## Table of contents for An introduction to numerical methods : a MATLAB approach / Abdelwahab Kharab, Ronald B. Guenther.

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```1 Introduction                               1
1.1 ABOUT THE SOFTWARE MATLAB . ......... 1
1.2 AN INTRODUCTION TO MATLAB . ......... 2,
1.2.1  Matrices and matrix computation  . .......  2
1.2.2  Polynom ials  ... ................. .  7
1.2.3  Output format  ...................  8
1.2.4  Planar plots.... ................. .  9
1.2.5  3-D  mesh  plots  .................. .  10
1.2.6  Function  files  ................... .  11
1.2.7  Defining functions  . . . . . . . . . . . . .....  12
1.2.8  Relations and loops  . . . . . . . . . . . .....  13
1.3  TAYLOR  SERIES  .....................  17
2 Number System and Errors                   25
2.1 FLOATING-POINT ARITHMETIC . . . . . . . . . . . 25
2.2 ROUND-OFF ERRORS ..................    30
2.3  TRUNCATION ERROR... .............  .  36
2.4 INTERVAL ARITHMETIC  ............... 38
3 Roots of Equations                         45
3.1 THE BISECTION METHOD  .............. 47
3.2 THE METHOD OF FALSE POSITION ....... . 55
3.3 FIXED-POINT ITERATION ............... ..62
3.4  THE SECANT METHOD ............... ....  70
3.5 NEWTON'S METHOD .................. 76
3.6 CONVERGENCE OF THE NEWTON AND
SECANT METHODS ................... 87
7.3.2  Hyperbolic form  . ...... ........... . 262
7.4 TRIGONOMETRIC LEAST SQUARES
POLYNOMIAL  ............ ..  ......... 269
APPLIED PROBLEMS  ............ ...........  272
8 Numerical Optimization               275
8.1 ANALYSIS OF SINGLE-VARIABLE FUNCTIONS . 276
8.2  LINE SEARCH METHODS ................ .. 278
8.2.1  Bracketing the minimum. ............ .  278
8.2.2  Golden section search  ............. . . 279
8.2.3  Fibonacci Search  ................ .  283
8.2.4  Parabolic Interpolation  . ............286
8.3 MINIMIZATION USING DERIVATIVES ........ 294
8.3.1  Newton's method  ................ .  294
8.3.2  Secant method  ..................  . 295
APPLIED  PROBLEMS  .....................  298
9 Numerical Differentiation            301
9.1  NUMERICAL DIFFERENTIATION ........ . . . 301
9.2 RICHARDSON'S FORMULA. .... . ........  309
APPLIED  PROBLEMS  ....... ............ . 316
10 Numerical Integration               321
10.1 TRAPEZOIDAL RULE  . ............... .  322
10.2 SIMPSON'S RULE ................. . . 333
10.3 ROMBERG ALGORITHM .   .............. 344
APPLIED PROBLEMS .. ................... 365
11 Numerical Methods for Differential Equations  371
11.1 EULER'S METHOD  ............. ......... 372
11.2  ERROR  ANALYSIS  ........ ........... . 380
11.3 HIGHER ORDER TAYLOR SERIES METHODS ... 385
11.4 RUNGE-KUTTA METHODS ............... .390
11.5 MULTISTEP METHODS . ............... ..406
11.6 ADAMS-BASHFORTH METHODS . ......... 406
11.7 PREDICTOR-CORRECTOR METHODS ........ 417
11.9 NUMERICAL STABILITY ................ ..427
11.10 HIGHER ORDER EQUATIONS AND SYSTEMS
OF DIFFERENTIAL EQUATIONS . ......... 431
11.11 IMPLICIT METHODS AND STIFF SYSTEMS . .. 438
11.12 PHASE PLANE ANALYSIS: CHAOTIC
DIFFERENTIAL EQUATIONS ............. .441
APPLIED  PROBLEMS  .......................  447
12 Boundary-Value Problems             457
12.1 FINITE-DIFFERENCE METHODS ........... 458
12.2 SHOOTING METHODS  ......... ........ 467
12.2.1 The nonlinear case . ............... .. 467
12.2.2  The linear case  . ......... .........  472
APPLIED PROBLEMS .. ................... 480
13 Eigenvalues and Eigenvectors        485
13.1 BASIC  THEORY  ....................... . 485
13.2 THE POWER METHOD  ......... ........ 490
13.3 THE QUADRATIC METHOD .............. .494
13.4 EIGENVALUES FOR BOUNDARY-VALUE
PROBLEMS  . . ............ .......... . 505
13.5 BIFURCATIONS IN DIFFERENTIAL
EQUATIONS  ......  ....  . ............  508
APPLIED  PROBLEMS  .....................  513
14 Partial Differential Equations      515
14.1 PARABOLIC EQUATIONS .......... ...... 516
14.1.1 Explicit methods .  ............... 516
14.1.2  Implicit methods  ............. ..... .  521
14.2 HYPERBOLIC EQUATIONS . .............  529
14.3 ELLIPTIC EQUATIONS  ........... ......  536
14.4 INTRODUCTION TO FINITE-ELEMENT METHOD. 543
14.4.1 Theory  .... .................... .  543
14.4.2  The Finite Element Method ............  551
APPLIED PROBLEMS  .. ............. ..... . 557
Bibliography and References             559
Appendix                               565
A Calculus Review                                      565
A.1  Limits and continuity  .......... ...... . . . . 565
A.2 Differentiation  .....................       . .566
A.3 Integration ................ ........            567
B MATLAB Built-in Functions                            569
C Text MATLAB Functions                                573