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| Scarborough J.B. — Numerical Mathematical Analysis |
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| Предметный указатель |
Absolute error 3
Accuracy in determination of arguments 24
Accuracy in evaluation of formulas 19
Accuracy of addition 9
Accuracy of averages 10
Accuracy of division 13 17
Accuracy of interpolation formulas 84
Accuracy of linear interpolation 94
Accuracy of logs and antilogs 15 18
Accuracy of multiplication 12 17
Accuracy of powers and roots 14 18
Accuracy of products and quotients 12 13 17
Accuracy of subtraction 11 12
Adams, J.C. 267
Adams, J.C., method of 267
Adams, J.C., method of, remarks on 272
Addition, errors of 9
Adopted values (of physical constants) 20
Alternating series, error in 30
Antilogarithms, accuracy of 16 17
Approximate calculations 1
Approximate numbers 1
Arguments, accuracy in determination of 24
Arguments, exponential functions 26
Arguments, logarithms 24
Arguments, trigonometric functions 25
Asymptotic series 140 298
Average deviation 323
Average deviation, error 320
Averages, accuracy of 10
Averages, method of 357
Ballistic equations 251
Bessel's formula of interpolation 64
Bessel's formula of interpolation for interpolating to halves 64
Bessel's formula of interpolation, power series form of 81
Bessel's formula of interpolation, symmetrical form of 65
Binomial series 30
Binomial series, remainder term of 31
Carvallo 217
Caution in use of empirical formulas 405
Caution in use of quadrature formulas 142
Central-Difference formulas of interpolation 56
Central-Difference formulas, quadrature 124
Central-Difference formulas, quadrature, geometric significance of 128
Central-Difference formulas, quadrature, remainder terms in 165
Charlier, C.L. 169
Check formulas, for 12 ordinates 396 397
Check formulas, for 24 ordinates 401
Chevilliet's formula 161
Complex roots, computation of, by Graeffe's method 205
Complex roots, detection of 206
Convergence of iteration process, for algebraic and transcendental equations 186 193
Convergence of iteration process, for differential equations 256 257
Cubature, formula for 146
Cubature, general statement concerning 148
Cubature, mechanical 117 146
Derivatives and differences, relation between 86
Derivatives, nth 33
Derivatives, partial, of tabulated functions 117
Detection of complex roots 206
Deviation, average 323
Deviation, standard 323
Diagonal difference table 40
Difference of a polynomial 45
Difference, double 103
Differences 40
Differential equations, numerical solution of, accuracy of 264 275
Differential equations, numerical solution of, by Adams's method 267
Differential equations, numerical solution of, by iteration method 218
Differential equations, numerical solution of, by Milne's method 280
Differential equations, numerical solution of, by successive approximations 218
Differential equations, numerical solution of, by the Runge — Kutta method 273
Differential equations, numerical solution of, principle of 218
Differential equations, numerical solution of, starting the solution 222 267
Differentiation, Numerical 114
Differentiation, partial, of tabulated functions 117
Direct measurements 317
Division, accuracy of 13 17
Double differences 103
Double interpolation, by repeated single interpolation 96
Double interpolation, formula for 106
Double interpolation, formula for, remainder term of 111 112
Empirical formulas 351
Empirical formulas, caution in use of 405
Empirical formulas, finding best type of 383
Empirical formulas, finding constants in, by method of averages 357
Empirical formulas, finding constants in, by method of least squares 363
Empirical formulas, finding constants in, by plotting 351
Empirical formulas, finding constants in, when both variables are subject to error 380
Empirical formulas, finding constants in, when residuals are weighted 370
Empirical formulas, general case of non-linear formulas 374
Equal effects, method of 340
Equal effects, principle of 21
Equations, algebraic and transcendental 171
Equations, ballistic 251
Equations, differential, numerical solution of, accuracy of 264 275
Equations, differential, numerical solution of, by Adams's method 267
Equations, differential, numerical solution of, by Milne's method 280
Equations, differential, numerical solution of, by successive approximations 218
Equations, differential, numerical solution of, by the Runge — Kutta method 273
Equations, differential, numerical solution of, principle of 218
Equations, differential, numerical solution of, starting the solution 222 267
Equations, locating the roots of 171
Equations, solution of, by iteration 184 191
Equations, solution of, by method of false position 174
Equations, solution of, by Newton — Raphson method 178 187
Equations, solution of, by repeated plotting 177
Error function 288
Error in difference table, effect of 43 44
Error, absolute 3
Error, average 320
Error, inherent, in Euler's formula 169
Error, inherent, in Gauss's formula 167
Error, inherent, in Newton — Raphson method 183
Error, inherent, in Simpson's one-third rule 155 156 163
Error, inherent, in Simpson's three-eighths rule 155
Error, inherent, in Weddle's rule 155
Error, mean square 318
Error, mean, of gun 301
Error, of measurement 308
Error, percentage 3
Error, probable 319
Error, probable, percentage 338
Error, probable, relative 338
Error, relative 3
Errors, accidental 284
Errors, general formula for 7
Errors, of addition 9
Errors, of subtraction 11
Errors, relatively small 8
Errors, systematic 284
Euler 218
Euler's formula of quadrature 139
Euler's formula of summation 140
Euler's formula, compared with Simpson's rule 169
Euler's formula, error in 169
Euler, method of 223
Evaluation of formulas, accuracy in 19
Evaluation of probability integral 296
Exponential series, remainder term in 31 32
Extrapolation 53
False position, method of 174
Function G(v) 252
Function H(y) 251
Function table 96
Functional equation 290
Gauss 113 131
Gauss's quadrature formula 132
Gauss's quadrature formula, expressions for error in 167
Geometric significance of central-difference quadrature formulas 128
| Geometric significance of Simpson's rule 120
Geometric significance of Weddle's rule 121
Geometric significance of Weierstrass's theorems 39
Graeffe's root-squaring method 198
Graeffe's root-squaring method, for complex roots 205
Graeffe's root-squaring method, for equal roots 214
Graeffe's root-squaring method, for real roots 201
Graeffe's root-squaring method, principle of 198
Graphical determination of constants in empirical formulas 351
Graphical solution of equations 177
Hermite's formula 112
Horizontal difference table 41
Index of precision 292
Indirect measurements 317 337
Indirect measurements, fundamental problems of 339
Integrating ahead, formula for 227
Integration, numerical See "Numerical integration"
Interpolation series 84
Interpolation, accuracy of 84
Interpolation, Bessel's formulas for 64 65
Interpolation, definition of 38
Interpolation, double 96 104
Interpolation, inverse 75
Interpolation, Lagrange's formula for 73
Interpolation, Stirling's formula for 60
Interpolation, trigonometric 112
Inverse interpolation 75
Inverse interpolation, by Lagrange's formula 75
Inverse interpolation, by reversion of series 79
Inverse interpolation, by successive approximations 76
Iteration process for algebraic and transcendental equations 184 191
Iteration process for algebraic and transcendental equations, convergence of 186 193
Iteration process for differential equations 218 232
Iteration process for differential equations, convergence of 256 257
Iteration process for differential equations, maximum error due to 266
Lagrange's formula of interpolation 73
Lagrange's formula of interpolation, remainder term in 86 91
Lagrange's formula of interpolation, uses of 74
Law of accidental errors 284
Law of error of a function 296
Least squares, method of 363
Least squares, principle of 304
Linear interpolation, accuracy of 94
Linear interpolation, when permissible 95
Logarithmic series, remainder term in 32
Logarithms, accuracy of 15 16
Maclaurin's series 28
Maclaurin's series, remainder term in 28
Magnitude factor 329
Maxima and minima of tabulated functions 116
Mean error of gun 301
Mean square error 318
Measures of precision 318
Measures of precision, computation of, from residuals 324
Measures of precision, geometric significance of 322
Measures of precision, relations between 321
Merriman, Mansfield 380
Method of averages 357
Method of equal effects 340
Method of false position 174
Method of iteration see "Iteration"
Method of successive approximations 218
Milne's method 280
Milne, W.E. 280
Modulus of complex roots, theorem relating to 212
Moors, B.P. 134 135
Moulton, F.R. 218 252
Multiplication, accuracy of 12 17
Negligible effects, criterion for 341
Newton — Raphson method 179
Newton — Raphson method, geometric significance of 181
Newton — Raphson method, inherent error in 183
Newton's formula for backward interpolation 50 51
Newton's formula for forward interpolation 48 49
Newton's formula, as a power series 80
Normal equations 364 372
Normal equations, rule for writing down 364 372
Numerical differentiation 114
Numerical integration 117
Numerical integration, by central-difference quadrature formulas 124
Numerical integration, by Euler's formula 139
Numerical integration, by Gauss's formula 131
Numerical integration, by Simpson's rule 119
Numerical integration, by Weddle's rule 121
Numerical solution of differential equations, accuracy of 266 275
Numerical solution of differential equations, advantages and disadvantages of 282
Numerical solution of differential equations, by Adams's method 267
Numerical solution of differential equations, by Milne's method 280
Numerical solution of differential equations, by Runge — Kutta method 273
Numerical solution of differential equations, by successive approximations 218 232
Numerical solution of differential equations, starting the 222 267
Partial derivatives of tabulated functions 117
Pearson, Karl 112
Percentage error 3
Percentage error, probable error 338
Picard, E. 218
Polynomial, differences of a 45
Polynomial, formula when nth differences are constant 385
Polynomials, use of approximating 225
Powers and roots, accuracy of 14 15 18
Precision and accuracy, difference between 317
Precision measures 318
Precision measures, computation of, from residuals 324
Precision measures, geometric significance of 322
Precision measures, relations between 321
Principle of equal effects 21
Principle of Graeffe's method 198
Principle of iteration process 218
Principle of least squares 304 305 307
Probability equation 292
Probability integral and its evaluation 296
Probability of errors lying between given limits 286
Probability of hitting a target 299
Probable error and weight, relation between 323
Probable error in indirect measurements 338
Probable error of a function 338
Probable error of a gun 301
Probable error of arithmetic and weighted means 324
Probable error, computation of, from residuals 324
Probable error, definition of 319
Probable error, formulas for 324 327 338
Probable error, meaning of 337
Product, accuracy of 12 17
Product, relative error of 12
Propagation of errors 338
Quadrature formulas in equidistant ordinates 117
Quadrature formulas, caution in use of 142
Quadrature formulas, central-difference 124
Quadrature formulas, Euler's 139
Quadrature formulas, Gauss's 131
Quadrature, mechanical 117
Quotient, accuracy of 13 17
Quotient, relative error of 13
Reciprocals of roots, relations between coefficients and 208
Regula falsi method 174
Rejection of observations and measurements, rule for 346
Relation between derivatives and differences 86
Relation between probable error and weight 323
Relation between roots and coefficients 201 208
Relative accuracy of Simpson's rules 155
Relative error and significant figures 3
Relative error and significant figures, theorems concerning 3 6
Relative error of a product 12
Relative error of a quotient 13
Relative error probable 338
Remainder term, in Bessel's formulas 89 90
Remainder term, in central-difference quadrature formulas 166
Remainder term, in Euler's formula 169
Remainder term, in formula for double interpolation 111 112
Remainder term, in formula for interpolating to halves 90 91
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