Francis Scheid Schaums Outline of NuSvcmerical Analysis, Second Edition 1989

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Text of Francis Scheid Schaums Outline of NuSvcmerical Analysis, Second Edition 1989

  • SCHAUM'S ouTlines

    NUMERICAL ANALYSIS

    Second Edition

    Covers all course fundamentals

    Use this book with any text or as a self-study guide

    Teaches effective problem-solving

    846 fully worked problems

    The perfect aid for better grades!

    Use with these courses: 1.!6 Lmear Algebra~ Numerical AnalySIS 1.!1' Differential Calculus

  • SCHAUM'S OUTUNB OF

    THEORY AND PROBLEMS

    ..

    NUMERICAL ANALYSIS

    Second Edition

    ..

    FRANCIS SCHEID, Ph.D. Pro(~ EIMriW. of Mo.tltt.motia

    &.ton Ur~it.'f'.t'fity

    SCHAUM'S OUTU NESF.RIE.'i McGrtiWllUI

    New Yor" San FraMtSCO Wastunaton, O.C Auckland Bogotli Ca.rata!l Usboo l..ood01t MIM.Irid Muico City Milan

    Momreal New Delhi Sun Ju'n Sln111)01~ Sydney rokyo Ti'lfoniO

  • F R A N C I S S C H E I D i s E m e r i t u s P r o f e s s o r o f M a t h e m a t i c s a t B o s t o n

    U n i v e r s i t y w h e r e h e h a s b e e n a f a c u l t y m e m b e r s i n c e r e c e i v i n g h i s

    d o c t o r a t e f r o m M I T i n 1 9 4 8 , s e r v i n g a s d e p a r t m e n t c h a i r m a n f r o m 1 9 5 6

    t o 1 9 6 8 . I n 1 9 6 1 - 1 9 6 2 h e w a s F u l b r i g h t P r o f e s s o r a t R a n g o o n U n i v e r s i t y

    i n B u r m a . P r o f e s s o r S c h e i d h a s l e c t u r e d e x t e n s i v e l y o n e d u c a t i o n a l

    t e l e v i s i o n a n d t h e v i d e o t a p e s a r e u s e d b y t h e U . S . N a v y . H i s r e s e a r c h

    n o w c e n t e r s o n c o m p u t e r s i m u l a t i o n s i n g o l f . A m o n g h i s p u b l i c a t i o n s a r e

    t h e S c h a u m ' s O u t l i n e s o f N u m e r i c a l A n a l y s i s , C o m p u t e r S c i e n c e , a n d

    C o m p u t e r s a n d P r o g r a m m i n g .

    S c h a u m ' s O u t l i n e o f T h e o r y a n d P r o b l e m s o f

    N U M E R I C A L A N A L Y S I S

    C o p y r i g h t 1 9 8 8 , 1 9 6 8 b y T h e M c G r a w - H i l l C o m p a n i e s , I n c . A l l r i g h t s r e s e r v e d . P r i n t e d

    i n t h e U n i t e d S t a t e s o f A m e r i c a . E x c e p t a s p e r m i t t e d u n d e r t h e C o p y r i g h t A c t o f 1 9 7 6 , n o

    p a r t o f t h i s p u b l i c a t i o n m a y b e r e p r o d u c e d o r d i s t r i b u t e d i n a n y f o r m o r b y a n y m e a n s , o r

    s t o r e d i n a d a t a b a s e o r r e t r i e v a l s y s t e m , w i t h o u t t h e p r i o r w r i t t e n p e r m i s s i o n o f t h e p u b l i s h e r .

    1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 V F M V F M 9 8 7 6 5

    I S B N 0 - 0 7 - 0 5 5 2 2 1 - 5

    S p o n s o r i n g E d i t o r , J o h n A l i a n o

    P r o d u c t i o n S u p e r v i s o r , L e r o y Y o u n g

    E d i t i n g S u p e r v i s o r , M a r t h e G r i c e

    P r o j e c t S u p e r v i s i o n , T h e T o t a l B o o k

    L i b r a r y o f C o n g r e s s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a

    S c h e i d , F r a n c i s J .

    S c h a u m ' s o u t l i n e o f t h e o r y a n d p r o b l e m s o f n u m e r i c a l a n a l y s i s / b y

    F r a n c i s S c h e i d . - 2 n d e d .

    p . c m . - ( S c h a u m ' s o u t l i n e s e r i e s )

    I S B N 0 - 0 7 - 0 5 5 2 2 1 - 5

    1 . N u m e r i c a l a n a l y s i s - P r o b l e m s , e x e r c i s e s , e t c . I . T i t l e .

    I I . T i t l e : T h e o r y a n d p r o b l e m s o f n u m e r i c a l a n a l y s i s .

    Q A 2 9 7 . S 3 7 1 9 8 9

    5 1 9 . 4 - d c 1 9 8 7 - 3 5 3 4 8

    C I P

    M c G r a w - H i l l g z

    A D i v i s i o n o f T h e M c G r a w - H i U C o m p a n i e s

  • Preface

    The main goal of numerical analysis remains what it has always been, to find approximate solutions to complex problems using only the simplest operations of arithmetic. In short, it is a business of solving hard problems by doing lots of easy steps. Rather clearly, this means finding procedures by which computers can do the solving for us. The problems come from a variety of mathematical origins, particularly algebra and analysis, the boundaries being at times somewhat indistinct. Much background theory is borrowed from such fields by the numerical analyst, and some must be included in an introductory text for clarity. It is also true that our subject returns more than raw numbers across the boundaries. Numerical method has made important contributions to algebraic and analytic theory.

    Many new topics have been added for this second edition. Included are backward error analysis, splines, adaptive integration, fast Fourier transforms, finite elements, stiff differential equations, and the QR method. The chapter on linear systems has been completely rewritten. A number of older topics have been shortened or eliminated, but a representative portion of classical numerical analysis has been retained partly for historical reasons. Some of the cuts have brought a tear to the author's eye, especially that of the constructive proof for the existence of solutions to differential equations. On the whole the new edition is a bit more demanding, but the same can be said of the subject itself.

    The presentation and purposes remain the same. There is adequate material for a year course at beginning graduate level. With suitable omissions an introductory semester course can easily be arranged. The problem format allows convenient use as a supplement to other texts and facilitates independent study. Each chapter still begins with a summary of what is to come and should be taken as a table of contents for the chapter. It is not intended to be self-explanatory, and supporting detail is provided among the solved problems.

    To repeat the closing paragraph of my original preface, there is no doubt that, in spite of strenuous effort, errors do remain within the text. Nmrierical analysts are among the world's most error conscious people, probably because they make so many of them. I will be grateful to hear from readers who discover errors. (The response to this request in the first edition was humbling.) There is still no reward except the exhilaration that accompanies the search for the all-too-elusive "truth."

    FRANCIS SCHEID

  • Contents Chapter Page

    1 WHAT IS NUMERICAL ANALYSIS? . . . . . . . . . . . . . . . . . . . . . . 1 2 THE COLLOCATION POLYNOMIAL...................... 17 3 FINITE DIFFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4 FACTORIAL POLYNOMIALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5 SUMMATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6 THE NEWTON FORMULA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7 OPERATORS AND COLLOCATION POLYNOMIALS...... 48 8 UNEQUALLY SPACED ARGUMENTS..................... 62 9 SPLINES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

    10 OSCULATING POLYNOMIALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 11 THE TAYLOR POLYNOMIAL............................. 86 12 INTERPOLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 13 NUMERICAL DIFFERENTIATION. . . . . . . . . . . . . . . . . . . . . . . . . 108 14 NUMERICAL INTEGRATION ............................. 118 15 GAUSSIAN INTEGRATION ............................... 136 16 SINGULAR INTEGRALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 17 SUMS AND SERIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 18 DIFFERENCE EQUATIONS ............................... 184 19 DIFFERENTIAL EQUATIONS ............................. 197 20 DIFFERENTIAL PROBLEMS OF HIGHER ORDER. . . . . . . . 232 21 LEAST-SQUARES POLYNOMIAL APPROXIMATION...... 241 22 MIN-MAX POLYNOMIAL APPROXIMATION ............. 275 23 APPROXIMATION BY RATIONAL FUNCTIONS .......... 292 24 TRIGONOMETRIC APPROXIMATION. . . . . . . . . . . . . . . . . . . . . 305 25 NONLINEAR ALGEBRA .................................. 326 26 LINEAR SYSTEMS ........................................ 354 27 LINEAR PROGRAMMING ................................ 405 28 OVERDETERMINED SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 29 BOUNDARY VALUE PROBLEMS ......................... 427 30 MONTE CARLO METHODS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

    ANSWERS TO SUPPLEMENTARY PROBLEMS ............ 457 INDEX .................................................... 467

  • Chapter 1

    What Is Numerical Analysis? ALGORITHMS

    The objective of numerical analysis is to solve complex numerical problems using only the simple operations of arithmetic, to develop and evaluate methods for computing numerical results from given data. The methods of computation are called algorithms.

    Our efforts will be focused on the search for algorithms. For some problems no satisfactory algorithm has yet been found, while for others there are several and we must choose among them. There are various reasons for choosing one algorithm over another, two obvious criteria being speed and accuracy. Speed is clearly an advantage, though for problems of modest size this advantage is almost eliminated by the power of the computer. For larger scale problems speed is still a major factor, and a slow algorithm may have to be rejected as impractical. However, other things being equal, the faster method surely gets the nod.

    EXAMPLE