2 Conjugate Gradient M e t h o d s for Solving
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Algebraic Eigenproblems J. C. NASH Research Division, Economics Branch, Agriculture Canada, OttawaK1A0C5,Canada S. G. NASH Mathematics Department, University of Alberta, Edmonton, Alberta, Canada The problem which is addressed in this paper is that of find ing eigensolutions (e,x) which satisfy Ax=e Β
x
(1)
where A and Β are real symmetric matrices and Β is positive defin ite. The particular case where Β is the identity is also of inter est. This study will focus on methods using the conjugate gradient algorithm because this is very frugal of storage when used to solve linear equations [1] or function minimization problems [2]. The algorithm can be used in either fashion to solve the eigen problem (1): 1) By solution of the equations (A - kB) y = Β
x
i
(2)
i
with x = y / || y || (3) i+1
i
i
where the double vertical lines mean "the norm of", it is possible to find the eigenvector x (or y) having eigenvalue e closest to the shift k. This is the process of inverse iteration [3,4]· 2) By minimization of the Rayleigh quotient R
T
=
x
A
x
/
T
x
B
x
(4)
T
with respect to x, where the denotes transposition, the eigen vector x corresponding to the most negative eigenvalue e is found [5]· R takes on the value e at its minimum. Note that if Β is the identity, minimization of 1
R
=
T
x
(A
-
2
k1)
x
/
T
x
x
24 Lykos; Minicomputers and Large Scale Computations ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
(5)
2.
NASH
AND
25
Conjugate Gradient Methods
NASH
g i v e s the e i g e n v e c t o r c o r r e s p o n d i n g to the e i g e n v a l u e w h i c h i s c l o s e s t t o t h e s h i f t k. A l t e r n a t i v e l y , r o o t - s h i f t i n g o r o r t h o g o n a l i z a t i o n a s d i s c u s s e d by S h a v i t t et^ a]_. 16] may be u s e d t o f i n d some o f t h e h i g h e r e i g e n s o l u t i o n s .
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The a p p r o a c h e s a b o v e s u g g e s t ways o f f i n d i n g one e i g e n v a l u e and v e c t o r a t a t i m e . No d e f l a t i o n o r o r t h o g o n a l i z a t i o n t e c h n i q u e i s needed f o r p r o b l e m s w i t h d i s t i n c t e i g e n v a l u e s . O n l y a few v e c t o r s o f s t o r a g e a r e r e q u i r e d and t h e p r o g r a m s we have w r i t t e n run i n v e r y s m a l l memories and c o u l d be u s e d on v e r y s m a l l m a c h ines s u c h as the programmable desk top c a l c u l a t o r s . The a i m o f t h e work r e p o r t e d h e r e was t o f i n d a r e l i a b l e y e t s i m p l e method f o r s o l v i n g t h e s y m m e t r i c m a t r i x e i g e n p r o b l e m when t h e m a t r i c e s i n v o l v e d would n o t n e c e s s a r i l y f i t in main memory in t h e c o m p u t i n g d e v i c e . In t h i s r e g a r d , t h e methods o f N e s b e t [7] and S h a v i t t [6J had p r o v e d u n r e l i a b l e , w h i l e t h a t o f D a v i d s o n [8j was t o o compl i c a t e d f o r t h e t a r g e t m a c h i n e s . Conjugate
Gradients
B e a l e [9] has g i v e n a s h o r t but l u c i d d e r i v a t i o n o f t h e c o n j u g a t e g r a d i e n t s (eg) f a m i l y o f a l g o r i t h m s . The f u n d a m e n t a l idea i s to g e n e r a t e a set o f l i n e a r l y independent s e a r c h d i r e c t i o n s by means o f a r e c u r r e n c e r e l a t i o n s h i p . T h a t i s , g i v e n a f u n c t i o n S(x) and i t s g r a d i e n t £ ( x _ ) t o g e t h e r w i t h t h e j ' t h s e a r c h d i r e c t i o n t_., t h e n e x t s e a c h d i r e c t i o n i s d e f i n e d a s
ij+i
=
ij - £
α
6
where α i s some p a r a m e t e r , t h e f o r m u l a f o r w h i c h d e t e r m i n e s t h e m e t h o d . Once s e a c h d i r e c t i o n s e x i s t , p r o v i s i o n o f a mechanism f o r a c t u a l l y performing a l i n e seach completes the s p e c i f i c a t i o n of an a l g o r i t h m e x c e p t f o r s t a r t o r r e s t a r t c o n d i t i o n s . W i t h i n t h i s work we c h o s e a l w a y s t o s e t
ί ο - - a.
(7)
and i f t h e a l g o r i t h m had n o t c o n v e r g e d in η s t e p s , where η i s t h e o r d e r o f t h e p r o b l e m , t o r e s t a r t in t h e same way f r o m t h e c u r r e n t p o i n t o r v e c t o r .x. M i n i m i z a t i o n of the p a r t i c u l a r f u n c t i o n S(x)
= x
T
C x
+ 2 x
T
w +
(any
where C i s a g i v e n p o s i t i v e d e f i n i t e s y m m e t r i c a given v e c t o r s o l v e s the l i n e a r equations C χ = -
scalar) m a t r i x and w
is
(9)
w
M o r e o v e r , in t h i s c a s e i t can be shown t h a t c o n v e r g e in no more t h a n η s t e p s , w h i l e t h e
(8)
the a l g o r i t h m l i n e search
Lykos; Minicomputers and Large Scale Computations ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
should
26
MINICOMPUTERS
simplifies
to
the computation o f h. J
=
g
T
g /
A N DLARGE
the
step
t! "~J
C t. -J
SCALE
COMPUTATIONS
length
(10)
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S i m i l a r s i m p l i f i c a t i o n s e x i s t f o r t h e R a y l e i g h q u o t i e n t (k). T h e s e are d e a l t w i t h below. B o t h t h e s p e c i a l i z e d and g e n e r a 1 - p u r p o s e eg m i n i m i z a t i o n a l g o r i t h m s are n o t o r i o u s l y s e n s i t i v e to implementation d e t a i l s w h i c h b e d e v i l any c o m p a r i s o n s . T h e s e d e t a i l s a r e d i s c u s s e d a t g r e a t e r l e n g t h i n [10] where t h e a l g o r i t h m s a r e p r e s e n t e d i n s t e p - d e s c r i p t i o n f o r m . C h a p t e r 19 o f [10] i s p a r t l y f o u n d e d on the p r e s e n t s t u d y . Direct
Rayleigh
Quotient
Minimization
As a r e s u l t o f o t h e r r e s e a r c h , a c o n j u g a t e g r a d i e n t s program f o r g e n e r a l f u n c t i o n m i n i m i z a t i o n was a v a i l a b l e [11] and o u r i n i t i a l hope was t o a p p l y t h i s d i r e c t l y t o t h e m i n i m i z a t i o n o f (4) o r (5). The m a j o r d i f f i c u l t y t h i s r e v e a l s i s t h a t b o t h t h e s e f u n c t i o n s a r e homogeneous o f d e g r e e z e r o , i m p l y i n g t h a t t h e s c a l e o r n o r m a l i z a t i o n o f x_ i s a r b i t r a r y . T h i s c a u s e s t h e m a t r i x o f s e c o n d p a r t i a l d e r i v a t i v e s , o r H e s s i a n , o f t h e f u n c t i o n R(x) to be o n l y p o s i t i v e s e m i d e f ï n i t e , v i o l a t i n g some t o t h e a s s u m p t i o n s w h i c h u n d e r l y t h e eg a l g o r i t h m . B r a d b u r y and F l e t c h e r [5] a d o p t e d a s t r a t e g y w h i c h r e s t r i c t s x_ t o I i e on a c o n v e x s u r f a c e , t h u s r e m o v i n g t h e t r o u b l e s o m d e g r e e o f f r e e d o m a t t h e e x p e n s e o f some e x t r a work w i t h i n t h e p r o g r a m . One c o u l d a l s o f i x one o f t h e e l e m e n t s o f x_ and a l l o w a l l t h e o t h e r s t o v a r y , e x c e p t t h a t t h e c h o s e n component s h o u l d p e r h a p s be v e r y s m a l l compared t o t h e r e s t , r e s u l t i n g i n v e r y l a r g e a l t e r a t i o n s in t h e s e a t e a c h s t e p o f t h e a l g o r i t h m . R a t h e r t h a n employ s p e c i f i c c o n s t r a i n t s , we t r i e d v a r i o u s p e n a l t y f u n c t i o n s t o impose a n o r m a l i z a t i o n on x. T h e s e a r e summarized in T a b l e I. U n f o r t u n a t e l y we must r e p o r t t h a t none o f t h e s e t e c h n i q u e s c a n be c o n s i d e r e d g e n e r a l l y u s e f u l due t o t h e s l o w c o n v e r g e n c e t o t h e minimum o f t h e f u n c t i o n . Our t e s t s were p r i m a r i l y made u s i n g t h e b i h a r m o n i c m a t r i x d e f i n e d by Ruhe [12] w i t h Β t h e i d e n t i t y . P r i m a r i l y the general m i n i m i z a t i o n is i n e f f i c i e n t because t h e l i n e s e a r c h i s i n e x a c t . F o r t h e R a y l e i g h q u o t i e n t w i t h o u t any p e n a l t y f u n c t i o n t o m a i n t a i n n o r m a l i z a t i o n t h i s can be c o r r e c t e d s i n c e t h e g r a d i e n t o f (k) i s
£ = 2 (A χ -
R Β χ ) / x
T
Β χ
(11)
so t h a t t h e one d i m e n s i o n a l m i n i m i z a t i o n o f R(x_ + h t_) w i t h r e s p e c t t o t h e s t e p l e n g t h h i s a c c o m p l i s h e d by t h e s o l u t i o n o f a q u a d r a t i c e q u a t i o n . T h i s i s s t r a i g h t f o r w a r d , t h o u g h n o t t o be u n d e r e s t i m a t e d ( s e e A c t o n [13]). A p a r t from t h e s e a r c h , the R a y l e i g h q u o t i e n t m i n i m i z a t i o n can be o r g a n i z e d so t h e r e c u r r e n c e (6) g e n e r a t e s s e a r c h d i r e c t i o n s
Lykos; Minicomputers and Large Scale Computations ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
2.
NASH
AND
27
Conjugate Gradient Methods
NASH
which a r e conjugate w i t h r e s p e c t ot the l o c a l Hessian m a t r i x . T h i s a v o i d s s e a r c h d i r e c t i o n s which c a u s e l a r g e growth in the e l e m e n t s o f x. The a p p r o a c h i s due t o G e r a d i n [ 1 h]. H o w e v e r , he d o e s n o t s p e c i f y t h e d e t a i l s o f h i s i m p l e m e n t a t i o n . T h a t w h i c h we have a d o p t e d i s g i v e n in f u l l by Nash [ 1 0 ] .
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Inverse
Iteration
by C o n j u g a t e
Gradients
E x c e p t by t h e u s e o f o r t h o g o n a 1 i z a t i o n o r p r o j e c t i o n t e c h niques R a y l e i g h q u o t i e n t m i n i m i z a t i o n cannot f i n d o t h e r than the e x t r e m e e i g e n s o l ut i o n s o f ( 1 ) . H o w e v e r , i n v e r s e i t e r a t i o n can f i n d an e i g e n s o l u t i o n w i t h e i g e n v a l u e c l o s e s t t o any p r e s c r i b e d s h i f t k p r o v i d e d a s t a r t i n g v e c t o r n o t t o t a l l y d e f i c i e n t in t h e r e q u i r e d e i g e n v e c t o r i s g i v e n , ( i n some c a s e s t h i s i s a n o n t r i v i a l p r o v i s i o n . F u r t h e r m o r e , f o r m u l t i p l e e i g e n v a l u e s we c a n o n l y f i n d one e i g e n v e c t o r f r o m t h e r e l e v a n t s u b s p a c e u n l e s s some o r t h o g o n a l i z a t i o n scheme i s e m p l o y e d . ) C o m p a r i n g e q u a t i o n s ( 2 ) and ( 9 ) shows method we s h o u l d make t h e identification C =
(A
-
kB)
;
w = -
Β χ.
;
that
to
u s e t h e eg
χ = y.
(12)
However, C was s u p p o s e d t o be p o s i t i v e d e f i n i t e , w h i c h i s now i m p o s s i b l e i f k i s g r e a t e r t h a n t h e s m a l l e s t (most n e g a t i v e ) e i g e n v a l u e o f o u r p r o b l e m ( 1 ) . Then one may t r y t o s o l v e t h e l e a s t squares problem (A
-
kB) (A T
-
kB)
y.
=
(A
-
kB)
T
Β χ.
(13)
w h i c h d o e s have a p o s i t i v e s e m i d e f i n i t e c o e f f i c i e n t m a t r i x but w h i c h i n c r e a s e s t h e amount o f work and w o r s e n s t h e n u m e r i c a l c o n d i t i o n of the equation system. Following Ruhe and W i b e r g [k] we have o p t e d t o i g n o r e t h e r e q u i r e m e n t t h a t C be p o s i t i v e d e f i n i t e , t h e r e b y d i s c a r d i n g t h e c o n v e r g e n c e r e s u l t s w h i c h have been p r o v e n f o r t h e eg a l g o r i t h m a p p l i e d t o l i n e a r e q u a t i o n s . To c o m p e n s a t e f o r t h i s , we must c h e c k t h a t t h e s t e p l e n g t h computed in ( 1 0 ) i s n o t t o o l a r g e , s i n c e t h i s w o u l d i m p l y t h a t t h e s h i f t k i s t o o c l o s e t o an e i g e n v a l u e . T h i s p o s e s t h e d i l e m m a : 1 ) For r a p i d convergence o f i n v e r s e i t e r a t i o n i t i s d e s i r a b l e have k a s c l o s e t o an e i g e n v a l u e a s p o s s i b l e . 2 ) F o r t h e eg a l g o r i t h m t o work w i t h o u t o v e r f l o w , k s h o u l d n o t be t o o c l o s e t o an e i g e n v a l u e . to
Ruhe and W i b e r g [k] n e v e r t h e l e s s u s e d e g - i n v e r s e iteration to r e f i n e e i g e n v e c t o r s , using s h i f t s v e r y c l o s e to the e i g e n v a l u e , a f t e r w a r d s r e f i n i n g t h e e i g e n v a l u e i t s e l f by means o f t h e R a y l e i g h q u o t i e n t . We have a p p l i e d t h e i r method t o t h e more d i f f i c u l t case where t h e s t a r t i n g v e c t o r may c o n t a i n o n l y a s m a l l component o f t h e d e s i r e d e i g e n v e c t o r and o n l y a c r u d e e s t i m a t e o f t h e e i g e n v a l u e may be a v a i l a b l e . In t h e c a s e where t h e s t e p l e n g t h ( 1 0 ) e x c e e d s some t o l e r a n c e ( t h e r e c i p r o c a l o f t h e s q u a r e r o o t o f t h e
Lykos; Minicomputers and Large Scale Computations ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
28
MINICOMPUTERS
A N DLARGE
SCALE
COMPUTATIONS
m a c h i n e p r e c i s i o n ) o u r p r o g r a m h a l t s , s i n c e we have y e t t o be c o n v i n c e d t h a t a s u i t a b l e a u t o m a t i c s h i f t i n g p r o c e d u r e c a n be devi sed. As a c o n t r o l on t h e p r o g r e s s o f t h e i n v e r s e i t e r a t i o n we compute t h e r e s i d u a l s
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r_
=
(A -
sB) χ
(14)
where s i s t h e c u r r e n t e s t i m a t e o f t h e e i g e n v a l u e . The sum o f s q u a r e s rjr_ p r o v i d e s a c o n v e n i e n t c o n v e r g e n c e t e s t . Unfortunately d i f f e r e n t n o r m a l i z a t i o n s a r i s e in t h e i m p l e m e n t a t i o n s o f t h i s a l g o r i t h m and t h a t o f G e r a d i n , so t h a t t h e c o n v e r g e n c e c r i t e r i a a r e n o t d i r e c t l y c o m p a r a b l e . We have s o f a r c h o s e n n o t t o p e r f o r m t h e e x t r a c o m p u t a t i o n n e e d e d t o make t h e c o n v e r g e n c e c r i t e r i a ident i c a l . Hybrid
Algorithms
H a v i n g now s e v e r a l p r o g r a m s a t o u r d i s p o s a l , i t i s t e m p t i n g to c o n s i d e r , s a y , combining a f u n c t i o n m i n i m i z a t i o n w i t h i n v e r s e i t e r a t i o n to r e f i n e the v e c t o r . E a r l y though q u i t e e x t e n s i v e t r i a l s with the general minimization a l g o r i t h m using f u n c t i o n D o f T a b l e I f o l l o w e d by i n v e r s e i t e r a t i o n a s i n t h e p r e v i o u s s e c t i o n showed t h a t t h i s c o m b i n a t i o n c o u l d be more e f f e c t i v e t h a n f u n c t i o n m i n i m i z a t i o n a l o n e . However, i f o n l y t h e extreme e i g e n s o l u t i o n s a r e d e s i r e d , t h e G e r a d i n a l g o r i t h m can in o u r e x p e r i e n c e s u p p l y an a c c u r a t e s o l u t i o n more e f f i c i e n t l y than any c o m b i n a t i o n m e t h o d , i n c l u d i n g G e r a d i n w i t h i n v e r s e iteration. D i s c u s s i o n and Examples T a b l e l i l i s t s t h e programs e i t h e r d e v e l o p e d o r used f o r c o m p a r i s o n p u r p o s e s w i t h i n t h i s s t u d y . We t e n t a t i v e l y recommend t h e a l g o r i t h m s w h i c h u n d e r l y t h o s e c a l l e d GER and INVIT. T h e r e a d e r i s a d v i s e d t h a t t h i s c a n be no more t h a n a t e n t a t i v e r e c o m m e n d a t i o n s i n c e work i s s t i l l g o i n g on i n t h i s a r e a . F o r i n s t a n c e , the a u t h o r s understand that P r o f . G.W.Stewart o f the U n i v e r s i t y o f M a r y l a n d has been d e v e l o p i n g a l g o r i t h m s f o r t h e sparse eigenproblem. His bibliography [16] p r o v i d e s a w e a l t h o f p o s s i b l e d i r e c t i o n s f o r r e s e a r c h . A t t h e t i m e o f w r i t i n g , we a r e s t i l l i n v e s t i g a t i n g t h e eg methods o f F r i e d [YJ]· T h e r e r e m a i n t o be s e t t l e d t h e q u e s t i o n s o f (1) c o n v e r g e n c e c r i t e r i a and (2) s t a r t i n g v a l u e s f o r t h e s h i f t a n d t h e i n i t i a l v e c t o r . F o r t h e program GER c o n v e r g e n c e i s assumed when t h e g r a d i e n t norm s q u a r e d £ 0
C)
x (A-kl) x
D)
x7(A-kl) x/x x +
T
2
2
+
(x x-l) T
T
Least s q u a r e s . T h i s i s not a Rayleigh quotient. Difficulties in f i n d i n g c e r t a i n s o l u t i o n s unless starting vector "good".
2
(x x-l) T
Extremely
2
poorly E)
x x/(x (A-kl) x)
Table
T
T
II:
to s o l v e the
Comments
Funct ion A)
29
Conjugate Gradient Methods
NASH
2
+
(x x-l) T
Programs a p p e a r i n g
2
s l o w when
eigenvalues
separated.
An u n s u c c e s s f u l a t t e m p t t o separate close eigenvalues. Slow c o n v e r g e n c e .
in t h e
study.
GER
The G e r a d i n a l g o r i t h m
INVIT
Inverse i t e r a t i o n using conjugate g r a d i e n t s s o l u t i o n of the l i n e a r equations (2).
TQL
A BASIC v e r s i o n o f TQL1 i n [15] f o r t h e e i g e n v a l u e s o n l y o f a symmetric t r i d i a g o n a l m a t r i x .
SSE
Eigenvalues only using
a s implemented
o f a symmetric
Sturm s e q u e n c e s
tridiagonal
in [ 1 0 ] . for
matrix
[3].
NES
Nesbet's algorithm
MOR
Method o f o p t i m a l
CGT(B)
General conjugate g r a d i e n t s penalized Rayleigh quotient
[J]. relaxation
[6]. function minimizer using (B) o f T a b l e I w i t h ζ = 1.
Lykos; Minicomputers and Large Scale Computations ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
30
MINICOMPUTERS
(This
is
lacking
not
normalized.)
a component
in
However,
the
in
A N DL A R G E
SCALE
some c a s e s
direction
of
the
this
desired
e s p e c i a l l y in c a s e s where t h e r e a r e s y m m e t r i e s (other than t h a t about the p r i n c i p a l d i a g o n a l )
in or
COMPUTATIONS
vector
is
eigenvector,
the the
matrices matrix
elements are i n t e g e r s . In s u c h c a s e s we have u s e d a p s e u d o random number g e n e r a t o r t o p r o d u c e e l e m e n t s in t h e interval if
( - 0 . 5 , 0 . 5 )
the
vector
of
ones
seemed
inappropriate.
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The a l g o r i t h m s have been t e s t e d i n BASIC i n a 6 0 0 0 b y t e p a r t i t i o n o f a Data G e n e r a l NOVA o p e r a t i n g i n 2 3 b i t binary f l o a t i n g p o i n t a r i t h m e t i c . T h i s c o r r e s p o n d s q u i t e c l o s e l y t o some o f t h e s m a l l d e s k t o p c o m p u t e r s in s i z e o f memory. Some o f t h e programs IBM 3 7 0 / 1 6 8 digits). 150
GER, INVIT, and CGT( ) - were a l s o t e s t e d on an a n d / o r Amdahl V6 i n s i n g l e p r e c i s i o n ( s i x hexadecimal
GER and l i n e s of
INVIT a r e q u i t e s h o r t , BASIC d e p e n d i n g on t h e
being a p p r o x i m a t e l y 1 0 0 to code f o r initialization,
n o r m a l i z a t i o n and p r i n t i n g o f r e s u l t s . GER was i m p l e m e n t e d and t e s t e d o n c e in l e s s t h a n two h o u r s f r o m a s t e p - d e s c r i p t i o n r e c i p e . T h a t i s , l e s s t h a n two h o u r s e l a p s e d f r o m t h e moment t h e f i r s t l i n e o f BASIC was w r i t t e n u n t i l t h e c o m p u t e r p r i n t e d c o r r e c t r e s u l t s f o r a t e s t p r o b l e m . INVIT i s s l i g h t l y l o n g e r in l i n e s o f program, for vectors, took 4 9 6 2 programs
2022
but r e q u i r e s o n l y 1 9 1 8 b y t e s o f s t o r a g e compared t o GER. M o r e o v e r , i t n e e d s o n l y 6 i n s t e a d o f 7 w o r k i n g so t h a t the o r d e r 1 0 0 problem r e p o r t e d in T a b l e III b y t e s t o run w i t h GER but o n l y 4 4 2 6 w i t h INVIT. B o t h r e q u i r e s u b - p r o g r a m s t o compute t h e r e s u l t s o f t h e
mult i p i i c a t ions ν By way o f d e f i n e d by
=
A x ;
illustration,
Α..
=
£
=
Β x_
consider
- 2 + l/[(n+l)
2
+
the
i ] 2
(15)
tridiagonal
for
matrix
i=l,2,...,n. (16)
A
i,i
+
1
=
A
i
+
l,i
"
1
f
o
r
i - 1 . 2 , - . n - l .
w h i c h we c a l l t h e F r o b e r g m a t r i x . Β w i l l be s e t t o t h e i d e n t i t y . On t h e NOVA we computed t h e s m a l l e s t e i g e n s o l u t i o n o f t h i s m a t r i x f o r o r d e r s η = 4 , 1 0 , 5 0 , and 1 0 0 . The r e s u l t s , w i t h c o m p a r a b l e values f o r o t h e r programs, r e s u l t s were f o u n d on t h e
a r e g i v e n in T a b l e 1 1 1 . V e r y s i m i l a r IBM 3 7 0 . T h e y i l l u s t r a t e what we have
o b s e r v e d i n a l l c a s e s t o d a t e , t h a t i s , t h a t when o n l y t h e s m a l l e s t (or l a r g e s t ) e i g e n v a l u e i s d e s i r e d t h e p r o g r a m GER i s t h e most e f f i c i e n t y e t p r o d u c e s a c c u r a t e s o l u t i o n s . U s i n g t h i s program w i t h t h e m a t r i x (A to o b t a i n
a solution
-
k
I)
(17)
2
with eigenvalue
closest
to
k causes a
Lykos; Minicomputers and Large Scale Computations ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
very
2.
NASH
31
Conjugate Gradient Methods
A N D NASH
s u b s t a n t i a l slow-down in t h e r a t e o f c o n v e r g e n c e . T h i s i s h a r d l y s u r p r i s i n g s i n c e we have s q u a r e d t h e c o n d i t i o n number o f t h e m a t r i x a s w e l l a s d o u b l i n g t h e work i n e a c h m a t r i x - v e c t o r product s t e p o f t h e p r o g r a m . T h e g e n e r a l f u n c t i o n m i n i m i z e r CGT(B) p e r f o r m s r e a s o n a b l y w e l l on t h e s e m a t r i c e s but u s e s r o u g h l y f i v e t i m e s a s much e f f o r t . Inverse i t e r a t i o n (INVIT) i s a l m o s t a s t e d i o u s . However, t h e r a t e o f c o n v e r g e n c e h e r e i s g o v e r n e d by t h e rat io
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ρ
=
(e
}
-
k)/(e
2
-
(18)
k)
where e^ i s t h e e i g e n v a l u e c l o s e s t t o k a n d e « t h e n e x t c l o s e s t . INVIT i n i t s m a j o r i t e r a t i o n c o n v e r g e s a s f a s t a s powers o f ρ tend to z e r o . F o r t u n a t e l y , as t h e e i g e n v e c t o r begins to dominate
Table Program
III :
Minimal
Order
GER INVIT TQL SSE CGT(B) NES MOR
4 4 4 4 4 4 4
GER INVIT TQL SSE CGT(B) NES MOR
10 10 10 10 10 10 10
GER INVIT TQL SSE CGT(B)
50 50 50 50 50
GER INVIT TQL SSE
100 100 100 100
eigensolution
Eigenvalue
0.350142
of
Rayleigh
Froberg's matrix. quotient
0.350144 0.350144
Matrix
produc
5 27
(7)
0.350145 0.350144 0.350144 0.350144 0.350505
7.44379E-2 7.44434E-2 7.44405E-2
3.48618E-3 3.49247E-3 3.48753E-3
8.86582E-4 9.64072E-4 8.89313E-4
36 10 >1000
7.44406E-2 7.44407E-2
1 1 43 (6)
7.44407E-2 f a i 1 ed 7.48813E-2
57 >1000
3.48733E-3 #.4873E-3
26
3.48748E-3
123
8.89216E-4
51 242 (4)
8.89202E-4
147
(5)
The f i g u r e i n p a r e n t h e s e s a f t e r t h e number o f m a t r i x p r o d u c t s i s t h e number o f i n v e r s e i t e r a t i o n s . In a l l c a s e s t h e s h i f t k=0.
Lykos; Minicomputers and Large Scale Computations ACS Symposium Series; American Chemical Society: Washington, DC, 1977.
32
MINICOMPUTERS AND LARGE SCALE COMPUTATIONS
x^ , the eg linear equations solution is usually accomplished in many fewer than η matrix-vector products. This has also been observed by Ruhe and Wiberg [4]. The number of inverse iterations, starting form a shift k=0, has been included in parentheses behind the number of matrix-vector products in Table III. Acknowledgement s
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While this study used very minimal computing resources, we wish to acknowledge time provided on the Data General NOVA and IBM 370 (Datacrown Ltcl. ) at Agriculture Canada and the Amdahl V6 at the University of Alberta. Literature Cited [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Hestenes M.R. and Stiefel E . , J. Res. Nat. Bur. Standards Section Β (1952) 49, 409-436. Fletcher R. and Reeves C.M., Computer Journal (1964) 7, 149-154. Wilkinson J.H., "The algebraic eigenvalueproblem ,Clarendon Press, Oxford, 1965. Ruhe A. and Wiberg T., BIT (1972) 12, 543-554. Bradbury W.W. and Fletcher R., Numer. Math. (1966) 9.,259-267. Shavitt I., Bender C.F., Pipano Α., and Hosteny R.P., J. Computational Physics (1973) 11, 90-108. Nesbet R.K., J. Chem. Phys. (19657) 43, 311-312. Davidson E.R., J. Computational Physics (1975) 17, 87-94. Beale E.M.L., in Lootsma F.A., "Numerical methods for nonlinear optimization", 39-44, Academic Press, London, 1972. Nash J.C., "Compact numerical methods: linear algebra and function minimization", To be published, probably in 1978. Nash J.C., "Function minimization with small computers", submitted to ACM Trans. Math. Software, 1976. Ruhe Α., in Collatz L., "Eigenwerte Probleme", 97-115, Birkhäuser Verlag, Basel, 1974. Acton F.S., "Numerical methods that work", 58-59, Harper & Row, New York, 1970. Geradin M., J. Sound. Vib. (1971) 19, 319-331. Bowdler H., Martin R.S., Reinsch C., and Wilkinson J.H., Numer. Math. (1968) 11, 293-306. Stewart G.W., in Bunch J.R. and Rose D.J., "Sparse matrix computations", 113-130, Academic Press, New York, 1976. Fried I. J . Sound. Vib. (1972) 20, 333-342. 11
Lykos; Minicomputers and Large Scale Computations ACS Symposium Series; American Chemical Society: Washington, DC, 1977.