Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi-110001 and Printed by Meenakshi Art Printers, Delhi-110006.
To my wife Leela
Chap Ch apter ter 11 Contents
Preface
xi
Preface to the Second Edition
1.
Basics in Computing 1.1 1.2 1.3
1.4
2.
xiii
1–7
Introduction 1 Repre Re prese sent ntat atio ion n of Nu Numbe mbers rs 1 1.2.1 1.2 .1 Flo Floati ating-p ng-poin ointt Rep Repres resent entati ation on 4 Erro Er rors rs in Co Comp mput utat atio ions ns 4 1.3.1 1.3 .1 Inh Inher erent ent Err Errors ors 4 1.3.2 1. 3.2 Loc Local al Round Round-of -offf Errors Errors 5 1.3.3 1.3 .3 Loc Local al Trunca Truncati tion on Error Error 6 Prob Pr oble lemm-so solvi lving ng usin using g Compu Compute ters rs 6
Solu So luti tion on of Al Alge gebr brai aicc an and d Tran ansc scen ende den nta tall Equ quat atiion onss 2.1 Introduction 8 2.2 Bi Bise seccti tion on Me Meth thod od 10 2.3 Re Regu gula la–F –Fal alsi si Me Meth thod od 12 2.4 Me Meth thod od of of Iter Iterat atio ion n 15 2.5 Ne Newt wton on–R –Rap aphs hson on Me Meth thod od 17 2.6 Mu Mull ller er’s ’s Me Meth thod od 22 2.7 Gr Grae aeffe ffe’s ’s Root Root Sq Squa uari ring ng Met Method hod 2.8 Ba Bair irst stow ow Me Meth thod od 27 2.9 Sy Syst stem em of of Non-L Non-Lin inea earr Equa Equati tion onss Exer Ex erci cise sess 34
3.
26 31
Soluti Solu tion on of Li Line near ar Sy Syst stem em of Eq Equa uati tion onss and and Matrix Inversion 3.1 3.2 3.3 3.4 3.5 3.6 3.7
Introduction 37 Gaus Ga ussi sian an El Elim imin inat atio ion n Met Metho hod d 38 Gaus Ga uss– s–Jor Jorda dan n Elimi Elimina nati tion on Meth Method od 43 Crou Cr out’ t’ss Redu Reduct ctio ion n Meth Method od 44 Jaco Ja cobi bi’s ’s Me Meth thod od 48 Gaus Ga uss– s–Se Seid idel el Ite Itera rati tion on Meth Method od 50 Thee Rel Th Relax axat atio ion n Met Metho hod d 52 vii
8–36 8– 36
37– 61
viii
Contents
3.8
Matrix Matr ix In Inve vers rsio ion n 54 3.8.1 Gaus Gaussian sian Elimi Eliminati nation on Metho Method d 3.8.2 3.8 .2 Gau Gauss– ss–Jord Jordan an Met Method hod 57 Exer Ex erci cise sess 59
4.
Eigenvalue Problems 4.1 Introduction 62 4.2 Power Method 63 4.3 Ja Jaco cobi bi’s ’s Me Meth thod od 66 4.4 Ge Gers rsch chgo gori rin’ n’ss Theor Theorem em Exer Ex erci cise sess 73
5.
55
62– 74
72
Curve Fitting Cu
75– 93
5.1 5.2 5.3
Introduction 75 Mettho Me hod d of of Gro Group up Ave vera rage gess 76 Thee Lea Th Least st Sq Squa uare ress Me Meth thod od 81 5.3.1 5.3 .1 Fit Fittin ting g a Stra Straight ight Lin Linee 82 5.3.2 5.3 .2 Fit Fittin ting g a Par Parabo abola la 84 5.3.3 5.3 .3 Fit Fittin ting g a Curve Curve of the the Form Form y = ax b 86 5.3.4 5.3 .4 Fit Fittin ting g an Exponen Exponentia tiall Curve Curve 88 5.4 Method of of Mo Moments 89 Exer Ex erci cise sess 92
6.
Interpolation 6.1 6.2
94 – 137
Introduction 94 Finit Fi nitee Dif Diffe fere renc ncee Ope Opera rato tors rs 94 6.2.1 6.2. 1 Forw Forward ard Diff Differe erence ncess 94 6.2.2 6.2 .2 Bac Backwa kward rd Diff Differe erence ncess 97 6.2.3 6.2 .3 Cen Centra trall Dif Differe ference ncess 99 6.3 New Newton ton’s ’s Forwar Forward d Differe Difference nce Inte Interpol rpolati ation on Formul Formulaa 104 6.4 New Newton ton’s ’s Backwa Backward rd Differe Difference nce Inter Interpol polati ation on Formula Formula 108 6.5 La Lagra grang nge’ e’ss Inter Interpo pola lati tion on Form Formul ulaa 110 6.6 Di Divi vide ded d Dif Diffe fere renc nces es 113 6.6.1 New Newton’s ton’s Divided Divided Differen Difference ce Interpolat Interpolation ion Formula Formula 115 6.6.2 6.6. 2 New Newton’ ton’ss Divided Divided Differenc Differencee Formula Formula with Error Term 11 119 9 6.6.3 6.6. 3 Error Term Term in Inte Interpol rpolatio ation n Formulae Formulae 119 6.7 In Inte terpo rpola lati tion on in in Two Two Dime Dimens nsio ions ns 120 6.8 Cu Cubi bicc Sp Spli line ne Int Inter erpo pola lati tion on 122 6.8.1 6.8 .1 Con Constr struct uction ion of Cubi Cubicc Spline Spline 123 6.8. 6. 8.2 2 En End d Cond Condit itio ions ns 125 6.9 Max Maxima ima and Mini Minima ma of a Tab Tabula ulated ted Fun Functi ction on 129 6.10 Hermi Hermite te Interp Interpolati olation on 132 Exer Ex erci cise sess 13 134 4
ix
Contents
7.
Numerical Differentiation and Integration
138–174
7.1 7.2 7.3 7.4 7.5 7.6
Introduction 138 Differ Dif ferent entiat iation ion usi using ng Diff Differe erence nce Ope Operat rators ors 138 Differe Dif ferenti ntiati ation on usi using ng Inte Interpo rpolat lation ion 145 Rich Ri char ards dson’ on’ss Extrap Extrapola olati tion on Metho Method d 147 Nume Nu meri rica call Integ Integra rati tion on 150 Newt Ne wton on–Co –Cote tess Integra Integrati tion on Formu Formula laee 150 7.6.1 The Trapez Trapezoidal oidal Rule (Compo (Composite site Form) 154 7.6.2 Simps Simpson’s on’s Rule Ruless (Compos (Composite ite Forms Forms)) 155 7.7 Ro Romb mber erg’ g’ss Int Integ egra rati tion on 159 7.8 Do Doub uble le In Inte tegr grat atio ion n 161 7.9 Ga Gaus ussi sian an Qu Quad adra ratu ture re Form Formul ulae ae 164 7.10 Mul Multip tiple le Intege Integers rs 169 Exer Ex erci cise sess 17 171 1
8.
Ordinary Differential Equations Introduction 175 Tayl Ta ylor or’s ’s Se Seri ries es Me Meth thod od 177 Euler Method 179 8.3.1 8.3 .1 Mod Modifi ified ed Eule Euler’s r’s Meth Method od 181 8.4 Ru Rung nge– e–Ku Kutt ttaa Met Metho hods ds 183 8.5 Pr Pred edic icto tor–C r–Corr orrec ecto torr Met Method hodss 191 8.5. 8. 5.1 1 Mi Miln lne’ e’ss Meth Method od 192 8.5.2 8.5 .2 Ada Adam–M m–Moul oulton ton Met Method hod 196 8.6 Nu Nume meri rica call St Stab abil ilit ity y 200 8.6.1 Stabi Stability lity of Modifie Modified d Euler’s Euler’s Method Exer Ex erci cise sess 20 206 6
Introduction 210 Basic Bas ic Conc Concept eptss in Fin Finite ite Dif Differe ference nce Met Methods hods 211 Expl Ex plic icit it Me Meth thod odss 216 9.3. 9. 3.1 1 Sc Schm hmid idtt Meth Method od 216 9.3.2 9.3. 2 Durf Durfort– ort–Fran Frankel kel Metho Method d (1953) (1953) 220 9.4 Im Impl plic icit it Me Meth thod odss 221 9.4.1 Clas Classica sicall Impli Implicit cit Metho Method d 221 9.4.2 9.4. 2 Cran Crank–Ni k–Nicols colson on Meth Method od (1947 (1947)) 222 9.4.3 Weigh Weighted ted Avera Average ge Implic Implicit it Method 227 9.5 Th Thee Con Conce cept pt of Sta Stabi bili lity ty 227 9.6 Met Methods hods for Two Two-di -dimen mensio sional nal Equ Equati ations ons 232 9.6.1 9.6 .1 Exp Explic licit it Me Metho thods ds 232 9.6.2 9.6 .2 Imp Implic licit it Met Method hodss 233 9.6.3 Alter Alternate nate Direc Direction tion Implic Implicit it Method Method 234 Exer Ex erci cise sess 23 237 7
210 –2 –239
Numerical Methods For Scientists And Engineers
25% OFF
Publishe Publi sherr : PHI : PHI Learning
ISBN : 9 : 97881203321 788120332171 71
Author : SANKARA : SANKARA RAO, K.
Type the URL : : http://www.kopykitab.com/product/7394
