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| Fike C.T. — Computer Evaluation of Mathematical Functions |
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| Предметный указатель |
Moler, C. B. 10 18
Monotonicity requirement 4
Moody, W. T. 90
Morrison, D. R. 22 35
Motzkin, T. S. 60
Moursund, D. G. 35
Murnaghan, F. D. 70 86 90 215
Near-minimax approximations, polynomial 65 76—78 81 89 92 108 115 127 130 132 136
Near-minimax approximations, rational 164—166 178 180 187 198
Nemeth, G. 109
Nested division see Rational-function evaluation
Nested multiplication see Polynomial evaluation
Newton — Raphson method 23 32 35 36
Newton’s method, cube-root evaluation 31—34 37 92 176
Newton’s method, square-root evaluation 22—31
Newton’s method, square-root evaluation, convergence 23
Newton’s method, square-root evaluation, number of iterations 23 24 28—31
Newton’s method, square-root evaluation, starting values (starting approximations) 23—28 34 35 75 76 80 86—88 91 93 167—169 173— 175 176 179
Nodes, minimax approximations with see Minimax polynomial approximations and Minimax rational approximations
Noise level, of rounding errors 11
Notation, summary of xii
Novodvorskii, E. P. 90
Objectives for function evaluation routines see Requirements for function evaluation routines
Odd functions 72 73 85 96 103 123 134 143 145 162 163 192 194 197 199
Operation times for arithmetic operations on IBM System 360 146
Operations, arithmetic, number required to evaluate, notation for operation counts 51 142
Operations, arithmetic, number required to evaluate, polynomials 52—55 134
Operations, arithmetic, number required to evaluate, rational functions 142
Optimal approximations see Minimax polynomial approximations and Minimax rational approximations
Orthogonal polynomials 97
Osborne, M. R. 172 176 214 215
Ostrowski, A. 60
Pad approximations 180 181—185 195 196 199 201 203 204
Pad table 183
Pan, V. Ya 60
Paszowski, S. 81 90
Performance standards 2—5 11
Perlin, I. E. 47 109 137
Perron, O. 183 190 193 195 202
Perturbation of argument 12—14 15 16
Picken, S. M. 108 136
Pinsker, I. S. 90
PL/I see Programming languages
Polynomial evaluation 1 51—63
Polynomial evaluation, Chebyshev series form 53 134 135
Polynomial evaluation, comparison of methods 54 59 135
Polynomial evaluation, economical methods 53—60 142
Polynomial evaluation, economical methods, 4th-degree 55 56 61
Polynomial evaluation, economical methods, 5th-degree 56 57
Polynomial evaluation, economical methods, 6th-degree 57—59
Polynomial evaluation, economical methods, higher degrees 54 55 62
Polynomial evaluation, nested multiplication 51—53 135 142 161
Polynomial evaluation, operation counts for 52 54 135
Polynomial evaluation, rounding error in 52 53 55 59 60 135
Pomentale, T. 177
Powell, M. J. D. 127 132 137
Power series 8 11 109 115 117 118— 122 131 135 138—140 181 183 185 186 193 195 202 210
Precision, double 2 10 14 16 17 19—21 25 31 34 45 46 58 76 137 146
Precision, extended 22
Precision, single 2 10 14 16 19—21 25 60 65 146 167 183
Precision, triple 14 16
Precision, variable 3 18 46 122
Program libraries see Libraries
Programming languages, Algol 1 4 136
Programming languages, assembly 4 25
Programming languages, Formac 63 114 134 139 153 205
Programming languages, FORTRAN 1 4 18 31 35 42 45 47 58 61 65 76 91 92 122 125 134 137 138 146 148 150 167 177 183 196 199 203 215
Programming languages, Jovial 62 122
Programming languages, PL/I 5 31
Quadratic convergence see Convergence
Quasi-arithmetic, computer 8 9 14 16
Ralston, A. 2 18 155 163 165 167 172 173 176 177 179 199 202
Random numbers 16 17 20 21
Range reduction see Reduction of argument range
Rational-function evaluation 141—153
Rational-function evaluation, continued fraction form 142—153 161 195
Rational-function evaluation, fractional form 141 142 147 151
Rational-function evaluation, nested division 143
Rational-function evaluation, operation counts for 142
Rational-function evaluation, rounding error in 149 150
Raymann, G. 91
Reduction of argument range 38—50 206
Reduction of argument range, propagation of rounding error in 45 46
Reduction of argument range, reasons for 38—40
| Reduction of argument range, specific techniques 40—43
Reduction of argument range, specific techniques, arcsine 41
Reduction of argument range, specific techniques, arctangent 48—50
Reduction of argument range, specific techniques, exponential 41—45 48 79
Reduction of argument range, specific techniques, gamma 42
Reduction of argument range, specific techniques, hyperbolic sine 40
Reduction of argument range, specific techniques, logarithm 48
Reduction of argument range, specific techniques, sine 41 45 46 48
Reduction of argument range, specific techniques, tangent 48
Reduction of argument range, subdividing range for piecewise approximation 42 43
Reduction of argument range, to (-1, 1) 115 116 139 180 181
Reduction of argument range, to arbitrary small interval 43—45 50 79 92
Relative error see Error
Remez, E. Ya 83 90
Remez’ method, for polynomial approximations 83—89 93 94 119 140
Remez’ method, for rational approximations 163 170 179 214
Requirements for function evaluation routines 2—7 11 14—16 79 166
Rice, J. 46 60 61 89 173 176 177
Rolle’s Theorem 74
Rounded arithmetic 2 9
Rounding error 8—12 (see also Error)
Rounding error, bounds for 16
Rounding error, control of 10 14 16 24 46
Rounding error, in function argument 11—16
Rounding error, in polynomial evaluation 52 53 59 60 135
Rounding error, in range reduction 45 46
Rounding error, in rational-function evaluation 149 150
Rounding error, instability and 12—16 19 20 46
Rounding error, propagation of 10 12—16 40 46 52 53 55 59 60 149 150
Rounding error, use of extra precision to control 10 14 16 17 46 60 86 150 173
Rounding error, use of fixed-point to control 10 60
Shifted Chebyshev polynomials see Chebyshev polynomials and Chebyshev series
Significant digits 5 6
Single precision see Precision
Singularities 40 43 103 131 136 137
Snyder, M. A. 97 102 105 108 109
Speed, program execution 3 31 45 51 141 145 146
Spielberg, K. 47 188 202
Square root see Functions and Newton’s method
Stability see Instability
Standard error functions see Minimax polynomial approximations and Minimax rational approximations
Starting values (starting approximations, for square-root and cubic-root evaluation) see Newton’s method
Stegun, I. 108 136 193 201 208 215
Sterbenz, P. H. 28 35
Stiefel, E. L. 90
Stoer, J. 173 176 177
Streamlined polynomial evaluation methods see Polynomial evaluation methods
Swarztrauber, P. N. 35 81 90
Synthetic division 51 (see also Nested multiplication)
Szeg , G. 96 109
Tables, use of large 44 47 50
Taylor series see Power series
Teichroew, D. 143 150
Telescoping 123 196 198 202
Testing of function evaluation routines 16—18
Thacher, H. C. 46 89 109 173 176 177
Thieleker, E. A. 18
Timings for IBM System 360
Timings for IBM System, arithmetic operations 146
Todd, J. 61 90 96 100 109
Transformation of argument see Reduction of argument range
Triple precision see Precision
Truncated (unrounded) arithmetic 9
Truncation error 8—12
Unrounded arithmetic see Truncated arithmetic
Variable precision see Precision
Variable-accuracy routines 122 135 136
Veidinger, L. 84 91
Vionnet, M. 61 62 107 109 137
Viskovatov’s method 193 194 196 203 205
Wall, H. S. 183 190 193 195 202
Watson, G. N. 208 215
Werner, H. 53 61 68 86 91
Wheeler, D. J. 2 35
Whittaker, E. T. 208 215
Wilkes, M. V. 2 35
Wilkinson, J. H. 10 18
Wimp, J. 109
Witzgall, C. 46 60 89 177
Wood, V. E. 137
Woodger, M. 46 136
Wrench, J. W. 70 86 90 215
Wynn, P. 202
Yanpol’skii, A. R. 35 47 60 90 109 137
Zeros, approximating a function near 70—72 74 93 116—118 121 156 160 181 188 200
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