CONTENTS iiⅲ 3 Galerkin Weighted Residual Methods 61 3.1 Mean Weighted Residual Methods 61 3.2 Completeness and Boundary Conditions 64 3.3 Inner Product Orthogonality 65 3.4 Galerkin Method 67 3.5 Integration-by-Parts.............................. 68 3.6 Galerkin Method:Case Studies 70 3.7 Separation-of-Variables&the Galerkin Method.,·········· 76 3.8 Heisenberg Matrix Mechanics.................. 77 3.9 The Galerkin Method Today.······················· 80 4 Interpolation,Collocation All That 81 4.1 4.2 Polynomial interpolation............ 8 4.3 Gaussian Integration Pseudospectral Grids .............. 86 4.4 Pseudospectral Is Galerkin Method via Quadrature 89 4.5 Pseudospectral Errors 93 5 Cardinal Functions 98 5.1 98 5.2 Whittaker Cardinal or "Sinc"Functions... 9 5.3 Trigonometric Interpolation.·· 100 5.4 Cardinal Functions for Orthogonal Polynomials 104 5.5 Transformations and Interpolation .................... 107 6 Pseudospectral Methods for BVPs 109 6.1 Introduction 109 6.2 Choice of Basis Set 109 6.3 Boundary Conditions:Behavioral Numerical.. 44 109 6.4 “Boundary-Bordering 111 6.5 "Basis Recombination" 112 6.6 Transfinite Interpolation.........·...·········· 114 6.7 The Cardinal Function Basis.................... 115 6.8 The Interpolation Grid 116 6.9 Computing Basis Functions&Derivatives...·....···· 116 6.l0 Higher Dimensions:Indexing...······.······· 118 6.11 Higher Dimensions 120 6.12 Corner Singularities.. 120 6.13 Matrix methods 121 6.14 Checking 121 6.15 Summary... 123 7 Linear Eigenvalue Problems 127 7.1 The No-Brain Method 127 7.2 QR/QZ Algorithm..... 128 7.3 Eigenvalue Rule-of-Thumb..··.·.·.·.·.······ 129 7.4 Four Kinds of Sturm-Liouville Problems.············ 134 7.5 Criteria for Rejecting Eigenvalues 137 7.6 “Spurious"Eigenvalues......·..············· 139 7.7 Reducing the Condition Number .. 142 7.8 3 The Power Method.......··...·.···········. 145 7.9 Inverse Power Method....·.·················: 149CONTENTS iii 3 Galerkin & Weighted Residual Methods 61 3.1 Mean Weighted Residual Methods . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Completeness and Boundary Conditions . . . . . . . . . . . . . . . . . . . . 64 3.3 Inner Product & Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4 Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.5 Integration-by-Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.6 Galerkin Method: Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.7 Separation-of-Variables & the Galerkin Method . . . . . . . . . . . . . . . . . 76 3.8 Heisenberg Matrix Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.9 The Galerkin Method Today . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4 Interpolation, Collocation & All That 81 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2 Polynomial interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3 Gaussian Integration & Pseudospectral Grids . . . . . . . . . . . . . . . . . . 86 4.4 Pseudospectral Is Galerkin Method via Quadrature . . . . . . . . . . . . . . 89 4.5 Pseudospectral Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5 Cardinal Functions 98 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.2 Whittaker Cardinal or “Sinc” Functions . . . . . . . . . . . . . . . . . . . . . 99 5.3 Trigonometric Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.4 Cardinal Functions for Orthogonal Polynomials . . . . . . . . . . . . . . . . 104 5.5 Transformations and Interpolation . . . . . . . . . . . . . . . . . . . . . . . . 107 6 Pseudospectral Methods for BVPs 109 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.2 Choice of Basis Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.3 Boundary Conditions: Behavioral & Numerical . . . . . . . . . . . . . . . . . 109 6.4 “Boundary-Bordering” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.5 “Basis Recombination” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.6 Transfinite Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.7 The Cardinal Function Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.8 The Interpolation Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.9 Computing Basis Functions & Derivatives . . . . . . . . . . . . . . . . . . . . 116 6.10 Higher Dimensions: Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.11 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.12 Corner Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.13 Matrix methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.14 Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.15 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7 Linear Eigenvalue Problems 127 7.1 The No-Brain Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.2 QR/QZ Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.3 Eigenvalue Rule-of-Thumb . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.4 Four Kinds of Sturm-Liouville Problems . . . . . . . . . . . . . . . . . . . . . 134 7.5 Criteria for Rejecting Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.6 “Spurious” Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.7 Reducing the Condition Number . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.8 The Power Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.9 Inverse Power Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149