Principles for Testing Polynomial Zerofmding Programs

Principles for Testing Polynomial Zerofmding Programs M.A. JENKINS Queen's University and J.F. TRAUB Carnegie-Mellon University

T h e s t a t e of t h e art in p o l y n o m i a l zerofmding a l g o r i t h m s a n d programs is briefly s u m m a r i z e d , w i t h emphasis on t h e principles for testing s u c h programs. T h e authors v i e w t e s t i n g as requiring four s t a g e s : (1) t e s t i n g program robustness, (2) t e s t i n g for convergence difficulties, (3) t e s t i n g for specific w e a k n e s s of t h e algorithms, (4) a s s e s s m e n t of program performance b y s t a t i s t i c a l testing. I t is emphasized t h a t t h e statistical testing m u s t b e d o n e w i t h care. T h e r e are m a n y w a y s t o generate 'random" p o l y n o m i a l s , b u t t w o classes of r a n d o m p o l y n o m i a l s w h i c h h a v e b e e n w i d e l y used are of o n l y l i m i t e d usefulness in terms of e v a l u a t i n g reliability or performance b e c a u s e t h e y produce p o l y n o m i a l s w i t h v e r y similar characteristics. Classes of r a n d o m p o l y n o m i a l s w h i c h s h o u l d b e used are discussed. 1

K e y W o r d s a n d P h r a s e s : p o l y n o m i a l zeros, m a t h e m a t i c a l software, statistical testing, program testing C R C a t e g o r i e s : 5.15, 5 . 1 0

1.

INTRODUCTION

T h e n u m e r i c a l s o l u t i o n of p o l y n o m i a l e q u a t i o n s h a s b e e n t h e s u b j e c t of i n t e n s i v e r e s e a r c h i n r e c e n t y e a r s w i t h p r o p o s a l s of n e w a l g o r i t h m s a p p e a r i n g f r e q u e n t l y

in

t h e n u m e r i c a l a n a l y s i s l i t e r a t u r e . W h i l e m a n y of t h e a l g o r i t h m s w h i c h a r e p r o p o s e d a r e m a t h e m a t i c a l l y i n t e r e s t i n g , o n l y a f e w a r e s e r i o u s c o n t e n d e r s a s t h e b a s i s of f a s t , r e l i a b l e , m a t h e m a t i c a l s o f t w a r e for t h i s p r o b l e m . T h e p u r p o s e of t h i s p a p e r i s t o p r o p o s e m e a n s of t e s t i n g programs

for p o l y n o m i a l z e r o f m d i n g i n o r d e r t o s e p a -

C o p y r i g h t © 1975, Association for C o m p u t i n g M a c h i n e r y , I n c . General permission t o republish, b u t n o t for profit, all or p a r t of this material is g r a n t e d p r o v i d e d t h a t A C M ' s c o p y r i g h t n o t i c e is g i v e n a n d t h a t reference is m a d e t o t h e publication, t o i t s d a t e of issue, a n d t o t h e fact t h a t reprinting privileges were granted b y permission of t h e Association for C o m p u t i n g M a c h i n e r y . A version of this paper w a s presented a t M a t h e m a t i c a l Software I I , a conference held a t P u r d u e U n i v e r s i t y , W e s t L a f a y e t t e , Indiana, M a y 2 9 - 3 1 , 1974. T h e work reported in this paper w a s s u p p o r t e d in part ( M . A . Jenkins) b y t h e D e f e n s e R e s e a r c h B o a r d of C a n a d a under Grant 9540-32, a n d b y t h e N a t i o n a l Research Council of C a n a d a under G r a n t A 7 8 9 2 , a n d in part (J.F. T r a u b ) b y t h e N a t i o n a l Science F o u n d a t i o n under G r a n t G J 3 2 1 1 1 and t h e Office of N a v a l Research under C o n t r a c t N 0 0 1 4 - 6 7 - A - 0 3 1 4 - 0 0 1 0 , N R 0 4 4 - 4 2 2 . A u t h o r s ' addresses: M . A . Jenkins, D e p a r t m e n t of C o m p u t i n g a n d I n f o r m a t i o n Science, Queen's U n i v e r s i t y , K i n g s t o n , O n t . , C a n a d a ; J . F . T r a u b , D e p a r t m e n t of C o m p u t e r Science, CarnegieM e l l o n U n i v e r s i t y , P i t t s b u r g h , P A 15213. ACM Transactions on Mathematical Software, Vol. 1, No. 1, March 1975, Pages 26-34.

Principles for Testing Polynomial Zerofinding Programs

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r a t e t h o s e w h i c h s h o u l d b e r e c o m m e n d e d a s c o n t e n d e r s for i n c l u s i o n i n p r o g r a m libraries a n d those which should be considered only m a t h e m a t i c a l novelties. T h e e m p h a s i s h e r e is o n t h e t e s t i n g of a s i n g l e p r o g r a m w h i c h i m p l e m e n t s a p a r t i c u l a r a l g o r i t h m , b u t s i m i l a r t e c h n i q u e s m a y b e u s e d for c o m p a r i n g t h e r e l a t i v e m e r i t s of s e v e r a l a l g o r i t h m s . W e e m p h a s i z e t h a t s t a t i s t i c a l t e s t i n g m u s t b e d o n e w i t h c a r e if i t is t o b e m e a n i n g f u l . T h e p o l y n o m i a l z e r o f m d i n g p r o b l e m is a difficult c o m p u t a t i o n a l p r o b l e m . I t c a n b e v i e w e d a s a n o n l i n e a r p r o b l e m or, i n t e r m s of l i n e a r a l g e b r a , a s a s p e c i a l c a s e of a n o n - H e r m i t i a n e i g e n v a l u e p r o b l e m w i t h (in g e n e r a l ) n o n l i n e a r d i v i s o r s . M a n y a l g o r i t h m s w h i c h w o r k e a s i l y o n e x a m p l e s w i t h w e l l - s e p a r a t e d s i m p l e z e r o s fail o n m o r e difficult e x a m p l e s . A l g o r i t h m s c a n b e classified a s t h o s e w h i c h find a s i n g l e z e r o o r a p a i r of z e r o s a t a t i m e o r t h o s e w h i c h d e t e r m i n e all t h e z e r o s s i m u l t a n e o u s l y . T h e f o r m e r r e q u i r e t h e u s e of s o m e d e f l a t i o n t e c h n i q u e , e i t h e r e x p l i c i t o r i m p l i c i t [ 1 6 ] , t o r e m o v e t h e zeros already d e t e r m i n e d . W e assume here t h a t we are discussing a p r o g r a m which is i n t e n d e d t o find all t h e z e r o s of a n a r b i t r a r y p o l y n o m i a l w i t h r e a l o r c o m p l e x coefficients. T h e p r o c e e d i n g s of t h e R u s c h l i k o n s y m p o s i u m [ 6 ] p r o v i d e a n i l l u s t r a t i o n of the diverse approaches which have been applied to this fundamental problem. A d i s c u s s i o n of t h e b a s i c a p p r o a c h e s m a y b e f o u n d i n [ 1 1 ] . O u r e x p e r i e n c e i n t e s t i n g polynomial zerofinding p r o g r a m s has been gained while developing algorithms w h i c h u s e a c o m b i n a t i o n of a p p r o a c h e s ( s e p a r a t i o n of z e r o s b y p o w e r i n g f o l l o w e d b y f u n c t i o n a l i t e r a t i o n [ 9 , 1 0 ] t o find l i n e a r o r q u a d r a t i c f a c t o r s ) a n d w h i c h u s e e x p l i c i t d e f l a t i o n t o r e d u c e t o a s m a l l e r p r o b l e m . T o s o m e e x t e n t t h i s v i e w of t h e z e r o f i n d i n g p r o c e s s b i a s e s o u r p r e s e n t a t i o n of t h e p r i n c i p l e s for t e s t i n g ; h o w e v e r , w e a t t e m p t t o discuss t h e issues in general t e r m s . 2.

OBJECTIVES

W e a r e c o n c e r n e d w i t h a l m o s t foolproof a l g o r i t h m s a n d p r o g r a m s for t h e f o l l o w i n g m a t h e m a t i c a l p r o b l e m . G i v e n t h e d e g r e e n a n d t h e coefficients a , a . . . , a of t h e polynomial 0

P(z)

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= aoZ + aiz ~

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a

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2

= a (z

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0

a

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)

m

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0,

0

w e a r e t o find t h e n u m b e r of d i s t i n c t z e r o s , k, t h e z e r o s a c o r r e s p o n d i n g m u l t i p l i c i t i e s m m , . . . , mu s u c h t h a t P(z) tored form as

h

h

a , . . • , a*, a n d t h e is e x p r e s s e d i n a f a c 2

m

... ( z -

a) K k

I n c o m p u t a t i o n a l w o r k t h e s h a r p distinction b e t w e e n m u l t i p l e zeros a n d closely c l u s t e r e d d i s t i n c t z e r o s is b l u r r e d , a n d w e u s u a l l y s e t t l e for f i n d i n g n a p p r o x i m a t i o n s ai, a , . . . , a t o t h e z e r o s w h i c h s a t i s f y a t l e a s t o n e of t h e f o l l o w i n g t w o o b j e c t i v e s : (i) a s e t of a p p r o x i m a t i o n s