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An application of linear analysis to initial value problems Law, Alan Greenwell 1961

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AN APPLICATION OF LINEAR ANALYSIS TO INITIAL VALUE PROBLEMS by ALAN GREENWELL LAW B,A 8, U n i v e r s i t y of B r i t i s h Columbia, 1958 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of MATHEMATICS We accept t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d . THE UNIVERSITY OF BRITISH COLUMBIA September, 1961 c In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 3, Canada, Date i i -ABSTRACT C e r t a i n p r o p e r t i e s of an unknown element u i n a H i l b e r t space are i n v e s t i g a t e d . For u s a t i s f y i n g c e r t a i n l i n e a r c o n s t r a i n t s , i t i s shown that approximations to u and e r r o r bounds f o r the approximations may be obtained i n terms of f u n c t i o n a l r e p r e s e n t e r s . The ge n e r a l approximation method i s a p p l i e d to homogeneous systems of o r d i n a r y l i n e a r d i f f e r e n t i a l equations and v a r i o u s formulae are d e r i v e d . An Alwac I I I - E d i g i t a l com-puter was used to compute optimal approximations and e r r o r bounds wi t h the a i d of these formulae. Numerous a p p l i c a t i o n s to p a r t i c u l a r systems are mentioned. On the b a s i s of the numerical r e s u l t s , c e r t a i n remarks are given as a guide f o r the numerical a p p l i c a t i o n of the method, at l e a s t i n the framework of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s . From the cases s t u d i e d i t i s seen that t h i s can be a p r a c t i c a b l e method f o r the numerical s o l u t i o n of d i f f e r e n t i a l e q u a t i o n s . i i i -TABLE OF.'..'CONTENTS Page Number 1. INTRODUCTION 1 2. HILBERT SPACE AND BOUNDED LINEAR FUNCTIONALS 3 3 . OPTIMAL APPROXIMATION OF LINEAR FUNCTIONALS 5 4 a INITIAL VALUE PROBLEM FOR AN HOMOGENEOUS SYSTEM 9 OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS 5 . SINGLE EQUATION. COMPUTATIONAL METHODS 13 6 . ORTHOGONAL POLYNOMIALS 15 7 . EXPERIMENTAL RESULTS, SINGLE EQUATION 16 8 . EXPERIMENTAL RESULTS. TWO EQUATIONS 17 9 . SYSTEM OF EQUATIONS. EXPERIMENTAL RESULTS 18 10. REFERENCES 22 11. TABLES TABLE I To Follow Page 21 TABLE II To Follo w Page 21 TABLE I I I To Follo w Page 21 - i v -ACCTOWLEDGEMENTS I wish to acknowledge the i n v a l u a b l e guidance and a s s i s t a n c e extended to me by Dr. C. A. Swanson; without h i s counsel t h i s t h e s i s would never have m a t e r i a l i z e d . I should a l s o l i k e to thank Dr. T. E, H u l l f o r h i s h e l p f u l remarks and Dr. C« C l a r k f o r h i s a s s i s t a n c e i n p r e p a r i n g the f i n a l m anuscript. 1 1„ INTRODUCTION The h i g h l y d i v e r s e f i e l d of approximation theory has been c o n s i d e r e d by such authors as Courant [ 4 ] / Kantorovich [ 1 1 ] , Lanczos [13] and S t r u t t [l8]<> Much of the t h e o r e t i c a l approach depends on f u n c t i o n a l a n a l y s i s o In the l a s t few years, with the advent of high speed computing d e v i c e s , f u n c t i o n a l a n a l y s i s has become of i n c r e a s i n g p r a c t i c a l importance [ l , 3» 6, 7] and, i n f a c t , appears to be d e v e l o p i n g under the c o n s i d e r a -t i o n of the c a p a b i l i t i e s of such d e v i c e s . In our r e s e a r c h we intended to study methods f o r s o l v i n g p a r t i a l d i f f e r e n t i a l equations, with the hope that v a r i o u s c l a s s i c a l approaches [11 , 13], such as the R a y l e i g h -R i t z method [ 4 , 5, 11, 18], c o u l d be used to advantage with the a i d of computers. We a l s o c o n s i d e r e d more modern methods [ 2 , 5# 7, 10] and i t was d e c i d e d that a method based on that of Golomb and Weinberger [ 7 ] deserved f u r t h e r c o n s i d e r a t i o n . T h i s method holds much promise s i n c e i t i s p e r t i n e n t f o r a var-i e t y of approximation problems [ 7 , p» 117] and, a l s o , p r a c t i c a l e x pressions f o r the maximum e r r o r i n c u r r e d i n the approximation may be generated [ 7 , p. 134]= In s t u d y i n g a numerical method f o r s o l v i n g d i f f e r e n -t i a l e quations, i t i s o f t e n f r u i t f u l to c o n s i d e r the o r d i n a r y case before the p a r t i a l case. Thus, i n t h i s t h e s i s , we have r e s t r i c t e d our r e s e a r c h to a system of o r d i n a r y d i f f e r e n t i a l equations of the form (4»3)" The r e s e a r c h r e q u i r e d e x t e n s i v e numerical experimentations T h i s was accomplished with the a i d 2 of an Alwac III-E d i g i t a l computer at the University of B r i t i s h Columbia Computing Centre; without an electronic computer our numerical procedures could not be u t i l i z e d . In order to evaluate the method, we considered i t in the framework of simple models (see, e.g., ( 9 . 2 ) ) . On the basis of the results obtained, some points are c l a r i f i e d so that the numerical success of the process might be increased. Sections 2 and 3 a r e devoted to a general develop-ment of the theory. It i s shown that, for an unknown vector u, which i s subject to certain l i n e a r constraints, approximations to u and error bounds for the approximations may be obtained in terms of functional representers. Section 4 deals with an evaluation of the results derived in section 3 f°r the case of a l i n e a r system of d i f f e r e n -t i a l equations. The unknown vector u i s the solution of an i n i t i a l value problem, where the constraints imposed on u are the i n i t i a l values. E x p l i c i t expressions for the approximation F q ( U ) and the error estimate E are obtained in terms of i n i -t i a l values and functional representers. In sections 5 and 6 the pertinent formulae of section 4 are presented in a form which i s suitable as a guide for com-puter programming. Also, a c r i t e r i o n i s given for computing polynomials which are orthogonal with respect to the scalar product adopted in this t h e s i s . In sections 6 and 7 there i s some discussion of the numerical results obtained for a single equation and a system of two equ a t i o n s . These r e s u l t s are c o n s i d e r e d i n determining the d e t a i l s f o r a p p l y i n g the procedure to the system of s e c t i o n 8. 2. HILBERT SPACE AMD BOUNDED LINEAR FUNCTIONALS D e f i n i t i o n 2.1 a A ( r e a l ) H i l b e r t space 0 i s a set of a b s t r a c t elements u, v, w, c a l l e d v e c t o r s , which s a t i s f y the f o l l o w i n g c o n d i t i o n s : ( i ) vf? i§. i l 1 i n e a r v e c t o r space over the f i e l d of r e a l numbers: (a) Jp i s an A b e l i a n group (b) J 5 admits m u l t i p l i c a t i o n by r e a l numbers a, 0 , . so t h a t , i f u, v £ J ^ , then (1) au £ 5 (2) (a+|3)u = au + 0u (3) (a(3)u = a(3u) (4) a(u+v) = au + av ; ( i i ) i s ji normed space whose norm i s d e r i v e d from _a s c a l a r  p r o d u c t: (a) with every p a i r of elements u, v i n J5 there i s a s s o c i a t e d a r e a l number, c a l l e d the s c a l a r product and denoted by (u, v ) , i n such a way that the f o l l o w i n g r u l e s are s a t i s f i e d : (1) (au,v) = a(u,v) f o r every number a (2) (u+v,w) = (u,w) + (v,w) (3) (u,v) = (v #u) 4 (4) (u,u) > 0 and (u,u) =0 i f and only i f u = 0; (b) with every element u in Jp there i s associated a re a l number ||u||, the norm of u, which i s defined by = ( u , u ) 1 / 2 ; ( i i i ) £3 i s .a complete space: any Cauchy sequence of elements in converges in the norm to an element of S? * Thus, i f ( u n l * s a sequence of elements in , and I |u - u I | >0 as n, m —> CD , then there exists an 1 ' n m 1 1 element u i n ^ so that | |u - U r | j >0 as n >oon It follows from condition ( i i ) that the norm has the properties of a metric, i 0 e 0 , (a) j|u|j > 0, with equality i f and only i f u = 0 (positive d e f i n i t e property); (b) ||au|| = J a | | | u j | (homogeneous property); (c) j|u+vj| < |ju|| + ||v|| (Minkowski i n e q u a l i t y ) ; for any u, v 6 £i and any number a„ Properties (a) and (b) are immediate and (c) i s a consequence of the well-known [4, P» 49] Schwarz inequality | (u,v) | < | |u| | | | v| | , u,v £ . D e f i n i t i o n 2.2. A li n e a r functional F i s a mapping u—»F(u) from Jp into the rea l numbers such that the following conditions are s a t i s f i e d : (a) F(au+pv) = aF(u) + £F(v), for a l l u,v&5p and a l l numbers a and 3 (b) there exists a positive number M such that, for a l l u G jp , |F(u)| < M ||u||. De f i n i t i o n 2.3. The l i n e a r functionals F^, F^, F^, defined on S?, are said to be 1inearly independent i f : n T" a, F.(u) = 0 for every u £ £i .—; l l i = l = a 0 = « » » = a =0. implies  n Remark 2.4. If F i s a li n e a r functional on a Hilbert space ^  , then by the Riesz representation theorem [16, p. 6l] there exists an element z in such that F(u) = (u,z) for every u €: . Moreover, the representer z i s uniquely determined by the functional F. 3. OPTIMAL APPROXIMATION OF LINEAR FUNCTIONALS Suppose there are given, a p r i o r i , n+1 lin e a r functionals F Q, F^, F n which are defined on a known Hilbert space S? • We consider an unknown element u £ which s a t i s f i e s (3-D F.(u) = £., i = 1, 2, n for fix e d known numbers e^, s^, £^ and seek an approxima-tion to the value F Q ( u ) . It s h a l l always be assumed that the 6 l i n e a r f u n c t i o n a l s FQ, F^, F^ are l i n e a r l y independent. Under t h i s assumption, i t can be shown [ 2 0 , p. 151] that there e x i s t s an element v i n S? such that CF (v) = 1 ( 3 - 2 ) ) ° I F . ( v ) = 0 , i = 1, 2 , n. For any number a, w = u+av i s i n and F.(w) = F.(u) + <xF.(v) = e., i = 1, 2 , n. Thus, ^ Q ( u ) c a n assume a r b i t r a r y values f o r u s a t i s f y i n g (3«l)« We wish u to be r e s t r i c t e d to some subset ^ of on which the l i n e a r f u n c t i o n a l s F q , F^, F r are bounded. Suppose then, that u s a t i s f i e s (3«l) and ( 3 - 3 ) | |u| | < A. where A. i s some known number, and l e t ( 3 . 4 ) = | ||v|| <X ; F.(v) = e,, i = l , 2 , . . . , n ] . Then F , F.. , F are bounded f o r v £ br » Let the centre o 1 ' n u of the h y p e r c i r c l e be that element which s a t i s f i e s . ,. = i n f 11v] ( 3 . 5 ) F . (v) - e. F i ^ = £ i ' i = 1/ 2 , n. The f a c t t h a t there e x i s t s a unique vi s a t i s f y i n g c o n d i t i o n s (3«5) i s w e l l known [ 1 6 , p. 7 1 ] . Since F^(u) = e^, i = 1,2,...,n, I l ^ | I < I l u l I < *^ and hence 'u £ 1r , By t a k i n g the f i r s t v a r i a -t i o n of (v,v) we f i n d that ( 3 . 6 ) (u,v) = 0 7 for a l l v £ s) , where ( 3 - 7 ) D = [ Y G £ | F.(v) = 0 , i = 1, 2, n}, i . e . , "u i s in the orthogonal complement £ ) of in « The conditions ( 3 « 6 ) and the second of ( 3 « 5 ) determine u* uniquely. Let y be that element of unit norm in X) f ° r which F (v) attains i t s upper bound when v varies under the condi-tions | |v| | = 1, v€ *D . This element exists and i s unique [ 1 6 , p. 6 2 ] . Following Golomb and Weinberger [ 7 / p. 134] we may state Theorem 1. If u varies over the set &" defined in ( 3 « 4 ) , then the range of a l l possible values of F q ( U ) i s an i n t e r v a l of length 2F Q(y)[A. 2 - | | u | | 2 ] 1 / 2 centered about F (u) . Thus, the maximum error E incurred in approximating to F Q(u) with F q ( U ) i s given by (3.9) E = * F Q(y)[A. 2 - | | u | | 2 ] 1 / 2 . F q ( U ) i s the optimal approximation to the value F q ( U ) in the sense that no smaller^interval length can be found. By ( 2 . 4 ) , there exist unique elements Z q , z^, z^ of such that ( 3 . 1 0 ) F.(v) = (z.,v), i = 0 , 1 n) for every v6J? « Since F q , F^, F n are assumed l i n e a r l y independent, the elements Z q , z^, . 0 0, Z^ are l i n e a r l y indepen-8 dent i n the sense that no one of them i s a l i n e a r combination of the o t h e r s . The space _S) , d e f i n e d i n ( 3 « 7 ) , can now be charac-t e r i z e d as the subspace of a l l v 6 jp which are orthogonal to each of z^, Zg# « o « , z^, i . e . , (3-11) J ) = [ v £ S? | (v,z.) =0, i = 1, 2 n} . Sin c e u6 ] D , i t i s a l i n e a r combination of z,, zn, .... z . ^ 1' 2 n Thus, n z (3.12) 'u = a. j and, u s i n g (3 «5) , = l J J (3.13) e, = F,(u) = F, ( E. o , z J = H a . F j z J 1 j = l J J j = l j 1 j = H a (z . ,z ) . j = l J 3 T h i s i s a l i n e a r a l g e b r a i c system i n ex., , a 0 , .... a . Let i <L n B = ( P . - ) = ( ( z . , z , ) ) denote the c o e f f i c i e n t m a t r i x of t h i s 13 -L 3 system. B i s n o n - s i n g u l a r s i n c e z^, z^, Z r are l i n e a r l y -1 ^ independent [ 4 , P. 62]. If B ^ i j ^ denotes the i n v e r s e of B then, from (3.12), u = > 3 . .£ . z . i , j = l and hence n (3.14) F (u) = 21 £,p\,(z .z,) . i , j = l 1 1 3 ° ^ A l s o , ( 3 . 1 5 ) ( u , u ) = T P ± . e . e . i , j = l and i t can be shown [ 7 , P» 142] that ( 3 - 1 6 ) F o ( y ) 2 = (zQlz ) - Z_ (5 ( a o , z ) ( z f z ) . i , j = l J 4 . INITIAL VALUE PROBLEM FOR AN HOMOGENEOUS SYSTEM OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS In t h i s and a l l subsequent s e c t i o n s , the H i l b e r t space S? w i l l be s p e c i a l i z e d to the space'*" c o n s i s t i n g of N-vec t o r s of the form r 1 2 V = L V , V , a . . , V j , where each component v 1 = v 1 ( x ) i s an a b s o l u t e l y continuous, s i n g l e v alued f u n c t i o n on 0 < x < 1, having a Lebesgue square i n t e g r a b l e d e r i v a t i v e . The inne r product i n t h i s space i s de-f i n e d by ( 4 » 7 ) (below). We a l s o use N x N mat r i c e s A = (a.^) "*"To show the completeness of when N = 1: every Cauchy sequence (^n) * n & n a s "the p r o p e r t y that the sequence 2 of i t s d e r i v a t i v e s converges i n the L norm to a square summable f u n c t i o n F [ 1 6 , p. 5 9 ] . Since F i s a l s o summable, i t i s the d e r i v a t i v e almost everywhere of an a b s o l u t e l y continuous f u n c t i o n G [ 1 6 , p. 4S and p. 5 3 ] « By d e f i n i n g 6(0) = l i m F (0) , G be-comes u n i q u e l y d e f i n e d , G ^ S% and | | F R - G | | ^ 0 . The proof extends to a r b i t r a r y N. 10 whose elements a „ ( x ) are piece-wise continuous f u n c t i o n s . For any u, v and A we d e f i n e the f o l l o w i n g p r o d u c t s : N (a) u»v = 21 u 1 1 l v i = l N N N (b) Av = [ zL a ^ * 1 * 2T a 2 i v 1 ' "•" ^ a N i v 1 - ' i ^ l ; i = l i = l N N N (c) VA = [ E a i i y 1 ' E a i 2 v l » TL a i N v l 3 -i = l i = l i = l 1/2 In p a r t i c u l a r , (v«v) i s c a l l e d the norm of the N-vector v and i t s h a l l be denoted by <v>. D e f i n i t i o n 4.1. The d e r i v a t i v e of any v e c t o r v (or m a t r i x A) i s d e f i n e d to be a v e c t o r v* (or matrix A f) of the same form whose components (or elements) are the d e r i v a t i v e s of the corres-ponding components (or elements) of the v e c t o r v (or m a t r i x A), We s h a l l assume an operator norm <A> such that (4.2) <Av> < <A><v>, f o r every v £ 55 . The unknown N-vector u i s the s o l u t i o n [ 1 2 , p. 48] of the problem C u» + A(x)u = 0 , 0 < x < 1 (4-3) i C u(0) = u Q , where U q i s a given v e c t o r [(i',u. , ix l w] and the m a t r i x A = ( a i : j ) i s g i v e n . We seek to approximate u 1 ( l ) , the f i r s t component of the s o l u t i o n v e c t o r u ev a l u a t e d at x = 1; i . e . , we seek an approximation to F q ( u ) where the l i n e a r f u n c t i o n a l 11 F q i s d e f i n e d by ( 4 . 4 ) F (v) = v 1 ( l ) , . o F q i s e a s i l y seen to be a (bounded) l i n e a r f u n c t i o n a l ; t h i s a l s o f o l l o w s from ( 4 . 1 2 ) below. If there e x i s t s a p o s i t i v e i n t e g r a b l e f u n c t i o n b(x) on 0 < x < 1 such that ( 4 « 5 ) - [ A ( x ) v ] " v < b(x) <v> 2, 0 < x < 1 and a l l v & S , then [ 2 , p. °S and 7 , p. 1 6 5 ] ( 4 - 6 ) ^ <u« (x)> 2dx < <u Q> 2 ^ <A(x)> 2 exp {2^* b ( t ) d t } d x . o o o Thus, we i n t r o d u c e the s c a l a r product ( 4 . 7 ) (v,w) = ^ v'(x)»w»(x)dx + v ( 0 )-w ( 0 ) o and (4=6) y i e l d s an a p r i o r i bound f o r (see ( 3 . 3 ) ) . The s c a l a r product ( 4 » 7 ) i s chosen f o r two reasons: f i r s t , i t induces a p o s i t i v e d e f i n i t e norm j |v|| and, second, i t allows e x p l i c i t formulae f o r the r e p r e s e n t e r s to be developed (see ( 4 . 1 2 ) ) . 2 Choose n - N l i n e a r l y independent N-vectors " " ' ^ n * n ^ a n < * define "^ n e l i n e a r f u n c t i o n a l s F N + l ' F N + 2 ' 8 8 B ' F n b y 2 I f ^N+l'^N+2'"° 0'^n a r e a"*"^  J - i n e a r l y inde-pendent then the corres p o n d i n g r e p r e s e n t e r s are not a l l l i n e a r l y independent and hence the m a t r i x B i s s i n g u l a r ; i . e . , F o ' F l ' ° ° ° ' F n a r e l i n e a r l y dependent. 12 f l (4=8) F.(v) = ) f.(x)°{v»(x)+A(x)v(x)}dx, i = N+l,...,n. o For any n > N and for any choice of f ^£ S} , we have the data (4*9) F i ( u ) " °' 1 = N + 1 ' N + 2 ' *•" n ° In addition, employ the N li n e a r functionals F^, F^, » B», F^ determined by (4.10) F.(v) = v j ( 0 ) , j = 1, 2, .„„, N. From the i n i t i a l conditions of (4»3) we have (4.11) F j ( u ) = ^ ' J = 1 ' 2 ' ° 0 0 ' N" The following e x p l i c i t formulae for the functional representers zo' z l ' 0 0"' zn c a n ' r e a c * i - * - y obtained [7, P« 166] : z o Z l [1+x, 0, 0, „„„, 0, 0] [1, 0, 0, „„., 0, 0] (4-12) J z2 = [0, 1, 0, „„„, 0, 0] Z:N z . 1 [0, 0, 0, 0, 1] f {f ( t ) + ( l * t ) f , ( t ) A ( t ) } d t o +(l+x) f , ( t ) A ( t ) d t , x i = N+1, N+2, s o . , n„ (The equations F^(v) = (v,z^) are easily checked using (4»12); since representers are unique, th i s proves (4 = 12))«> 13 5 . SINGLE EQUATION. COMPUTATIONAL METHODS In t h i s section we develop e x p l i c i t expressions for the representers z N + I ' Z N + 2 ' • ° "' z n a n c* a j - s o f ° r t~he scalar product ( z ^ , Z j ) of any two representers in the form of power se r i e s . These formulae are derived for a single equation but, in view of remarks ( 5 « 8 ) , they are also applicable for a system of N equations. If we assume that the chosen vectors f„ ,,f„T ~ , . . . , f N+l N+2 have polynomial components and the c o e f f i c i e n t matrix A has polynomial elements, then, because of ( 4 « 1 2 ) , the corresponding representers z J J + T/ ZN+ 2 ' •**' z n w i l l have polynomial component in this case the series (5«l) below w i l l be f i n i t e and hence there i s no question of convergence. Under th i s assumption, formulae ( 5 « 5 ) and ( 5 . 7 ) are of f i n i t e e x p l i c i t format and thus are a suitable guide for computer evaluation of ( 3 . 1 4 ) - ( 3»l6). Let N = 1 in the previous section. Assume that oo ( 5 . 1 ) z = l_ Vi**' i = 2 , 3 , n, l — l r r=o Then ( 5 . 2 ) z [ m ) ( o ) = mla. m, m = 0 , 1, 2 , . . . From ( 4 . 1 2 ) z ( 5 . 3 ) »(x) = f.(x) + ^ f (t)A(t)dt x z»(x) = f!(x) - {f,(x)A(x)} l D i f f e r e n t i a t i n g the second member of ( 5 - 3 ) k times and using Leibniz* formula we f i n d that 14 CO (5.4) z< k* 2 )U) - f j t + 1 ) («) - E ( J ) f p U)A( f c-^ u ) i k = 0, 1, 2, • We deduce f r o m (4.12), (5.2), (5.3) and (5.4) t h a t ttio = S f i ( t ) A ( t ) d t , o a... = a. +f.(0), i l i o 1 v ' (5.5) ^ «i,k +2 -(^IJT - £ ( J ) * l J ) ( 0 ) A f e j ) J . i = 2, 3/ .«», ^# ^ = 0, 1, 2, • c • Thus, t h e c o e f f i c i e n t s a.^^ i n (5.1) can be computed u s i n g (5.5) Suppose t h a t any two r e p r e s e n t e r s z - j / z j J = 1, 2, . n) a r e known i n t h e f o r m (5.1); l e t CO oo (5.6) z. = ? a - x r , z. = Z_ a, x S , I 4 — l r j — i s ' r=o J s=o J where t h e numbers a. , a. a r e known, r , s = 0, 1, 2, ... . 1 T J S D i f f e r e n t i a t i n g (5.6) t e r m by te r m , s u b s t i t u t i n g i n (4.7)', t h e n i n t e g r a t i n g t e r m by term, we f i n d t h a t (4.7) can be e x p r e s s e d i n t h e f o r m co (5.7) ( z . , z ) = a a + r a i r s a i s , i , j = 1,2,...,n. J J r , s = l r+s-1 F o r m u l a (5.5) and (5.7) have been d e v e l o p e d f o r N = l . However, t h e y a r e a p p l i c a b l e t o t h e s y s t e m (4.3) i n v i e w o f t h e f o l l o w i n g o b v i o u s r e m a r k s : 15 ( 5 . 8 ) (a) ( z o , z . ) = z * ( l ) ; i = 0 , 1, 2 , n (b) ( z . , z . ) = z | ( 0 ) ; i = 0 , 1, n, ^ j = l , 2 , ».», N (c) ( z . , z . ) = Z__ ( z, , z . ) ; i , i J K=l 1 J i , j - 0 , 1, s o . , n 6 . ORTHOGONAL POLYNOMIALS A set of polynomials g ^ ( x ) , g^(x), ... which are m u t u a l l y orthogonal with respect to the s c a l a r product ( 4 . 7 ) w i l l be found u s e f u l i n ensuing s e c t i o n s . Let g^(x) 5 1 a n d suppose that i - 1 ( 6 . 1 ) g. (x) = Y H 1 " * ' 1 = 2 ' 3 , -j=o 1 J Impose the c o n d i t i o n s (g,,g v) = 0 ; k = 1, 2 , ( i - 1 ) ( 6 . 2 ) i  1 k (gi,g±) * o . When ( 6 . 1 ) i s s u b s t i t u t e d i n t o ( 6 . 2 ) and (5-7) i s used (or ( 4 . 7 ) ) , we o b t a i n a system of ( i - l ) l i n e a r homogeneous equations i n Y i o / Y n ' ••»/ Y^ F o r e a c n i = 2 , 3 , 4 , s o l u t i o n s i n the form of r e l a t i v e l y prime i n t e g e r s are e a s i l y found. A set of polynomials ( 6 . 1 ) which s a t i s f y ( 6 . 2 ) i s then ( 6 . 3 ) g l = = 1 g 2 = = X g 3 = 2 = X -X g 4 = 3 2 = 2 x p - 3 x +x g 5 = = 5 x f - 1 0 x 3 + 6 x 2 * 16 7 - EXPERIMENTAL RESULTS. SINGLE EQUATION Suppose the problem is u» + xu = 0 , 0 < x < 1 ( 7.D u( 0 ) = 1, and an approximation to u(l) i s sought. 1 2 The exact solution of (7«l) i s u(x) = exp^-gX ) and u(l) ^  0 . 6 0 6 5 3 1 . Numerous values for n and choices of functions *2' * 3 ' ° * 0'^n w e r e adopted and the resu l t i n g approximations F q ( U ) to u(l) were obtained. The following remarks are based on extensive numerical experimentation. Remark 1. For any chosen set of functions ij^, f^, f ^ ^ , . . . , le t F.o^(u) denote the approximation F^iu) obtained for the choice of the k functions f^, f^» f ^ + ^ . Then, as may be expected for a numerical procedure, the sequence {|F o^(u)-u(l)|}, k = 1, 2 , ... i s not necessarily monotonically convergent. Remark 2 . In some cases the matrix B (see ( 3 « 1 3 ) et seq.) appeared to be not well-conditioned [ 8 , p. 4 3 9 , 14, 1 5 , 1 9 ] . For example, for n = 7 and choosing f^ = x* 2 , i = 2 , 3 , . . . , 7 , there was numerical d i f f i c u l t y in inverting B. However, in those cases in which f 2 , f , j , a . . , f n were a l l chosen from the set ( 6 . 3 ) # the matrix B was well-conditioned. Remark 3. On the basis of the experimentation, i t appeared to 17 be important that the constant function belong to the set {f 2 # ' f ^ # • • • / f n} • 8. EXPERIMENTAL RESULTS. TWO EQUATIONS Suppose the problem is f u» + Au = 0, 0 < x < 1 ( 8- L } I u(0) . u Q 1 2 0 1 where u = [u (x),u (x)], U Q = [1,0] and A = [ Q], and an approximation to u"*"(l) is sought. The exact solution [7, p. 166] involves a Bessel function of order 1/3; the value [9] of u X ( l ) i s O.8388I. There were numerous experiments with the method for system (8.1). It was found that for the choice (8.2) f 3 = [ g ^ O ] , ^ = [ 0 , g i ] , f 5 = [ g 2 , 0 ] , f f i = [ 0 , g 2 ] , , the approximations F q ( U ) to u^(l) were as follows: n F Q(u) 4 .794006 5f .781422 6 .839162 7 .839143 8 .838793 For fixed n, the approximation obtained for the choice (8.2) was numerically closer to u^(l) than the one obtained for any other choice for f , f , f . 3 4 n 18 9. SYSTEM OF EQUATIONS. EXPERIMENTAL RESULTS From s e c t i o n s 7 and 8 i t i s c l e a r t h a t , f o r s a t i s -f a c t o r y approximating, the choice of vec t o r s "^ N+2' ^ N i s , by no means, an a r b i t r a r y one. Since g^(x) i s a polynomial of degree ( i - l ) , i t can e a s i l y be seen that any polynomial has a unique r e p r e s e n t a -t i o n as a l i n e a r combination of the g , T s . I t then f o l l o w s from l the W e i e r s t r a s s Approximation Theorem [4, p. 65] that the system (g^(x)} of orth o g o n a l polynomials i s complete. An immediate g e n e r a l i z a t i o n i s that the system of N-vectors [g-^0, . . . , 0] , [0, g 1 # . . . ,0] , . . . ,[0,0, . . . ,g 1] , (9.1) i [ g 2 , 0 , . . . , 0 ] , [ 0 , g 2 , . . . , 0 ] , . . . , [ 0 , 0 , . . . , g 2 ] , [ g ^ , 0 , . . . , 0 ] , . . . i s complete. The members of t h i s system are mutually orthogonal with r e s p e c t to the s c a l a r product (4«7)« On the b a s i s of the experimental evidence i n s e c t i o n s 7 and 8, i t c o u l d be expected t h a t an a p p r o p r i a t e choice of fN+l' fN+2' "•*' ^ n F R O M T I L E S E T (9.1) co u l d r e s u l t i n (a) a w e l l - c o n d i t i o n e d m a t r i x B and (b) a n u m e r i c a l l y c l o s e a p p r o x i -mation F q ( U ) to F q ( U ) , In view of t h i s , suppose that the 1 2 3 where u = [u ,u , u ^ ] , U Q = [1,0,0] and A = and an approximation to u ^ ( l ) i s sought. 0 - 1 0 0* 0-1 x 0 0 19 F i r s t we c a l c u l a t e a bound f o r ||u|| i n order that the e r r o r estimate i n the approximation may be found. 1 2 3 d Let v = [ v (x) , v ( x ) , v J ( x ) ] be any v e c t o r i n Jp Then, f o r 0 < x < 1 , ( 9 - 3 ) <A(x)v> 2 < <v> 2, f o r a l l v € $ . By the Schwarz i n e q u a l i t y , |Avv| < <AvXv> and, u s i n g ( 9 * 3 ) , we conclude that ( 9 . 4 ) - A v v < <v> 2 . Thus, we can choose b(x) E l i n ( 4 . 5 ) and, si n c e <^uc/> ~ 1 and <A(x)> < 1 f o r 0 < x < 1 , we have from ( 4 . 6 ) ^ <u»(x)> 2dx < ^ <A(x)> 2 e 2 x d x < -| e 2 . o o F i n a l l y , u s i n g ( 4 . 7 ) , (u,u) = ||u|| 2 < ± e 2 + 1 = 4 - 6 9 4 5 2 9 . Hence we may take ( 9 . 5 ) A,2 = 4 . 6 9 4 5 2 9 i n ( 3 . 9 ) . Choose f ^ , f<-, f ^ , ... ac c o r d i n g to f 4 = [ 1 , 0 , 0 ] , f 5 = [ 0 , 1 , 0 ] , f 6 = [ 0 , 0 , 1 ] , ( 9 . 6 ) <J f ? = [ x , 0 , 0 ] , f 8 = [ 0 , x , 0 ] , f 9 = [ 0 , 0 , x ] , f l Q = [x - x , 0 , 0 ] , . . . where the non-zero components are chosen from the set ( 6 . 3 ) ; thus c o n s t r u c t the l i n e a r f u n c t i o n a l s ( 4 . 8 ) . Using ( 5 « 5 ) (or ( 4 « 1 2 ) ) the r e p r e s e n t e r s z,, z c , determined by the l i n e a r 4 5 f u n c t i o n a l s ( 4 . 8 ) , may then be found. Thus, i n view of ( 4 . 1 2 ) , 2 0 we have Z q, z^, . .., as given i n Table 1 . The symmetric m a t r i x B = (0 ) = ( ( z . , z . ) ) can be c o n s t r u c t e d with the a i d 1 j X J of ( 5 * 7 ) and ( 5 . 8 ) (or ( 4 - 7 ) ) and i s given i n Table I I . A l s o , from ( 5 . 8 ) ( a ) , ( z o ' z o } = 2 ' ( z o ' z l } = X< ( z o ' z 2 } = ° ' ( z Q , z 3 ) = 0 , ( z Q , z ^ ) = 1 , ( Z Q , Z 5 ) = 0 , (9.7) <( ( a 0 , z 6 ) = - 8 3 3 3 3 3 , ( v z 7 } = . 5 , ( z o , z 8 } = ° ' ( z Q , z 9 ) = . 5 8 3 3 3 3 , ( z o / 2 1 0 ) = - . 1 6 6 6 6 7 , ( z 0 » z u ) = 0 , ( z o , z 1 2 ) = - . 1 3 3 3 3 3 , . . . From ( 9 . 2 ) : CF,(u) = 1 ( 9 . 8 ) < 1 (u) = 0 , j = 2 , 3 , 4 , n. For each n = 4 , 5 / 6 , ... and the choice ( 9 * 6 ) , we can c a l c u -l a t e ( 3 . 1 4 ) from ( 9 . 7 ) , ( 9 . 8 ) and Table I I ; a l s o ( 3 . 1 5 ) and ( 3 . 1 6 ) may be e v a l u a t e d and, f o r the bound ( 9 . 5 ) , E can be computed a c c o r d i n g to ( 3 e 9 ) « The numerical r e s u l t s obtained are condensed i n t o Table I I I . Remark 1 . In a l l the cases considered, the m a t r i x B was w e l l - c o n d i t i o n e d . Remark 2 . Because of the accuracy of the c a l c u l a t i o n s , the numbers E of Table I I I are expected to be gross e s t i m a t e s . For example, f o r n = 1 2 , F q ( y ) A i n ( 3 . 1 6 ) i s . 0 0 0 0 0 & , where cT appears to be indeterminate because of the number of s i g n i -f i c a n t d i g i t s i n the computation. 2 1 Remark 3. A f t e r the r e s u l t s i n Table I I I had been o b t a i n e d . Dr. Z. A. Melzak p o i n t e d out to the author that u ^ ( l ) ~ .958457 may be c a l c u l a t e d from power s e r i e s . To Fol l o w Page 21 TABLE I Representers Corresponding to the Choice ( 9 « 6 ) f o r the Problem ( 9 . 2 ) z Q = [ l + x, 0, 0] z 2 = [1, 0, 0] z 2 = [0, 1, 0] z 3 = [0, 0, 1] z^ = [x, 0 5 x 2 - x - l , 0] z^ = [0, x, «5x -x-1] z 6 = [-.l66667x3+.5x+.5, 0, x] z ? - [.5x2, .166667X 3 -O5X'-.5, 0] z 8 = [0, -5x 2, 8l66667x 3-.5x -.5] z q = [- B083333x 4+.333333x+„333333, °* «5x 2] z10 = £"333333x 3-.5x 2, .083333x 4-.l66667x 3+.l66667x+»l66667, 0] z l l = °333333x 3-.5x 2, O083333x4-.l66667x3+.l66667x+.166667] z 9 = [-.05x5+.083333x4-o083333x-.O83333, 0, -333333x 3-.5x 2] TABLE I I M a t r i x B = (z.., z ..), i , j = 1, 2, ..», f o r t h e R e p r e s e n t e r s o f T a b l e I 1 0 1 0 0 1 0 -1 0 2.333333 0 0 -1 - . 5 .5 0 0 .333333 0 - . 5 0 1.208333 0 0 - . 5 -.166667 .333333 0 0 .25 0 .16666? 0 -.391669 0 0 .166667 .083333 -.O83333 0 0 - . 0 5 2.333333 - . 5 1.383333 -.333333 .125 . 7 1 6 6 6 7 I.2O8333 -.333333 - . 1 2 5 . 7 1 6 6 6 7 - . 1 6 6 6 6 7 =763888 .1 - . 1 2 5 .O83333 -.058333 -.202779 .024998 -.391669 .O83333 .058333 -.202779 .O83333 -.229364 -.016667 .058333 .515875 -.044447 .071429 .024998 -.013.886 .071429 -.126288 .008733 -.013886 .043647 Pi O <Q <D O To F o l l o w Page 21 TABLE I I I Approximation F Cu) and E r r o r E f o r (9»2) With the Choice (9.6) n F (u) E o v 6 .888889 - .528271 7 .885198 - .524532 8 .926869 - .251036 9 .957037 - .067486 10 .957186 - .067430 11 .957224 - .067353 12 .958447 - .002577 22 10. REFERENCES [ l ] Buck, Ro C. " L i n e a r spaces and approximation theory", _in R. E. Langer, ed., On numerical approximation; proceedings  of _a symposium conducted by the Mathematics Research Center, U n i t e d S t a t e s Army. at the U n i v e r s i t y of Wisconsin. Madison. A p r i l 2 1-23. 1958. Madison: U n i v e r s i t y of Wisconsin Press, 1 9 5 9 , 1 1 - 2 3 . [ 2 ] Coddington, E. A. and N. Levinson. Theory of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s . New York: McGraw-Hill Book Co., 1955. [ 3 ] Courant, R. "Remarks about the R a y l e i g h - R i t z method", i n R. E. Langer, ed.. Boundary problems in d i f f e r e n t i a l equations: proceedings of _a symposium conducted by the Mathematics  Research Center. U n i t e d S t a t e s Army. at the U n i v e r s i t y of  Wisconsin. Madison. A p r i l 20-22. 195?. Madison: U n i v e r s i t y of Wisconsin Press, I960, 273-277. [ 4 ] Courant, R. and D. H i l b e r t . Methods of mathematical p h y s i c s . v o l . 1. New York: I n t e r s c i e n c e P u b l i s h e r s , 1953-[ 5 ] Forsythe, G« E. and P. C. Rosenbloom. Numerical a n a l y s i s  and p a r t i a l d i f f e r e n t i a l e q u a t i o n s . New York: John Wiley and Sons, 1958. [ 6 ] F r i e d r i c h s , K. 0. F u n c t i o n a l a n a l y s i s and a p p l i c a t i o n s . [New Y o r k ] : New York U n i v e r s i t y , I n s t i t u t e of Mathematical S c i e n c e s , 1953« [ 7 ] Golomb, M. and H. F. Weinberger. "Optimal approximation and e r r o r bounds", i n R. E. Langer, ed., On numerical appro x i-mation; proceedings of _a symposium conducted by the Mathematics  Research Center. U n i t e d S t a t e s Army. at the U n i v e r s i t y of  Wisconsin. Madison. A p r i l 2 1 - 2 3 . 1958. Madison: U n i v e r s i t y of Wisconsin Press, 1959, 1 1 7 - 9 0 . -[ 8 ] H i l d e b r a n d , F. B. I n t r o d u c t i o n to numerical a n a l y s i s . New York: McGraw-Hill, Bao'E'Co ., 1956. [ 9 ] Jahnke, E. and F.-Emde. Tables of f u n c t i o n s with formulae  and c u r v e s . New York: Dover P u b l i c a t i o n s , S 1 3 3 « [ 1 0 ] John, F. Advanced numerical a n a l y s i s . [New Y o r k ] : New York U n i v e r s i t y , I n s t i t u t e of Mathematical S c i e n c e s , 1956. [1 1 ] Kan±o<rovich, L. V. and V. I. K r y l o v . Approximation methods of h i g h e r a n a l y s i s . T r a n s l a t e d by C. D. Benster. Groningen, The Netherlands: P. Nffordhoff, 1958. 23 [ 1 2 ] Kolmogorov, A. N. and S. V. Fomin. Elements of the  theory of f u n c t i o n s and f u n c t i o n a l a n a l y s i s , v o l . 1. T r a n s l a t e d by L. F. Boron. Rochester, New York: Graylock Press, 1957. [ 1 3 ] Lanczos, C. A p p l i e d a n a l y s i s . Englewood C l i f f s , New Je r s e y : P r e n t i c e H a l l , 1956. [ 1 4 ] von Neumann, J . and H. H. G o l d s t i n e . "Numerical i n v e r t i n g of m a t r i c e s of higher o r d e r " . B u l l e t i n of the American  Mathematical S o c i e t y . 53 (November 1947), 1021-99. [ 1 5 ] von Neumann, J . and H. H. G o l d s t i n e . "Numerical i n v e r t i n g of m a t r i c e s of higher o r d e r . I I " , Proceedings of the Ameri-can Mathematical S o c i e t y . 2 ( 1 9 5 1 ) , 188-202. [ 1 6 ] R i e s z , F. and B. Sz.-Nagy. F u n c t i o n a l a n a l y s i s . T r a n s l a t e d by L. F. Boron. New York: F r e d e r i c k Ungar, 1955. [ 1 7 ] Sneddon, I. N. S p e c i a l f u n c t ions of mathematical p h y s i cs  and c h e m i s t r y . New York: I n t e r s c i e n c e P u b l i s h e r s , 1 9 6 l . [ 1 8 ] S t r u t t , J . W. (Baron R a y l e i g h ) . The theory of sound, v o l . 1. London: Macmillan, 1894. [ 1 9 ] Todd, J . "The c o n d i t i o n of c e r t a i n m a t r i c e s . I", Q u a r t e r l y  J o u r n a l of Mechanics and A p p l i e d Mathematics. 2 ( 1 9 4 9 ) , 469-72. 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