3.2.3),

(7.4.9) s

t.

7.5. Eigenvectors x A x = Ax (7.5.2)

( A - AZ)x = 0. x # 0,

A - II

(n Fox

7.6. Bibliography and Comments

x.

CHAPTER

a

Change of Basis and Miscellaneous Topics

8.1. Introduction

by

by

al., 8.2, by 133

134

8 Change of Basis and MiscellaneousTopics

A'). 8.3. A 8.4.

A

U 2.2.

8.2.

The Result of Changes in a Column of A on A-' A

A A

(5.2.4),

by (2.4.1) A by 8,.

A- '

A-

'.

FIRST METHOD A-

'

A,

A-

I,,

A-'A

(8.2.1)

+ ( ~ - ' d ,- e,)e,' = I , + (a:+ - e,)e,', =

I,

')

a:+')

A-' = [ I , + ( a ,

(8.2.2)

=

=

A-'a,.

- e ) e 'l-lA-1 4

4

EA-',

5.2, (8.2.3)

(8.2.4)

=

@)

=

-di:'')/dr

I, + ($4)

'),

- e,)e,'

i#q

tp) =

'A

135

8.2. The Result of Changes in a Column of A on A -

A-'

A-'. %(4)

3. A

3)

a A AA

SECOND METHOD

a,

a4 U,

(2.4.1) by

(8.2.5)

by (8.2.3),

c$q)= -6$y/d::, a:)

(8.2.6)

=

# q

[$)

L,..-L,A

L, . . . L , A (2.2.5).

,

L, . . L A

on U , , . I . U,L,. . . L,A

U , , . . U,L,.

. . LIA

2.2, by L, . L , A^ (2.2.10),

3

(8.2.3),(8.2.5), (8.2.6) U,, . . . U,L,. . .L,A e4 < U q , * * U,L,. . . L , A

,

;

44

by

L;.. L,A

by (8.2.6).

=

u,,, . . . U,L, . . .~ ~ 6 , . A("+')

(2.2.1

by

t(4)

,

136

8 Change of Basis and Miscellaneous Topics

UqUq+

. . . U,L,. . . L,A I,

U , . . . U,L,-..L,A

=

= u,..'U,-,$u,,,.'A^-,

(8.2.7)

u, . . . uq-1 $ U q i

=

1

U,L,."L,A^

* . ' U,L,

* *.

L,.

q

q1 :

< q, $, U,, a;! = u4,+,... uq_,$uq,,...u,~,...~,aq,.

1.

4,

2.

q , > 4,

$,

$,

a:!

=

fpq+ . . . u,L,. . . L,aq,,

U,,

ul., 1969).

THIRD METHOD

1972). As

al., 1969;

A^("+ = L, . . L , A A("+') U

OqA^('+')

by (8.2.8)

0, n-q eq

U(q)

(2.2.9), (2.2.10), U,

A^-'

uq,

by dq A^ U = L, . . . L,A,

$

Uq)= $OqA^(n+l).

U by eq, by eq' (2.2.1l), U ( ¶ ) - ' CJ, . . . U , = 0, k > 4 , (2.2.11). = @?-'q+0 L q q n***L,.

8.2.

137

The Result of Changes in a Column of A on A -

$

0,

k'.

0, = I ,

(8.2.9)

e4 ' +

+

e,$q),

.. U ,

e,'U,+

& I = )

'

A("+')ej= U e1 .7 1' # q,

e , ' O 4 P + ' ) e j = (eq' + @)Uej

.. U,Uej

= e,'U,+ =

'

e4 ' eJ . = 0,

U,+ ',. . . , U , e j ) o , = ej), j # q,

n

0,

a;)

')

=

6;:.

0 , ~. . L,a,, ~ . ',

6,ACn+l)

$

2-1 =

A,

iqr,

dql

f+0 L q n***L1,

~(,*l?l)-l~10

q1 q

U(q)by

U(q*ql)

by ebl $%L,,. . . L 1 2 , L,A. e,, 1972).

by (8.2.5)

(8.2.3)

(8.2.10).

(8.2.11)

2.2).

A("+') 6;) =

eq'oq#''+

U

-q

I, by (8.2.9)

(8.2.10 )

on

j # q9

oql

$I

f+,

L, . . .

0,

o,$"+

4

cI

U(,.ql)

q

(8.2.11); 4'

')

')

138

8 Change of Basis and Miscellaneous Topics

8.3. Kron's Method of Tearing C

n x

x n,

A

=

A

+ KEC,

by

A-'

(8.3.2)

=

[ I , - A-'KE(I,

+ CA-'KE)-'C]A-'. k'

A-' :

1.

n x

Y

2.

+ CY)'Z' = Y',

3.

A-'

a,

A-'KE. I

x n

- ZC)A-'.

=

k', =

=

Z, C, - a,,

A-'

A-'

=

1

C = ei,

A-'.

r x r,

8.4.

139

Bifactorization

8.4. Bifactorization U

I , by

2.2

u8102...8, -1 = I ,

(8.4.1) = 1,.

. . ,n

- 1,

8,

~0~. . . 8k-

U D l ... O k - l ,

ek' by ek,

U U 1 ... 8 k - l

ii

8,

U.

8,= I , + eke(')

(8.4.2)

(8.4.3)

-k

by

=

@)

0, j

< k, (Ol . . . 0,- ,)U

(8.4.1), (2.2.6) (2.2.7),

= -ukj,

&)

=

I,,

by (2.2.2),(2.2.3),

A(k+') (2.2.7),

k (8.4.2)

(2.2.4),

8,

(8.4.3),

Lks

A(k+')

Ll no

9

0 1 9

L2

9

02 . . . L,- 1 0,- 1 L, 9

I

> k.

7

9

2.2.

U by (2.2.6)

0k.s

140

8 Change of Basis and MiscellaneousTopics

8.5. Bibliography and Comments 8.2

by

by et

by

;

8.2,

et by

8.2

7,

7,

Lr,

by PI-’.

by

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pp.

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Comm. ACM 9,802. A

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P.,

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SIAM Reo. 4,

145

References (1958).

:

C. (1971). IEEE Trans. Circuit Theory CT 18, 89-95. (1971). (1972).

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Compt. J . 9, 28 1-285. (1968). Internat. Comput. Math. 2, 1-21. (1971). In pp. 97-104. (1967). Proc. IEEE 55, 1997-2000. (1970). Struct. Diu. Proc. Amer. SOC.Civil Eng. 96, 4 9 4 4 . (1969).

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In pp. Struct. Div. ASCE 94 Quart. Appl. Math. 26,425432. G.

V. SIAM Rev.

V.

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On SIAM Rev.

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Nordisk. Tidskr. Informations

Behandling ( B I T ) Comput.

On

10,

Computing (Arch.

Elektron. Rechnen) Nordisk. Tidskr. Informations Behandling ( B I T ) 8,

R.

Comput.

Internat. J . Comput. Math.

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Ziewkiewin, 0. C. (1967). “The Finite Element Method in Structural and Continuum Mechanics.” McGraw-Hill, New York. Zollenkopf, K. (1971). Bi-Factorization : Basic computational algorithm and programming techniques. In “Large SparseSets Linear Equations” (J. K. Reid, ed.),pp. 75-96. Academic Press. New York.

Author Index on

A

121, 142 12, 142 31, 94, 98, 106, 133, 136, 140,142,145 142 140, 142 81,142,148 G., 12,41, 142

71,80,141 12, 30, 141 G. G., 12,80,141 141 12,94,141

B 46,141 133, 140,141 12, 81, 141 12, 106,141, 142 13,142 12, 106,142 R.,12,41,142 81, 142 12, 13,31, 142 31, 142 31,142

C

13,81, 142 31, 94, 142 12,31,142 12,94,143 W. 143 Y. 61,143 M. 12, 27, 30, 31, 143 20, 143 153

154

Author Index

46, 143 12,41, 142 13, 143 12, 69, 143 11, 96, 101, 106, 150

145 12, 144 12, 13, 31, 90, 94, 96, 98, 106, 133, 136, 140, 142, 145 12, 145 0.

H 11, 13, 31, 61, 81, 96, 105, 106, 133,140, 143 J., 107, 108, 121, 144 12, 13, 106, 140, 142 C., 11, 13, 101, 106, 144 A., 144 50, 79, 144 A. 12, 50, 54, 144-

E 12, 31,94, 144 A. 149 A. 94, 144 80, 144 12,144 147

F

J. J . 140, 144

96, 105, 144 96, 105, 144 12, 13, 31, 133, 136, 137,

12, 31, 144 31, 105, 123, 124, 132, 144 12, 13,144

G 105, 145 C. 12, 145 J. 12, 81, 145 S., 146 145 G. 12, 145 133, 140, 141

145 105, 145 148 12, 41, 45, 50, 55, 77, 80, 145, 148

11, 13,96, 106, 140, 143 145 27, 31, 149 105, 145 S., 114, 145 146 1 46, 146 146

J A,, 80, 144 A., 12, 13, 75. 80, 94, 146 12, 81, 146 A. 12, 146

K 12, 147 12, 50, 146 12, 145, 146 146 I., 146 12, 133, 140, 146 1 11,96, 101, 105, 146

155

H.

13, 146 146 146 12, 13, 30, 90, 94, 145, 146 12, 13, 146 A,, 12,41, 142 146

M 12,94,147 12, 69, 143 1 1 , 13,96, 106, 140, 143 147 12, 80, 147 11, 25, 31, 101, 103, 140,147 12, 80, 141 12, 147 76, 147 12, 50, 54, 144 12, 31, 144 147

N A,, 147 S., 147 12, 147

0 12, 30, 31, 76, 80, 147 11, 13, 35, 36, 96, 101, 104, 105, 106, 133, 143, 147,149

P 12, 147 G. A., 12, 81, 146 S., 80, 147 148 A. 148 147 143

R 61, 148 A., 105, 148 13, 31, 76, 143, 148 108, 121, 148 148 A., 2, 148 80, 81, 148 12,68,80, 148 1. 148 12, 140, 148 148 132, 148

5 12, 41, 142 12, 31, 148 140, 148 126, 132, 148 146 149 1 1 , 13, 96, 101, 105, 106, 140, 149 11, 13, 96, 106, 140, 143 141 27, 31, 149 G. 149 12,50, 56, 77. 149 12, 81, 141 141

T 11,20, 23, 31, 35, 36, 37, 45,73,75,79,80,87,96,101, 102, 104, 106, 107, 108, 115,123, 124, 132,143, 149,150 12, 27, 31, 76, 80, 90, 92, 94,147,148, 150 150 A., 12,13,31,133,136,137,140, 144, 150 A. 12,94, 146

156

Author Index

U Utku, S., 71, 80, 141

Wenke, V. K., 80, 144,150 Westlake, J. R., 12, 83, 90, 105, 150 Wilkinson, J. H., 12, 16, 19, 31, 83, 86, 87, 89,91,115,118,121,123,124,127,132,

V van der Sluis, A., 13, 150 Varga, R. A., 12, 13, 150

W Walker, J. W., 76, 80, 147 Warshall, S., 46, 150 Weaver, W., Jr., 150 Weil, R. L., Jr., 12, 50, 146,150

150 Willoughby, R. A., 12, 13, 30, 31, 90, 94, 96, 98, 106, 133, 136, 140, 142, 145, 146,148,150 Wolfe, P., 1 1 , 12, 13, 20, 81, 96, 101, 106, 143,144,150 Z

Ziewkiewicz, 0. C., I51 Zollenkopf, K., 138, 150

Subject Index

40

A

see

43

67- 74 see

see see

see

139

see

43 42

6&74

71 67

see

45-50 50-60 79-80

8.40, see of,126427

40

see

8-9 60-66

44, 45, 88

62

72 42, 44,46,48, 54, 88, 117, 129, 131

62

64 157

158

Subject Index 75-77 75-77

by by

79-80

21 21

30 see

C 91-93 92-93 92-93 43

404I , 45-80 22 87-90

42 23-27

84-87 89

109-1 13

8687 87 117-1 18

90

see see

G

43 42 42

1619 16, 18-19 19-20

55 43

20 57

27-28

see 77-78

1618 28-29

77-78

see

43

19 43

19

see

20

9G91

GE, see

75-78

95-97 105

75-78

96 101

E

101

41 see

97-101 2G22 134-138 by

(I

priori

123-125 126127 4144

34-39

43 4142

22-29, 33-39

107-109

159 R

H 129

108-109

131-132 129-132

114 14

see

123, 127-129 128-129

see

91-92 19-20

114-1 17 115-117

101 43 42

see

J 118-121 see

119 118-119, 124-125

see

1&11

K

row, 1 1 see

138

75

L

75-76

see

2 1-2

0

see

see

9-10 4

P

4-6, 10 2-10

see

7-10

43 43 77

44 8, 6 6 6 7 , 91-93, 123-

see

6U7.91 95-96 134-138

123-129

T

101-104 97-

101

124, see

77-79, 138 54

160

Subject Index

Triangular decomposition, definition, 83 desirable forms 93-94

V Vertex, 41 degree of, 44 ernmiter, 55 in-degree of, 44 isolated, 55 out-degree 44 attachment set. 76