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| Fox L., Parker I.B. — Chebyshev Polynomials in Numerical Analysis |
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| Предметный указатель |
 method for elliptic equations 174
method for integral equations 158 159
method for ordinary differential equations 96 120—125 134—142 148—150 153
method for parabolic equations 164—167 179
method for rational functions 87 88 92 93 119
Aitken iteration 197
Analytical properties of Chebyshev polynomials 53
Asymptotic series 110 149
Asymptotic series, economization of 73
Averaging operator 80
Backward recurrence 55 126
Backward recurrence for integrated differential system 103 105 109 111 114 134
Backward recurrence for mixed differential system 115 116
Backward recurrence for rational function 99 100
Baker, C.T.H., Fox, L., Mayers, D.F. and Wright, K. 163 198
Basis 9 40 41
Beasley, J.D. 64 198
Bessel interpolation formula 77
Bessel interpolation formula, economized 80
Birkhoff, G. and MacLane, S. 40 198
Boundary conditions for elliptic equations 171
Boundary conditions for parabolic equations 164—166 169
Boundary conditions for parabolic equations, incompatibility of, with initial conditions 170 171 179 181
Cancellation of powers of x 121 124 136 140 142
Canonical polynomials 149—151 164
Chebyshev acceleration of iterative method 186 187 197
Chebyshev behaviour 5 75 77
Chebyshev polynomial 4 14
Chebyshev polynomial in extended range 18
Chebyshev polynomial in range  50
Chebyshev polynomial of second kind 20 35 53
Chebyshev quadrature 36
Chebyshev series 16 25 31
Chebyshev series in two variables 172 178
Chebyshev series, determination of, for general function 65
Chebyshev series, determination of, for rational function 85 97
Chebyshev series, evaluation of 55 56 58
Clenshaw, C.W. 56 83 89 114 116 198
Clenshaw, C.W. and Curtis, A.R. 60 198
Clenshaw, C.W. and Hayes, J.G. 178 179 198
Clenshaw, C.W. and Norton, H.J. 145 198
Complete system 44 45 157
Complex plane 93 150
Comrie, L.J. 78 198
Conformal transformation 176
Convergence in the mean 44
Convergence of Chebyshev series 21 25 31 45 56 60 65
Convergence of Fourier series 21 23 24 45
Convergence of iterative method 186 196
Convergence, asymptotic 189
Convergence, point-wise 40 45
Convergence, uniform 45
Cosine series 23 24
Crank — Nicolson method 164
Curtis, A.R. See "Clenshaw C.W."
Curve fitting, one-dimensional 81
Curve fitting, two-dimensional 177
Davis, P.J. 42 198
Diagonal dominance 106 113 153
Differences 77
Differences, modified 78 80
Differential equations for Chebyshev polynomials 53
Differential equations with polynomial coefficients 95 101 102 144
Differential equations, elliptic 171 174 185
Differential equations, non-linear 97
Differential equations, ordinary, general linear 96 143
Differential equations, orthogonal polynomials 13 17
Differential equations, parabolic 163 167
Differential system See also "Integrated system"
Differential system, mixed 114—118 141
Differential system, original 111 121 124 135—137 142 148—150
Differentiation of Chebyshev polynomial 55
Differentiation, series 61 101 112
Direct iteration 107 190
Dirichlet problem 171
Discrete least-squares approximation 7 25 27 30—32 46 See
Discrete least-squares approximation, analogy of, with continuous case 33 34 46
Economization 64 72 77 81 110
Eigenvalue problem 106 163 175 190 194
Elimination 99 103 107 112 157 185
Elliott, D. 68 90 162 164 198
Elliptic equation 171 174 185
Error in backward recurrence 100 126
Error in Chebyshev approximation to differential system 96 129 132 135
Error in Chebyshev approximation to economized series 72
Error in Chebyshev approximation to general function 68
Error in Chebyshev approximation to integral equation 160
Error in Chebyshev approximation to rational function 87 88 95 126—128
Error in curve fitting 84
Error in differentiation of Chebyshev series 62 63
Error in elimination method 113
Error in evaluation of Chebyshev series 57
Error in Gauss quadrature 36 38 39
Error in iterative method for linear equations 189
Error in Lagrange interpolation 6 15
Error in Lagrange interpolation, quadrature 39
Error in rounding 36 62 112
Euler transformation 93
Everett's formula 77
Everett's formula, economized 81
Exponential function, approximations for 69
Exponential integral 73 110
Finite approximation to elliptic equation 173
Finite approximation to general function 66
Finite approximation to ordinary differential equation 118 121 148
Finite approximation to parabolic equation 166 179
Finite approximation to power series 72
Finite approximation to rational function 85—88
Forsythe, G.E. 82 198
Fourier coefficients 22 24 43 44
Fourier coefficients, series 21 22
Fourier coefficients, series, convergence of 22—24
Fourier coefficients, transform 192
Fox, L. 57 78 81 198 See C.T.H."
Fox, L. and Goodwin, E.T. 161 198
Fredholm equations 155 163
Functional equation 52 53
Gauss quadrature 34—37 39 46 60
Gauss — Seidel iteration 106 153 196
Generating function 51
Gershgorin's theorem 194
Goodwin, E.T. See "Fox L."
Gram — Schmidt method 9 10 27
Hastings, C. 3 198
Hayes, J.G. See "Clenshaw C.W."
Hermite interpolation 37 38
Hilbert matrix 85 92
Hildebrand, F.B. 12 198
Hypergeometric function 54
Ill-conditioning 84 163
Initial rows 100 102 104 105 119
Initial rows, state 170 179
Inner product 42—45
Integrated system, first-order equations 101 102 109 110 115 116 118 120 129 131—137 142
Integrated system, parabolic equations 167 170 179 181
Integrated system, second-order equations 101 105 108 117 118 121 125 135 138—141 144
Integrated system, second-order equations, of eigenvalue type 106
Integration by Gauss rule 60
Integration by mid-point rule 33
Integration by Simpson's rule 60
Integration by trapezium rule 33
Integration, definite of Chebyshev series 55 59 60
Integration, definite of Fourier series 22
Integration, indefinite of Chebyshev polynomial 54
Integration, indefinite of Chebyshev polynomial, series 58 59 88 89
Interchanges 103 110
Interpolation in mathematical tables 77
Interpolation in two dimensions 172
Interpolation, Chebyshev 32 146 159
Interpolation, equal interval 7 77
| Interpolation, Hermite 37
Interpolation, trigonometric 28
Inverse iteration 191
Iterative method for error analysis 130—133 137
Iterative method of Aitken 197
Iterative method, acceleration of 186 187 197
Iterative method, direct 107 190
Iterative method, Gauss — Seidel 106
Iterative method, inverse 191
Iterative method, Jacobi 188 196
Iterative method, non-stationary 188
Iterative method, stationary 185—189
Jacobi iteration 188 196
Kalaba, R. 147 199
Kantorovich, L.V. and Krylov, V.I. 160 199
Kernel 155
Kernel, degenerate 156 159 163
Kernel, discontinuous 158
Kernel, even 160
Kernel, non-degenerate 157
kernel, resolvent 160
Krylov, V.I. See "Kantorovich L.V."
L membrane 175
Lagrange interpolation 5 15 18 30 38 96 101
Lagrange interpolation, quadrature 34 35 37 39
Lanczos, C. 7 17 29 50 55 72 87 89 121 148 185 190—192 195 199
Least-squares approximations for curve fitting 81—84 178—179
Least-squares approximations, connexions between continuous and discrete cases 33 46
Least-squares approximations, continuous case 7 19 21 39 42 44
Least-squares approximations, continuous case, Chebyshev 15 19 25 31
Least-squares approximations, continuous case, Fourier 22 44
Least-squares approximations, continuous case, Legendre 15
Least-squares approximations, discrete case 25 26 28—30
Least-squares approximations, discrete case, Chebyshev 28 30—32
Least-squares approximations, discrete case, Fourier 27 31
Least-squares approximations, discrete case, Legendre 46
Legendre polynomial 13 15 17 19 46 63
Legendre series 16 17 46 63
Linear independence 9 10 40
Love's integral equation 161
MacLane, S. See "Birkhoff G."
Makinouchi, S. 126 199
Mason, J. 174—176 199
Matching points See also "Selected points" "Zeros"
Matching points in interpolation, Chebyshev 28 32 66 89 90 172
Matching points in interpolation, Fourier 28
Matching points in interpolation, Hermite 38
Matching points in interpolation, Lagrange 6 7 15 18 30 34 96
Matching points in interpolation, orthogonal polynomial 30
Matching points, quadrature, Chebyshev 36
Matching points, quadrature, Gauss 36 46
Matching points, quadrature, Lagrange 35
Matrix of eigenvalue problem 106 107
Matrix, codiagonal 29
Matrix, diagonal 8 25 26 82
Matrix, Hilbert 85 92
Matrix, infinite 98 100—102 119 125 163
Matrix, inverse 148 150
Matrix, iterative method 133 185 189 197
Matrix, normal equations 8 26 81 84
Matrix, orthogonal 29
Matrix, singular 109 110 123 154
Matrix, sparse 109 123 185
Matrix, symmetric 8 29
Matrix, triangular 79 99 100 102 103 105 109 122 148 150
Mayers, D.F. See "Baker C.T.H."
Miller, J.C.P. 78 126 199
Minimax, criterion 3 5 33
Minimax, linear fit 75—77 91
Minimax, polynomial 4 5 15 18 39 42 50 74—78 89 126 132 139 140
Minimax, theory 5 39 41 42 172
Mixed system 114—118 141 See
Nested multiplication 55
Neumann problem 171
Newton iteration 145 147
Newton — Cotes formula 37
Norm 41 44
Normal equations 8 26 81 84
Normalization 10 16 27 29 30 36 37 108
Norton, H.J. 147 199 See C.W."
Novodvorskii, E.P. and Pinsker, I.S. 74 199
Olver, F.W.J. 126 199
Orthogonal functions 21 39 43 44
Orthogonal matrix 29
Orthogonal polynomials 8 15 27—29 34—39 45 83
Orthogonal polynomials, Chebyshev 14 15 30
Orthogonal polynomials, differential equations for 13 19
Orthogonal polynomials, Gram — Schmidt method for 9 10 27
Orthogonal polynomials, Legendre 13 15 46
Orthogonal polynomials, recurrence relations for 11—14 27 28 56 63 82 83 177
Orthogonal polynomials, trigonometric 21 27
Orthogonal polynomials, ultra-spherical 16
Orthogonal polynomials, zeros of 10 19
Orthogonal series, Chebyshev 16 17 25 27 30
Orthogonal series, Legendre 16 17
Orthogonal series, Taylor 16 17
Orthogonality, continuous 8 21 35 43 71 79
Orthogonality, discrete 25—27 46 67 82
Parabolic equation 163 167
Parseval's formula 44 46
Periodicities, search for 192
Perturbation of associated conditions 131 133 145
Perturbation of differential equation 96 120 122—125 129—131 135
Perturbation of differential system 134—136 144 152
Perturbation of elliptic equation 174
Perturbation of integral equation 158 159
Perturbation of parabolic equation 166
Perturbation of rational function 86 88 119 120 127
Picard iteration 145 146
Pinsker, I.S. See "Novodvorskii E.P."
Point-wise convergence 40 45
Powers in terms of Chebyshev polynomials 49 51 71
Products in terms of Chebyshev polynomials 52 53 101 157
Rational functions 38 85—88 93 95 97 119 127 151
Recurrence relations See also "Backward recurrence"
Recurrence relations for Chebyshev acceleration of iterative process 187 188 190 192 196
Recurrence relations for Chebyshev coefficients of rational function 86 98
Recurrence relations for Chebyshev polynomials 4 5 20 48 50
Recurrence relations for differentiation of Chebyshev series 61
Recurrence relations for evaluation of Chebyshev series 56 57
Recurrence relations for evaluation of Chebyshev series, Legendre series 63
Recurrence relations for Legendre polynomials 13 14
Recurrence relations for orthogonal polynomials 11 12 28 37
Recurrence relations for orthogonal polynomials in curve fitting 82—84 177
Reid, J.K. and Walsh, J.E. 176 199
Scraton, R.E. 153 199
Selected points, method of See also "Matching points"
Selected points, method of, for elliptic equations 172—175 183 184
Selected points, method of, for integral equations 162
Selected points, method of, for ordinary differential equations 143—147
Simpson's rule 60
Sine series 23 24
Space 9 10 40—45
Spectral radius 186 190 191 196
Spectroscopic analysis 190
Stability of elimination method 103
Stability of evaluation of Chebyshev series 58
Stability of nested multiplication 55
Steady state 169 170 179 181
Steffenson, J.F. 39 199
Stirling approximation 20
Taylor's series 1 16
Taylor's series, connexion of, with Chebyshev series 89 149
Taylor's series, convergence of 2 17 73 86 93
Taylor's series, economized 72
Taylor's series, non-existence of 111
Terminal discontinuities 23 24
Throwback 78
Todd, J. 34 42 199
Torsion problem 173
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