MTH3051 Introduction to Computational Mathematics - User Web ...

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School of Mathematical Sciences Monash University 7.2.1 The whole polynomial or just its value? . . . . . . . . . . . . . . . . 85 7.2.2 Matlab code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.3 Newton polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.3.1 Horner’s form of the Newton polynomial . . . . . . . . . . . . . . . 92 7.4 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 7.5 Piecewise polynomial interpolation . . . . . . . . . . . . . . . . . . . . . . 93 7.5.1 Cubic Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.5.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.6 Non-polynomial interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.6.1 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 7.6.2 Estimating the Fourier coefficients . . . . . . . . . . . . . . . . . . . 98 7.6.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.7 Approximating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.7.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.7.2 Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 7.7.3 Generalised least squares . . . . . . . . . . . . . . . . . . . . . . . . 105 7.7.4 Variations on a theme . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.7.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.7.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.7.7 Matlab example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 8. Extrapolation Methods 110 8.1 Richardson extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.1.1 Example – computing π . . . . . . . . . . . . . . . . . . . . . . . . 111 8.1.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 9. Numerical integration 113 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 9.2 The Left and Right hand sum rules . . . . . . . . . . . . . . . . . . . . . . 114 9.2.1 The Left Hand Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9.2.2 The Right Hand Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9.2.3 The Mid Point rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 9.3 The Trapezoidal rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 9.3.1 Choices, choices, so many choices . . . . . . . . . . . . . . . . . . . 117 9.4 Simpson’s rule and Romberg integration . . . . . . . . . . . . . . . . . . . 118 16-Feb-2014 4
School of Mathematical Sciences Monash University 9.5 Simpson’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 9.5.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 9.6 Romberg integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 9.6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 10. Numerical differentiation 121 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 10.2 Finite differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 10.2.1 First derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 10.2.2 Second derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 10.3 Truncation and round-off errors . . . . . . . . . . . . . . . . . . . . . . . 125 10.3.1 Example – Forward finite differences . . . . . . . . . . . . . . . . . 125 10.3.2 Example – Centred finite differences . . . . . . . . . . . . . . . . . 127 11. Numerical Solutions of Ordinary Differential Equations 129 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 11.2 Initial value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 11.3 Euler’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 11.4 Improved Euler scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 11.5 Taylor series method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 11.6 Runge-Kutta schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 11.6.1 Second Order Runge-Kutta . . . . . . . . . . . . . . . . . . . . . . 136 11.6.2 Fourth Order Runge-Kutta . . . . . . . . . . . . . . . . . . . . . . 137 11.7 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 11.7.1 Discretization errors . . . . . . . . . . . . . . . . . . . . . . . . . . 138 11.7.2 Local discretization error . . . . . . . . . . . . . . . . . . . . . . . 138 11.7.3 Global Discretization Error . . . . . . . . . . . . . . . . . . . . . . 139 11.7.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 11.8 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 11.8.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 11.8.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 11.8.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 11.8.4 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 12. Optimisation 143 12.1 Golden Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 16-Feb-2014 5
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