Fike C.T. — Computer Evaluation of Mathematical Functions :: Электронная библиотека попечительского совета мехмата МГУ

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Fike C.T. — Computer Evaluation of Mathematical Functions

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Название: Computer Evaluation of Mathematical Functions

Автор: Fike C.T.

Аннотация:

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:

Рубрика: Computer science/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1968

Количество страниц: 238

Добавлена в каталог: 20.12.2012

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Предметный указатель

Abramowitz, M.      108 136 193 201 208 215
Absolute error      see Error
Accuracy objectives      3 11 14—16 28—30
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 Maehly’s 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      13—16
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 9—11 146
Arithmetic, quasi-arithmetic, computer      8 9 14 16
Arthurs, A. M.      136
Assembly language      see Programming languages
Asymptotic expansions      1 42 206—217
Asymptotic expansions, approximations similar to      212—215
Asymptotic expansions, definition      206
Asymptotic expansions, derivations of      208
Asymptotic expansions, examples      207
Asymptotic expansions, notation      207
Asymptotic expansions, truncation of, to obtain approximations      209—212
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      115 125—130 139
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 95—114
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      46 102—105 115 118 122 130—136 139 140 185—189 210 214
Chebyshev series, approximations obtained by truncating      115 117 118 130—136 139 140
Chebyshev series, compared with power series      131
Chebyshev series, computing numerical values for coefficients      105—108 112 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      53 134—136 140
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      104 111—114
Chebyshev series, relation to polynomial economization      140
Chebyshev series, shifted      102 103 105 111—1 13 116 130 140 214 216
Chebyshev, P. L.      66
Chebyshev’s theorem on minimax approximations, polynomials      66 67 70 73 80 81 100 154
Chebyshev’s 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      191—193
Continued fractions, convergents      180 191 194 196 204
Continued fractions, economization      180 196—201 204
Continued fractions, evaluation by nested division      143
Continued fractions, expanding functions in      191—194 196 203 205
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      180 190—196 204
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, Remez’ 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 Newton’s 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      180 196—201 202 204
Economization, polynomial      115 122—125 138—140
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      5—8 64 66 67 70 116—118 127 132 156 181 182 187 188
Error, conditions      4
Error, contrast between absolute and relative error      5—8 65 67 91 116—118 126

Error, functions      8 66 67 73 74
Error, generated      9
Error, in function argument      11 12 15
Error, logarithmic      8 20
Error, minimax      64—67 104 111 113
Error, relative      5—8 64 67 70—72 116- 126 156 181 182 184 188 189 200 210
Error, rounding      see Rounding error
Error, sources of      8—12
Error, statistical estimates      16
Error, truncation      8—12
Evaluating approximations      see Polynomial evaluation and Rational-function evaluation
Eve, J.      34 54 60 89
Even functions      72 73 85 88 91 96 103 123 127 129 134 140—142 143 145 162 163 175 177 188 191 192 194 197—200
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. — 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 77 102 109 116 119 131 137—139 147 186 202—204
Functions, Bessel      2 53 61 91 107 108 1 13 137
Functions, Bessel function integrals      176 177
Functions, complementary error function      177 207—214
Functions, cosine      1 3 14 15 19 21 34 47 68 69 75 79 82 88 93 104 107 109 113 115 119—121 124 125 130 133 134 137 149—151 157—159 161—163 175 178 193 202 203
Functions, cosine integral      109 136
Functions, cube root      22 31—34 37 38 92 94 176 178
Functions, Dawson’s      109
Functions, Debye      177
Functions, digamma      90
Functions, elliptic integrals      2 89 202
Functions, error (erf )      1 42 109 116 136 215
Functions, error function integrals      137
Functions, exponential      1 4 13 19 31 39—45 47—50 56 58 63 70 78 79 82 91—93 107 109 112 131 137— 139 148 149 152 165 166 176 178 183 192 194 196 202 215
Functions, exponential integral      109 207 215 216
Functions, Fermi — Dirac integrals      91 176
Functions, Fresnel integrals      2 109 136
Functions, gamma      1 20 38 42 43 49 68 81 92 111 112 114 157 177 179 206 207 215—217
Functions, hyperbolic cosine      1 21 205
Functions, hyperbolic sine      20 21 40 47 49 215
Functions, hyperbolic tangent      1 47 160 181 183—185 188—190 192 196 200— 201 204
Functions, Kelvin      124 136
Functions, logarithm      1 4 13 17 31 39 48 61 70—72 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 45—48 50 71 93 109 112 117 118 121 136—139 159 160 164 193 202
Functions, sine integral      109 136
Functions, square root      1 3—5 16 18 20 22—37 38 80 81 86—89 91 93 167—169 173 174 176 178 179
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
Hero’s method      see Newton’s method
Hillstrom, K. E.      18 176
Hitchcock, A. J. M.      137
Hornecker, G.      83 90 109 114 132 133 137 177
Horner’s 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      12—16 20 46
Invalid arguments      4
Irreducible rational functions      141
Iterative function evaluation methods, cube-root evaluation      22 31—38
Iterative function evaluation methods, other functions      22 34 35 38
Iterative function evaluation methods, square-root evaluation      22—38
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
Maehly’s method      180 185—190 204
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 64—94 126 127 132 136 137 154
Minimax polynomial approximations, and least-squares approximations      65
Minimax polynomial approximations, Chebyshev’s 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 73— 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      68—72 75 76 80—83 86-89 93
Minimax polynomial approximations, Hornecker’s method for obtaining      83 132 133
Minimax polynomial approximations, relation to near-minimax approximations      65 70 76—78 92 115
Minimax polynomial approximations, remarks on use of      65 115
Minimax polynomial approximations, Remez’ method for obtaining      83—89 140
Minimax polynomial approximations, standard error functions      73—75 83 86 91
Minimax polynomial approximations, to even or odd functions      72 73 85 88 91
Minimax polynomial approximations, when F () has a zero in (, )      70—72 74
Minimax polynomial approximations, with nodes or other constraints      79—83 85 89 90 93 94
Minimax rational approximations      147 154—179
Minimax rational approximations, admissible rational functions      154
Minimax rational approximations, Chebyshev’s theorem      155 156
Minimax rational approximations, comparison with polynomial approximations      161 162 165 167 168
Minimax rational approximations, contrast between absolute error, relative error      5—8 65 67 91 116—118 126
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      157—160 167—169
Minimax rational approximations, for infinite argument range      157 169
Minimax rational approximations, relation to near-minimax approximations      164—166 180
Minimax rational approximations, remarks on use of      180
Minimax rational approximations, Remez’ method for obtaining      163 170—179 214
Minimax rational approximations, standard error functions      163 164 170 178
Minimax rational approximations, to even or odd functions      162—164
Minimax rational approximations, when F () has a zero in (, )      156 160 164
Minimax rational approximations, with nodes or other constraints      79 166—169 175 177

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