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Fike C.T. Ч Computer Evaluation of Mathematical Functions
Fike C.T. Ч Computer Evaluation of Mathematical Functions



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Ќазвание: Computer Evaluation of Mathematical Functions

јвтор: Fike C.T.

јннотаци€:

This book developed from lecture notes for a class that I have taught for several years at the IBM Systems Research Institute. This is an eight-week class on mathematical methods used in function evaluation routines for digital computers. So far as I know, this is the first textbook to contain all the main topics relevant to this subject. Indeed, much of the material was not available in books at all and could be found only in original papers and in the documentation of existing computer programs. As the reader can see, the topics included in the book make a varied and interesting assortment and would probably not be grouped together under any other heading.


язык: en

–убрика: Computer science/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1968

 оличество страниц: 238

ƒобавлена в каталог: 20.12.2012

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
Abramowitz, M.      108 136 193 201 208 215
Absolute error      see Error
Accuracy objectives      3 11 14Ч16 28Ч30
Achieser, N. I.      66 77 89 155 165 176
Algol      see Programming languages
Alternating series      120
Analytic function      131
Approximation methods      see Asymptotic expansions Chebyshev Chebyshev Continued Economization MaehlyТs Minimax Minimax and
Argument range, infinite      39 157 169 206
Argument range, reduction      see Reduction of argument range
Argument range, singularities in      40 43 131 136 137
Argument range, transformation to (-1, 1)      115 139 181
Arguments      see also Argument range and Reduction of argument range
Arguments, invalid      4
Arguments, perturbation of      13Ч16
Arguments, special, handling of      3
Arithmetic      see also Precision and Floating-point base
Arithmetic, complex      2 138
Arithmetic, fixed-point      2 6 10 16
Arithmetic, floating-point      2 7 9Ч11 146
Arithmetic, quasi-arithmetic, computer      8 9 14 16
Arthurs, A. M.      136
Assembly language      see Programming languages
Asymptotic expansions      1 42 206Ч217
Asymptotic expansions, approximations similar to      212Ч215
Asymptotic expansions, definition      206
Asymptotic expansions, derivations of      208
Asymptotic expansions, examples      207
Asymptotic expansions, notation      207
Asymptotic expansions, truncation of, to obtain approximations      209Ч212
Bab$\check{u}$skova, R.      136
Baskervill, M.      150
Belaga, E. G.      60
Best-fit approximations      see Minimax polynomial approximations and Minimax rational approximations
Bisection, method of      85 89 173 175
Blanch, G.      143 150 207
Boas, R. P.      108
Boersma, J.      136
Buck, R. C.      108
Bulirsch, R.      136
Burgoyne, F. D.      124 136
Carslaw, H. S.      103 108
Chawla, M. M.      132 136
Chebyshev approximations      see Minimax polynomial approximations and Minimax rational approximations
Chebyshev interpolation      115 125Ч130 139
Chebyshev interpolation, error bounds      126 127 129
Chebyshev interpolation, even and odd functions      127 129
Chebyshev interpolation, nodes      126 127 129
Chebyshev interpolation, relation to Chebyshev series      128
Chebyshev polynomials      85 95Ч114
Chebyshev polynomials, critical points      85 97 98 111 113 172
Chebyshev polynomials, definition      95
Chebyshev polynomials, graphs      98 99
Chebyshev polynomials, minimax property      99 100 126
Chebyshev polynomials, orthogonality properties      96 98 128 140
Chebyshev polynomials, power series form (tables)      97 101
Chebyshev polynomials, shifted      100 101 138
Chebyshev polynomials, use of Formac to generate      114 134 139
Chebyshev polynomials, zeros      97 98 110 126 127
Chebyshev series      46 102Ч105 115 118 122 130Ч136 139 140 185Ч189 210 214
Chebyshev series, approximations obtained by truncating      115 117 118 130Ч136 139 140
Chebyshev series, compared with power series      131
Chebyshev series, computing numerical values for coefficients      105Ч108 112 1 13 128 136
Chebyshev series, convergence      102 103 131 137 186
Chebyshev series, conversion to power series      96 97 134 135
Chebyshev series, evaluation of finite      53 134Ч136 140
Chebyshev series, even and odd functions      103 134 140
Chebyshev series, expanding functions in      105 109 111 113
Chebyshev series, expressing polynomial in      96 97 110
Chebyshev series, relation to Chebyshev interpolation      129
Chebyshev series, relation to Fourier series      103
Chebyshev series, relation to minimax polynomial approximations      104 111Ч114
Chebyshev series, relation to polynomial economization      140
Chebyshev series, shifted      102 103 105 111Ч1 13 116 130 140 214 216
Chebyshev, P. L.      66
ChebyshevТs theorem on minimax approximations, polynomials      66 67 70 73 80 81 100 154
ChebyshevТs theorem on minimax approximations, rational functions      155 156
Cheney, E. W.      89 99 104 108 176 188 201
Chervonenkis, O. A.      35 47 90 109 137
Chopped arithmetic      see Truncated arithmetic
Circuitry, for square-root evaluation      23
Clark, N. A.      18
Clendenin, W. W.      136
Clenshaw, C. W.      46 105 107 108 112 1 13 136 139 214 215
Cody, W. J.      18 34 54 60 86 89 165 167 173 176
Collinge, R.      68 86 91
Compatibility of programs      5 36
Computers, BESM      56
Computers, CDC 1060      4 35 47 177
Computers, CDC 3600      34 81
Computers, CDC 7700      31 35 47 122 125 137 150 183 199 203
Computers, EE System/4      2
Computers, GE-724/734      122 137 188 203
Computers, IBM 172      18 122 138 145
Computers, IBM 7030 (Stretch)      23 35 65 91 122 138
Computers, IBM 704      188
Computers, IBM 709      188
Computers, IBM 7090/7094      57 61 122 130 138 149 150
Computers, IBM 9020      61 63 122 138
Computers, IBM System/360      2 18 22 31 34 35 42 45 47 54 58 61 91 92 130 134 138 146 149 150 151 167 183 196 203 215
Computers, IBM System/4 Pi      145 150
Computers, RCA Spectra 70      2 47
Constraints on minimax approximations      see Minimax polynomial approximations and Minimax rational approximations
Continued fractions      1 201 202
Continued fractions, convergence of expansions      191Ч193
Continued fractions, convergents      180 191 194 196 204
Continued fractions, economization      180 196Ч201 204
Continued fractions, evaluation by nested division      143
Continued fractions, expanding functions in      191Ч194 196 203 205
Continued fractions, expressing rational approximations as terminating      see Rational-function evaluation
Continued fractions, nonterminating      143 191
Continued fractions, Pad$\acute{e}$ approximations and      183 195 196
Continued fractions, terminating      142 191
Continued fractions, use of convergents as approximations      180 190Ч196 204
Continued fractions, value of      191
Convergence, Chebyshev series      102 103 130 131 137
Convergence, continued fraction expansions      193
Convergence, cube-root iterations      32 37
Convergence, cubic      32 36 37
Convergence, geometric      131
Convergence, linear      172
Convergence, order of      36
Convergence, power series      193
Convergence, quadratic      23 32 36 84 172
Convergence, RemezТ method      84 172 173
Convergence, square-root iterations      23 36
Convergents of continued fractions      see Continued fractions
Copson, E. T.      208 215
Critical points      see Chebyshev polynomials Minimax and
Cube root      see Functions and NewtonТs method
Cubic convergence      see convergence
Curtis, A.      172 176 214 215
De Bruijn, N. G.      208 215
de la Vall$\acute{e}$e Poussin, C. J.      90
Defective rational approximations      see Minimax rational approximations
Dekkers, A. J.      89 108 176
Dijkstra, E. W.      22 34
DOUBLE PRECISION      see Precision
Duijvestijn, A. J. W.      89 108 176
Economical polynomial evaluation methods      see Polynomial evaluation methods
Economization, continued fraction      180 196Ч201 202 204
Economization, polynomial      115 122Ч125 138Ч140
Eisman, S. H.      60
Elliott, D.      132 136 182 201
Equal-ripple graphs      see Chebyshev polynomials Minimax and
Erd$\ddot{o}$s, P.      89
Erdelyi, A.      208 215
Error, absolute      5Ч8 64 66 67 70 116Ч118 127 132 156 181 182 187 188
Error, conditions      4
Error, contrast between absolute and relative error      5Ч8 65 67 91 116Ч118 126
Error, functions      8 66 67 73 74
Error, generated      9
Error, in function argument      11 12 15
Error, logarithmic      8 20
Error, minimax      64Ч67 104 111 113
Error, relative      5Ч8 64 67 70Ч72 116- 126 156 181 182 184 188 189 200 210
Error, rounding      see Rounding error
Error, sources of      8Ч12
Error, statistical estimates      16
Error, truncation      8Ч12
Evaluating approximations      see Polynomial evaluation and Rational-function evaluation
Eve, J.      34 54 60 89
Even functions      72 73 85 88 91 96 103 123 127 129 134 140Ч142 143 145 162 163 175 177 188 191 192 194 197Ч200
Fair, W. G.      202
Fike, C. T.      28 34 35 60
Fixed-point arithmetic      see Arithmetic
Floating-point arithmetic      see Arithmetic
floating-point base      2 7 10 11 19 24 25 32 35 41 42 48 49
FORMAC      see Programming languages
Forsythe, G.      10 18
FORTRAN      see Programming languages
Fourier series, relation to Chebyshev series      103
Fox, L.      97 108
Fr$\ddot{o}$berg, C. Ч E.      147 150 151 176
Fractional form for rational approximations      see Rational-function evaluation
Fraser, W.      34 86 89 102 103 108 157 173 176
Functions, arccosine      1 20 47 62 104 109 137
Functions, arcsine      20 38 40 41 47 49 57 61 62 109 111 116 119 131 202
Functions, arctangent      1 4 20 47 48 50 56 70 77 102 109 116 119 131 137Ч139 147 186 202Ч204
Functions, Bessel      2 53 61 91 107 108 1 13 137
Functions, Bessel function integrals      176 177
Functions, complementary error function      177 207Ч214
Functions, cosine      1 3 14 15 19 21 34 47 68 69 75 79 82 88 93 104 107 109 113 115 119Ч121 124 125 130 133 134 137 149Ч151 157Ч159 161Ч163 175 178 193 202 203
Functions, cosine integral      109 136
Functions, cube root      22 31Ч34 37 38 92 94 176 178
Functions, DawsonТs      109
Functions, Debye      177
Functions, digamma      90
Functions, elliptic integrals      2 89 202
Functions, error (erf $x$)      1 42 109 116 136 215
Functions, error function integrals      137
Functions, exponential      1 4 13 19 31 39Ч45 47Ч50 56 58 63 70 78 79 82 91Ч93 107 109 112 131 137Ч 139 148 149 152 165 166 176 178 183 192 194 196 202 215
Functions, exponential integral      109 207 215 216
Functions, Fermi Ч Dirac integrals      91 176
Functions, Fresnel integrals      2 109 136
Functions, gamma      1 20 38 42 43 49 68 81 92 111 112 114 157 177 179 206 207 215Ч217
Functions, hyperbolic cosine      1 21 205
Functions, hyperbolic sine      20 21 40 47 49 215
Functions, hyperbolic tangent      1 47 160 181 183Ч185 188Ч190 192 196 200Ч 201 204
Functions, Kelvin      124 136
Functions, logarithm      1 4 13 17 31 39 48 61 70Ч72 92 103 109 113 119 131 139 140 152 161 162 192 202
Functions, logarithm of gamma      176 207 216
Functions, modified Bessel      108 109 136
Functions, normal probability integral      39 139 169 215
Functions, psi      90
Functions, roots, higher-order      2 31 34 39 47 150 202
Functions, sine      1 4 8 11 20 34 39 41 45Ч48 50 71 93 109 112 117 118 121 136Ч139 159 160 164 193 202
Functions, sine integral      109 136
Functions, square root      1 3Ч5 16 18 20 22Ч37 38 80 81 86Ч89 91 93 167Ч169 173 174 176 178 179
Functions, tangent      1 48 116 204
Gargantini, I.      173 176
Garrett, J. R.      47 109 137
Gaussian elimination      86 88
Gaussian quadrature      111
Generated error      9
Gilormini, C.      81 89 177
Gower, J. C.      34 36
Guard digits      10
Hammer, C.      18
Hamming, R. W.      99 108
Hart, J. F.      34 38 46 86 89 157 170 173 176
Hastings, C.      38 47 60 63 70 77 81 92
Henrici, P.      126 136
HeroТs method      see NewtonТs method
Hillstrom, K. E.      18 176
Hitchcock, A. J. M.      137
Hornecker, G.      83 90 109 114 132 133 137 177
HornerТs method      51 60
Householder, A. S.      16 18
Hummer, D. G.      109
IBM System/360 floating-point speeds      146 151
Infinite argument range      see Argument range
Instability and propagation of rounding error      12Ч16 20 46
Invalid arguments      4
Irreducible rational functions      141
Iterative function evaluation methods, cube-root evaluation      22 31Ч38
Iterative function evaluation methods, other functions      22 34 35 38
Iterative function evaluation methods, square-root evaluation      22Ч38
James, W.      34
Jarratt, P.      34
JOVIAL      see Programming languages
Khovanskii, A. N.      190 193 194 196 202
King, R.      34 90
Knuth, D. E.      60
Kogbetliantz, E. G.      22 34 47 137 150 188 196 202
Kuki, H.      17 18 20
Lagrange remainder formula      125
Lance, G. N.      137
Lanczos, C.      95 109
Lawson, C. L.      46 89 177
Least-maximum approximations      see Minimax polynomial approximations and Minimax rational approximations
Least-squares approximations      65 137
Lebedev, N. N.      208 215
Length, program      3 45
Libraries, manufacturer supplied      1 2 31 42 45 58 62 65 76 92 122 125 130 134 146 148 167 183 188 196 199 215
Logarithmic error      8 20
Luke, Y. L.      109 202
Lyusternik, L. A.      22 35 38 47 55 56 60 83 90 95 104 105 107 109 137 139 193
Maclaurin series      see Power series
Macon, N.      150
Maehly, H. J.      35 47 177 188 199 202
MaehlyТs method      180 185Ч190 204
McCarroll, R.      136
Meinardus, G.      78 84 86 90 177
Mesztenyi, C.      60 89 177
Miller, G. F.      46 47 132 136 137
Minimax polynomial approximations      27 34 56 62 63 64Ч94 126 127 132 136 137 154
Minimax polynomial approximations, and least-squares approximations      65
Minimax polynomial approximations, ChebyshevТs theorem      66 67 70 73 81 100 154
Minimax polynomial approximations, comparison with rational approximations      155 161 162
Minimax polynomial approximations, contrast between absolute error, relative error      65 67 91
Minimax polynomial approximations, critical points of error curves      67 73Ч 75 81 83
Minimax polynomial approximations, equal-ripple error curves      67 68 69 75 81
Minimax polynomial approximations, estimation of minimax error      78 79 104 111 113
Minimax polynomial approximations, examples      68Ч72 75 76 80Ч83 86-89 93
Minimax polynomial approximations, HorneckerТs method for obtaining      83 132 133
Minimax polynomial approximations, relation to near-minimax approximations      65 70 76Ч78 92 115
Minimax polynomial approximations, remarks on use of      65 115
Minimax polynomial approximations, RemezТ method for obtaining      83Ч89 140
Minimax polynomial approximations, standard error functions      73Ч75 83 86 91
Minimax polynomial approximations, to even or odd functions      72 73 85 88 91
Minimax polynomial approximations, when F ($x$) has a zero in ($a$, $b$)      70Ч72 74
Minimax polynomial approximations, with nodes or other constraints      79Ч83 85 89 90 93 94
Minimax rational approximations      147 154Ч179
Minimax rational approximations, admissible rational functions      154
Minimax rational approximations, ChebyshevТs theorem      155 156
Minimax rational approximations, comparison with polynomial approximations      161 162 165 167 168
Minimax rational approximations, contrast between absolute error, relative error      5Ч8 65 67 91 116Ч118 126
Minimax rational approximations, critical points of error curves      156 163 164 173 179
Minimax rational approximations, defective approximations      155 163 179
Minimax rational approximations, equal-ripple error curves      156 158 160 178 188
Minimax rational approximations, examples      157Ч160 167Ч169
Minimax rational approximations, for infinite argument range      157 169
Minimax rational approximations, relation to near-minimax approximations      164Ч166 180
Minimax rational approximations, remarks on use of      180
Minimax rational approximations, RemezТ method for obtaining      163 170Ч179 214
Minimax rational approximations, standard error functions      163 164 170 178
Minimax rational approximations, to even or odd functions      162Ч164
Minimax rational approximations, when F ($x$) has a zero in ($a$, $b$)      156 160 164
Minimax rational approximations, with nodes or other constraints      79 166Ч169 175 177
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