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Fox L., Parker I.B. — Chebyshev Polynomials in Numerical Analysis
Fox L., Parker I.B. — Chebyshev Polynomials in Numerical Analysis

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Название: Chebyshev Polynomials in Numerical Analysis

Авторы: Fox L., Parker I.B.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1968

Количество страниц: 214

Добавлена в каталог: 31.08.2013

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\tau$ method for elliptic equations      174
$\tau$ method for integral equations      158 159
$\tau$ method for ordinary differential equations      96 120—125 134—142 148—150 153
$\tau$ method for parabolic equations      164—167 179
$\tau$ 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 $0\leq x\leq 1$      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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