Nash - Compact Numerical Methods for Computers (523163), страница 60
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R., 144, 207One-dimensional problems, 148O’Neill, R., 171, 178One-sided transformation, 136Ones matrix, 254Operations,arithmetic, 5Optimisation, 142constrained, 3Ordering of eigenvalues, 127, 134Ordinary differential equations, 20Orthogonal vectors, 25, 32Orthogonalisation,by plane rotations, 32of matrix rows, 49, 54Orthogonality,of eigenvectors of real symmetric matrix, 119of search directions, 198of vectors, 26Osborne, M. R., 85, 142, 186, 226Paige, C. C., 234Parabolic interpolation, 151Parabolic inverse interpolation, 152, 199, 210formulae, 153Parameters, 142Parlett, B. N., 234Partial penalty function, 222Partial pivoting, 75Pascal, 12Pauling, L., 28Penalty functions, 222, 223Penrose, R., 26Penrose conditions for generalised inverse, 26Permutations or interchanges, 75Perry, A., 144, 230Peters, G., 105Pierce, B.
O., 139Pivoting, 75, 93, 95, 97Plane rotation, 32, 49, 54, 126formulae, 34Plauger, P. J., 12Plot or graph of function, 151Polak, E., 198, 199Polak-Ribiere formula, 199Polynomial roots, 143, 145Positive definite iteration matrix, 192Positive definite matrix, 22, 120, 188, 197, 211,235, 241, 243Positive definite symmetric matrix, 83inverse of, 24Powell. M. J. D., 185, 199Power method for dominant matrixeigensolution, 102Precision,double, 9, 14extended, 9, 14machine, 5, 46, 70Price, K., 90Principal axes of a cube, 125Principal components, 41, 46Principal moments of inertia, 125Product of triangular matrices, 74Program,choice, 14coding, 14compactness, 12maintenance, 14readability, 12reliability, 14testing, 14Programming,mathematical, 13structured, 12Programming language, 11, 15Programs,manufacturers’, 9sources of, 9Pseudo-random numbers, 147, 166, 240QR algorithm, 133QR decomposition, 26, 49, 50, 64Quadratic equation, 85, 244Quadratic form, 22, 89, 190, 198, 235Quadratic or parabolic approximation, 15 1Quadratic termination, 188, 199, 236Quantum mechanics, 28Quasi-Newton methods, 187R2 statistic, 45, 63Radix, 7Ralston, A., 95, 104, 121, 127, 218Rank, 20Rank-deficient case, 24, 25, 55IndexRayleigh quotient, 122, 123, 138, 200, 234, 242,244minimisation, 250minimisation by conjugate gradients, 243Rayleigh-Ritz method, 138Readability of programs, 12Real symmetric matrix, 119Reconciliation of published statistics, 204Recurrence relation, 166, 198, 235, 246Reduction,of simplex, 168, 170to tridiagonal form, 133Reeves, C.
M., 198, 199References, 263Reflection of simplex, 168, 169, 172Regression, 92stepwise, 96Reid, J. K., 234Reinsch, C., 13, 83, 97, 102, 110, 133, 137, 251Reliability, 14Re-numeration, 98Re-ordering, 99Residual, 21, 45, 250uncorrelated, 56, 70weighted, 24Residuals, 142, 144, 207for complex eigensolutions, 117for eigensolutions, 125, 128sign of, 142, 207Residual sum of squares, 55, 79computation of, 43Residual vector, 242Restart,of conjugate gradients for linear equations, 236of conjugate gradients minimisation, 199of Nelder-Mead search, 171Ribiere, G., 198, 199Ris, F.
N., 253Root-finding, 143, 145, 148, 159, 160,239Roots,of equations, 142of quadratic equation, 245Rosenbrock, H. H., 151, 182, 196, 208, 209Rounding, 7Row,orthogonalisation, 49, 54permutations, 75Ruhe, A., 234, 242Rutishauser, H., 127, 134Saddle points, 143, 146, 159, 208Safety check (iteration limit), 128Sampson, J. H., 74Sargent, R. W. H., 190277Saunders, M. A., 234Scaling,of Gauss -Newton method, 211of linear equations, 80Schwarz, H. R., 127Search,along a line, 143, 148directions, 192, 197Sebastian, D. J., 190Secant algorithm, 162Seidel, L., 131sgn (Signum function), 34Shanno, D. F., 190Shift of matrix eigenvalues, 103, 121, 136, 242Shooting method, 239Short word-length arithmetic, 159, 191Signum function, 34Simplex, 168size, 171Simulation of insurance scheme, 165Simultaneous equations,linear, 19nonlinear, 142, 144Single precision, 134, 159Singular least-squares problem, 240Singular matrix, 20Singular-value decomposition, 26, 30, 31, 54, 66,69, 81, 119algorithm, 36alternative implementation, 38updating of, 63Singular values, 30, 31, 33, 54, 55ordering of, 33ratio of, 42Small computer, 3Software,mathematical, 10Soland, R.
M., 144, 230Solution,least-squares, 22minimum-length least-squares, 22Sorenson, H. W., 198Sparse matrix, 20, 23, 102, 234Spendley, W., 168‘Square-root-free Givens’ reduction, 50Standardisation of complex eigenvector, 111Starting points, 146Starting vector,power method, 104Statistical computations, 66Steepest descent, 186, 199, 208, 209, 211Stegun, I. A., 4Step adjustment in success-failure algorithm, 154Step length, 178, 187, 197, 200, 242Step-length choice, 158278Compact numerical methods for computersStep length for derivative approximation, 219Stepwise regression, 96Stewart, G. W., 40, 234Structured programming, 12Styan, G.
P. H., 56Substitution for constraints, 221Success-failure,algorithm, 151, 153search, 152Success in function minimisation, 226Sum of squares, 22, 23, 39,42, 55, 79and cross products, 66nonlinear, 207total, 45Surveying-data fitting, 24, 240Swann, 182, 225Sweep or cycle. 35, 49, 126Symmetric matrix, 135, 243Symmetry.use in eigensolution program, 134Synge, J. L., 125System errors, 4Taylor serves, 190, 209Tektronix 4051, 156Test matrices, 253Test problems, 226Time series, 180Tolerance, 5, 15, 35, 40, 54for acceptable point search, 190for conjugate gradients least-squares, 240for deviation of parameters from target, 204for inverse iteration by conjugate gradients,243Total sum of squares, 45Transactions on Mathematical Software.
11Transposition, 22Traub, J. F., 143, 148Trial function, 28Triangle inequality, 22Triangular decomposition, 74Triangular matrix, 72Triangular system,of equations, 72of linear equations, 51Tridiagonal matrix, 251Truncation, 7Two-point boundary value problem, 238Unconstrained minimisation. 142Uncorrelated residuals, 56, 70Uniform distribution. 167Unimodal function, 149Unit matrix, 29Univac 1108, 56, 120Updating,formula, 190of approximate Hessian, 189, 192V-shaped triple of points, 152Values,singular, see Singular valuesVarga, R. S., 83Variable metric.algorithms, 198methods, 186, 187, 223, 228, 233Variables. 142Variance computation in floating-pointarithmetic. 67Variance of results from ‘known’ values, 241Variation method, 28Vector. 19, 30null, 20.
32residual. 21Weighting.for nonlinear least-squares, 207of constraints, 222in index numbers. 77Wiberg, T., 242Wilkinson, J. H., 13, 28, 75, 83. 86, 97, 102, 105,110, 119, 127, 133, 137, 251, 253, 254W+matrix, 254W- matrix, 108, 254Wilson, E. B., 28Yourdon.
E., 12Zambardino, R. A., 13.
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