EBOOK - Scientific Computing with MATLAB and Octave 3rd edition (Alfio Quarteroni) | Cộng đồng Kỹ thuật cơ điện Việt Nam EBOOKBKMT

EBOOK - Khoa học máy tính với MATLAB (Alfio Quarteroni) - 379 Trang.

This textbook is an introduction to Scientific Computing. We will illustrate several numerical methods for the computer solution of certain classes of mathematical problems that cannot be faced by paper and pencil. We will show how to compute the zeros or the integrals of continuous functions, solve linear systems, approximate functions by polynomials and construct accurate approximations for the solution of differential equations.
With this aim, in Chapter 1 we will illustrate the rules of the game that computers adopt when storing and operating with real and complexnumbers, vectors and matrices.

CONTENTS:

1 What can’t be ignored................................ 1
1.1 TheMATLABandOctaveenvironments ............. 1
1.2 Realnumbers ...................................... 3
1.2.1 Howwerepresent them ....................... 3
1.2.2 How we operate with floating-point numbers . . . . . 6
1.3 Complexnumbers .................................. 8
1.4 Matrices .......................................... 10
1.4.1 Vectors ..................................... 14
1.5 Realfunctions ..................................... 16
1.5.1 Thezeros ................................... 18
1.5.2 Polynomials ................................. 20
1.5.3 Integrationanddifferentiation ................. 22
1.6 To err is not only human ............................ 25
1.6.1 Talking about costs........................... 29
1.7 TheMATLABlanguage ............................ 30
1.7.1 MATLAB statements ......................... 32
1.7.2 ProgramminginMATLAB .................... 34
1.7.3 Examples of differences betweenMATLAB
andOctavelanguages......................... 37
1.8 What wehaven’t toldyou ........................... 38
1.9 Exercises .......................................... 38
2 Nonlinear equations.................................. 41
2.1 Somerepresentativeproblems........................ 41
2.2 Thebisection method............................... 43
2.3 TheNewton method................................ 47
2.3.1 Howto terminate Newton’s iterations........... 49
2.3.2 The Newton method for systems of nonlinear
equations.................................... 51
2.4 Fixedpoint iterations ............................... 54
2.4.1 How to terminate fixed point iterations . . . . . . . . . 60
XII Contents
2.5 Acceleration using Aitken’s method . . . . . . . . . . . . . . . . . . . 60
2.6 Algebraicpolynomials............................... 65
2.6.1 H¨orner’salgorithm ........................... 66
2.6.2 The Newton-H¨orner method ................... 68
2.7 What wehaven’t toldyou ........................... 70
2.8 Exercises .......................................... 72
3 Approximation of functions and data................. 75
3.1 Somerepresentativeproblems........................ 75
3.2 Approximationby Taylor’s polynomials ............... 77
3.3 Interpolation....................................... 78
3.3.1 Lagrangian polynomial interpolation . . . . . . . . . . . . 79
3.3.2 Stability of polynomial interpolation . . . . . . . . . . . . 84
3.3.3 Interpolationat Chebyshev nodes .............. 86
3.3.4 Trigonometric interpolation and FFT . . . . . . . . . . . 88
3.4 Piecewise linear interpolation . . . . . . . . . . . . . . . . . . . . . . . . 93
3.5 Approximationbysplinefunctions.................... 94
3.6 Theleast-squaresmethod............................ 99
3.7 What wehaven’t toldyou ........................... 103
3.8 Exercises .......................................... 105
4 Numerical differentiation and integration.............107
4.1 Somerepresentativeproblems........................ 107
4.2 Approximationof function derivatives................. 109
4.3 Numericalintegration............................... 111
4.3.1 Midpoint formula............................. 112
4.3.2 Trapezoidalformula .......................... 114
4.3.3 Simpsonformula ............................. 115
4.4 Interpolatoryquadratures ........................... 117
4.5 Simpsonadaptiveformula ........................... 121
4.6 What wehaven’t toldyou ........................... 125
4.7 Exercises .......................................... 126
5Linearsystems........................................129
5.1 Somerepresentativeproblems........................ 129
5.2 Linear systemandcomplexity........................ 134
5.3 TheLUfactorizationmethod ........................ 135
5.4 Thepivoting technique.............................. 144
5.5 Howaccurateis thesolution of a linear system? ........ 147
5.6 How to solve a tridiagonal system . . . . . . . . . . . . . . . . . . . . 150
5.7 Overdeterminedsystems............................. 152
5.8 What is hidden behind theMATLABcommand\ ..... 154
5.9 Iterativemethods................................... 157
5.9.1 How to construct an iterative method . . . . . . . . . . . 158
5.10 Richardsonandgradient methods .................... 162
5.11 Theconjugategradientmethod ...................... 166
5.12 When should an iterative method be stopped? . . . . . . . . . 169
5.13 To wrap-up:direct or iterative? ...................... 171
5.14 What wehaven’ttoldyou ........................... 177
5.15 Exercises .......................................... 177
6 Eigenvalues and eigenvectors.........................181
6.1 Somerepresentativeproblems........................ 182
6.2 Thepower method ................................. 184
6.2.1 Convergenceanalysis ......................... 187
6.3 Generalizationof thepower method .................. 188
6.4 Howto computetheshift............................ 190
6.5 Computationof allthe eigenvalues.................... 193
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