tonyart
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Lectures on Numerical Analysis
Dennis Deturck and Herbert S. Wilf
Department of Mathematics
University of Pennsylvania
Philadelphia, PA 19104-6395
Contents
1 Differential and Diference Equations 5
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Linear equations with constant coeffcients . . . . . . . . . . . . . . . . . . . 8
1.3 Difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Computing with difference equations . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Stability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Stability theory of difference equations . . . . . . . . . . . . . . . . . . . . . 19
2 The Numerical Solution of Differential Equations 23
2.1 Euler'smethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Software notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Systems and equations of higher order . . . . . . . . . . . . . . . . . . . . . 29
2.4 How to document a program . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 Themidpoint and trapezoidal rules . . . . . . . . . . . . . . . . . . . . . . . 38
2.6 Comparison of themethods . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.7 Predictor-corrector methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.8 Truncation error and step size . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.9 Controlling the step size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.10 Case study: Rocket to the moon . . . . . . . . . . . . . . . . . . . . . . . . 60
2.11 Maple programs for the trapezoidal rule . .. . . . . . . . . . . . . . . . . 65
2.11.1 Example: Computing the cosine function . . . . . . . . . . . . . . . 67
2.11.2 Example: Themoon rocket in one dimension . . . . . . . . . . . . . 68
2.12 The big leagues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.13 Lagrange and Adams formulas . . . . . . . . . . . . . . . . . . . . . . . . . 74
3 Numerical linear algebra 81
3.1 Vector spaces and linear mappings . . . . . . . . . . . . . . . . . . . . . . . 81
3.2 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3 Building blocks for the linear equation solver . . . . . . . . . . . . . . . . . 92
3.4 How big is zero? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.5 Operation count . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.6 To unscramble the eggs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.7 Eigenvalues and eigenvectors of matrices . . . . . . . . . . . . . . . . . 108
3.8 The orthogonal matrices of Jacobi . . . . . . . . . . . . . . . . . . . . . . . 112
3.9 Convergence of the Jacobi method . . . . . . . . . . . . . . . . . . . . . . . 115
3.10 Corbato's idea and the implementation of the Jacobi algorithm . . 118
3.11 Getting it together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.12 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Dennis Deturck and Herbert S. Wilf
Department of Mathematics
University of Pennsylvania
Philadelphia, PA 19104-6395
Contents
1 Differential and Diference Equations 5
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Linear equations with constant coeffcients . . . . . . . . . . . . . . . . . . . 8
1.3 Difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Computing with difference equations . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Stability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Stability theory of difference equations . . . . . . . . . . . . . . . . . . . . . 19
2 The Numerical Solution of Differential Equations 23
2.1 Euler'smethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Software notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Systems and equations of higher order . . . . . . . . . . . . . . . . . . . . . 29
2.4 How to document a program . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 Themidpoint and trapezoidal rules . . . . . . . . . . . . . . . . . . . . . . . 38
2.6 Comparison of themethods . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.7 Predictor-corrector methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.8 Truncation error and step size . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.9 Controlling the step size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.10 Case study: Rocket to the moon . . . . . . . . . . . . . . . . . . . . . . . . 60
2.11 Maple programs for the trapezoidal rule . .. . . . . . . . . . . . . . . . . 65
2.11.1 Example: Computing the cosine function . . . . . . . . . . . . . . . 67
2.11.2 Example: Themoon rocket in one dimension . . . . . . . . . . . . . 68
2.12 The big leagues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.13 Lagrange and Adams formulas . . . . . . . . . . . . . . . . . . . . . . . . . 74
3 Numerical linear algebra 81
3.1 Vector spaces and linear mappings . . . . . . . . . . . . . . . . . . . . . . . 81
3.2 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3 Building blocks for the linear equation solver . . . . . . . . . . . . . . . . . 92
3.4 How big is zero? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.5 Operation count . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.6 To unscramble the eggs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.7 Eigenvalues and eigenvectors of matrices . . . . . . . . . . . . . . . . . 108
3.8 The orthogonal matrices of Jacobi . . . . . . . . . . . . . . . . . . . . . . . 112
3.9 Convergence of the Jacobi method . . . . . . . . . . . . . . . . . . . . . . . 115
3.10 Corbato's idea and the implementation of the Jacobi algorithm . . 118
3.11 Getting it together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.12 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
