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```1   What can't be ignored ..............................   1
1.1 Real numbers ................ ...................  2
1.1.1  How  we represent them  .......................  2
1.1.2 How we operate with floating-point numbers .....  4
1.2 Complex numbers ................................   6
1.3 Matrices ................ .......................  8
1.3.1  Vectors  .....................................  14
1.4 Real functions ................ .................. 15
1.4.1 The zeros ................ ................ 16
1.4.2 Polynomials ................................. 18
1.4.3  Integration  and  differentiation  .................  21
1.5  To  err is not only  human  ............................  23
1.5.1  Talking  about costs ...........................  26
1.6 The MATLAB and Octave environments ............. 28
1.7  The MATLAB   language ...........................  29
1.7.1  MATLAB  statements  .........................  31
1.7.2 Programming in MATLAB .................... 32
1.7.3 Examples of differences between MATLAB
and  Octave languages ............... .........  36
1.8  W hat we haven't told  you  ............... ...........  37
1.9 Exercises ....................................... 37
2   Nonlinear equations .................................. 39
2.1 The bisection method ...........................  41
2.2 The Newton method ............................. 45
2.2.1 How to terminate Newton's iterations ........... 47
2.2.2 The Newton method for systems of nonlinear
equations .................................... 49
2.3 Fixed point iterations ............................. 51
2.3.1 How to terminate fixed point iterations ......... 55
2.4 Acceleration using Aitken method ................... . 56
2.5 Algebraic polynomials............................ 60
2.5.1  Hdrner's  algorithm  ...........................  61
2.5.2 The Newton-H6rner method ................... 63
2.6  W hat we haven't told  you  ...........................  65
2.7 Exercises ................. ....................... 67
3   Approximation of functions and data ................. 71
3.1 Interpolation ....................................... 74
3.1.1 Lagrangian polynomial interpolation ............ 75
3.1.2 Chebyshev interpolation..................... 80
3.1.3 Trigonometric interpolation and FFT ........... 81
3.2 Piecewise linear interpolation ......... . . . . ..... . . .  86
3.3 Approximation by spline functions .................. 88
3.4  The least-squares method............  .. ...........  92
3.5  W hat we haven't told  you  .............  .............  97
3.6 Exercises ...............  ....................... 98
4   Numerical differentiation and integration ............. 101
4.1 Approximation of function derivatives ................. 103
4.2 Numerical integration ............................  105
4.2.1 Midpoint formula............ .. ........... 106
4.2.2  Trapezoidal formula  .......................... 108
4.2.3  Simpson  formula  ............................. 109
4.4 Simpson adaptive formula ........................... 115
4.5 What we haven't told you ............ ............ 119
4.6 Exercises ............. ....................... 120
5   Linear systems.................. .  .......... ... 123
5.1  The LU  factorization  method  ........................ 126
5.2 The pivoting technique ............................ 134
5.3 How accurate is the LU factorization? ................. 136
5.4  How  to solve a tridiagonal system  .................... 140
5.5 Overdetermined systems.......................... 141
5.6 What is hidden behind the command \ ........... 143
5.7  Iterative methods............ ..... .................  144
5.7.1 How to construct an iterative method ........... 146
5.8 Richardson and gradient methods .................... 150
5.9  The conjugate gradient method  ...................... 153
5.10 When should an iterative method be stopped? ......... 156
5.11  To wrap-up: direct or iterative?  ...................... 159
5.12 What we haven't told you ........................ 164
5.13 Exercises ................ ....................... 164
6   Eigenvalues and eigenvectors ....................... 167
6.1  The  power method  .................................  170
6.1.1  Convergence analysis  ......................... 173
6.2 Generalization of the power method .................. 174
6.3  How  to compute the shift...................... ... . 176
6.4  Computation of all the eigenvalues .................... 179
6.5  W hat we haven't told  you  ........................... 183
6.6  Exercises  ...................................... ....  183
7   Ordinary differential equations ...................... 187
7.1 The Cauchy problem............................. 190
7.2 Euler methods ................. .................. 191
7.2.1  Convergence analysis  ......................... 194
7.3 The Crank-Nicolson method ....................... 197
7.4 Zero-stability ............. .. ................... 199
7.5  Stability on unbounded intervals  ................... .. 202
7.5.1 The region of absolute stability ................ 204
7.5.2 Absolute stability controls perturbations ........ 205
7.6 High order methods.............................. 212
7.7  The predictor-corrector methods  ..................... 216
7.8  Systems of differential equations ...................... 219
7.9 Some examples ............ ..................... 225
7.9.1 The spherical pendulum ............ ......... 225
7.9.2  The three-body problem  ..................... 228
7.9.3  Some stiff problems ........................... 230
7.10 What we haven't told you .......................... 234
7.11 Exercises ................... ........... ........ 234
8 Numerical methods for (initial-) boundary-value
problems ................... ....................... 237
8.1 Approximation of boundary-value problems ............ 240
8.1.1 Approximation by finite differences ............. 241
8.1.2 Approximation by finite elements ............... 243
8.1.3 Approximation by finite differences
of two-dimensional problems ................... 245
8.1.4 Consistency and convergence ................... 251
8.2 Finite difference approximation of the heat equation .... 253
8.3  The  wave  equation  ................................. 257
8.3.1 Approximation by finite differences ............. 260
8.4  W hat we haven't told  you  ........................... 263
8.5 Exercises ................ ....................... 264
9   Solutions of the exercises . . . . . . ......... .......... 267
9.1  Chapter  1  ................. ............ . . .  .  .   . .   . . 267
9.2  Chapter  2  . .  .  . ................ . . . . . . . . . .......   .  .   . . . 270
9.3  C hapter  3  . . . . . . . . . . . . . . .   . . . . . . . . . . . . . . . . . . . . . . .  .   276
9.4  Chapter 4  ...... ...........  .................  ..... 280
9.5  C hapter  5  ... . . . . . . . . . . . ... . .  . . . . . . . .  .  .  .  .... ... . . .. . . .  .  285
9.6  Chapter  6  . .................................... ... .  289
9.7  Chapter  7  . . . . . . . . . . . . . . . .  .. . . . . . . . . . . . . . . . . . . .  . . .. . 293
9.8  Chapter 8  . . . . . . . . ..... . . . . .. .. ....   . .... .. . . . . . .  ... . 301

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Library of Congress subject headings for this publication: Science Data processing, MATLAB