Lectures on Numerical Analysis

# Lectures on Numerical Analysis - Lectures on Numerical...

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Lectures on Numerical Analysis Dennis Deturck and Herbert S. Wilf Department of Mathematics University of Pennsylvania Philadelphia, PA 19104-6395 Copyright 2002, Dennis Deturck and Herbert Wilf April 30, 2002

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Contents 1 Diﬀerential and Diﬀerence Equations 5 1 . 1 In t rodu c t i on. ................................... 5 1 . 2 L in e a requa t i on sw i thc s t an tc o eﬃ c i en t s................... 8 1 . 3 D iﬀ e r c eequa t i s ............................... 1 1 1 . 4 Compu t ingw i thd e r c t i s...................... 1 4 1.5 Stability theory . ................................. 1 6 1.6 Stability theory of diﬀerence equations . . . . ................. 1 9 2 The Numerical Solution of Diﬀerential Equations 23 2 . 1 Eu l e r sm e thod . 2 3 2 . 2 So f tw a r eno t e s .................................. 2 6 2 . 3 Sy s t em sandequa t i so fh i gh e ro rd e r ..................... 2 9 2 . 4 H owt odo cum tap g ram . ......................... 3 4 2 . 5 Th idpo tandt rap e z o ida lru l e s....................... 3 8 2 . 6 Compa r i s ono fth e thod s ........................... 4 3 2.7 Predictor-corrector methods . . . ........................ 4 8 2 . 8 T run c a t i one r rands t eps i z e.......................... 5 0 2.9 Controlling the step size . . . . . ........................ 5 4 2 . 1 0Ca s es tudy :R o c k e tt oth emoon . ....................... 6 0 2 . 1 1M ap l ep g ram sf o rth et e z o l e .................... 6 5 2 . 1 1 . 1 Ex amp l e :Compu t ingth ec o s efun c t i on . .............. 6 7 2 . 1 1 . 2 Ex l e :Th emoonro c k e tinon ed im s i ............ 6 8 2 . 1 2Th eb i gl e a gu e s.................................. 6 9 2 . 1 3L a g rang eandAdam o rmu l a s ......................... 7 4

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4 CONTENTS 3 Numerical linear algebra 81 3.1 Vector spaces and linear mappings . ...................... 8 1 3 . 2 L in e a rsy s t em s .................................. 8 6 3 . 3 Bu i ld ingb l o c k sf o rth el e a requa t i ons o lv e r ................. 9 2 3 . 4 H owb i gi sz e ro ? ................................. 9 7 3 . 5 Op e ra t i onc oun t ................................. 1 0 2 3 . 6 T s c ramb l eth ee g g s ............................. 1 0 5 3.7 Eigenvalues and eigenvectors of matrices . . . ................. 1 0 8 3.8 The orthogonal matrices of Jacobi . 1 1 2 3 . 9 Con v e rg en c eo fth eJa c ob im e thod . 1 1 5 3.10 Corbat´o’s idea and the implementation of the Jacobi algorithm . . . . . . . 118 3.11 Getting it together . . . . . . . . ........................ 1 2 2 3 . 1 2R a rk s...................................... 1 2 4
Chapter 1 Diﬀerential and Diﬀerence Equations 1.1 Introduction In this chapter we are going to study diﬀerential equations, with particular emphasis on how to solve them with computers. We assume that the reader has previously met diﬀerential equations, so we’re going to review the most basic facts about them rather quickly. A diﬀerential equation is an equation in an unknown function, say y ( x ), where the equation contains various derivatives of y and various known functions of x . The problem is to “ﬁnd” the unknown function. The order of a diﬀerential equation is the order of the highest derivative that appears in it. Here’s an easy equation of ﬁrst order: y 0 ( x )=0 . (1.1.1) The unknown function is y ( x ) = constant, so we have solved the given equation (1.1.1). The next one is a little harder: y 0 ( x )=2 y ( x ) . (1.1.2) A solution will, now doubt, arrive after a bit of thought, namely y ( x )= e 2 x .Bu t ,i f y ( x ) is a solution of (1.1.2), then so is 10 y ( x ), or 49 . 6 y ( x ), or in fact cy ( x ) for any constant c . Hence y = ce 2 x is a solution of (1.1.2). Are there any other solutions? No there aren’t, because if y is any function that satisﬁes (1.1.2) then ( ye - 2 x ) 0 = e - 2 x ( y 0 - 2 y , (1.1.3) and so - 2 x must be a constant, C .

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Lectures on Numerical Analysis - Lectures on Numerical...

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