Nash - Compact Numerical Methods for Computers (523163), страница 59
Текст из файла (страница 59)
C., 154, 182, 223, 225Doolittle method, 75, 80Double precision, 9, 14, 81, 83, 91Dow Jones index, 77E (notation), 17Eason, E. D., 182Eberlein, P., 110, 117ECLIPSE, 52, 96, 128, 153, 156, 159Effect of Jacobi rotations, 126Eigenproblem,generalised, 104total or complete, 119Eigenproblem of a real symmetric matrix,comparison of methods, 133Eigensolutions, 28, 31by singular-value decomposition, 123of a complex matrix, 117of a real symmetric matrix, 119Eigenvalue, 28, 135degenerate, 103Eigenvalue approximation in inverse iteration,108Eigenvalue decomposition of matrix, 135Eigenvalue problem,matrix or algebraic, 102Eigenvector, 28, 135Elementary matrices, 73Elementary operations on matrices, 73Elimination method for linear equations, 72Elimination of constraints, 22 1choice in, 223IndexEquations,linear, 19, 20, 51Equilibration of matrix, 80Equivalent function evaluations (efe’s), 227Euclidean norm, 22Examples,list of, 256Execution time, 227Expenditure minimisation, 156Exponents of decimal numbers, 17Expression of algorithms, 15Extended precision, 14Extension of simplex, 168, 169, 172Extrapolation, 151False Position, 161Fenton, R.
G., 182Financial Times index, 77Finkbeiner, D. T., 87Fletcher, R., 190, 192, 198, 199, 215, 228, 244Fletcher-Reeves formula, 199FMIN linear search program, 153Ford B., 135Formulae,Gauss-Jordan, 98Forsythe, G. E., 127, 153FORTRAN, 10, 56, 63Forward difference, 2 19Forward-substitution, 86, 136Foster, R. M., 139Frank matrix, 250,253Fried, I., 246Fröberg, C., 21, 127, 238, 251Full-rank case, 23, 66Function evaluation count, 157, 164, 209, 217,227, 232Function minimisation, 142, 207Functions,penalty, 222Galle, 131Gauss elimination, 72, 79, 82, 93for inverse iteration, 105, 109variations, 80with partial pivoting, 75Gauss-Jordan reduction, 82, 93Gauss-Newton method, 209, 211, 228Gearhart, W.
B., 146, 232Generalised eigenvalue problem, 135, 234, 242Generalised inverse, 44, 662 and 4 condition, 26of a matrix, 24Generalised matrix eigenvalue problem, 28, 104Gentleman, W. M., 50273Geradin, M., 244, 246Gerschgorin bound, 136Gerschgorin’s theorem, 121Gill, P. E., 221, 225Givens’ reduction, 15, 49, 51, 63, 83and singular-value decomposition,implementation, 54for inverse iteration, 105, 109of a real rectangular matrix, 51operation of, 52singular-value decomposition and least-squaressolution, 56Givens’ tridiagonalisation, 133Global minimum, 146Golub, G. H., 56GOTO instructions, 12Gradient, 186, 188, 197, 208, 226computed, 226of nonlinear sum of squares, 209of Rayleigh quotient, 245Gradient calculation in conjugate gradients forlinear equations, 235Gradient components,‘large’ computed values of, 206Gram-Schmidt orthogonalisation, 197Gregory, R.
T., 117Grid search, 149, 156, 160Griffith, B. A., 125Guard digits, 7Hall, G., 135Hamiltonian operator, 28, 138Hammarling, S., 50Hanson, R. J., 64Hartley, H. O., 210, 211Harwell subroutine library, 215Hassan, Z., 223Healy, M. J. R., 88, 90Heaviside function, 222Hemstitching of function minimisation method,186, 208Henderson, B., 153Henrici, P., 127, 162Hermitian matrix, 137Hessian, 189, 197, 231for Rayleigh quotient, 244matrix, 187Hestenes, M. R., 33, 134, 235, 241Heuristic method, 168, 171Hewlett-Packard,computers, see HP9830pocket calculators, 5Hilbert segment, 108, 253Hillstrom, K. E., 227274Compact numerical methods for computersHomogeneity of a function, 244Hooke and Jeeves method, 182Householder tridiagonalisation, 133HP9830, 44, 56, 62, 70, 90, 92, 131, 164IBM 370, 120IBM 370/168, 56, 128, 167, 196, 239I11 conditioning of least-squares problem, 42Implicit interchanges for pivoting, 81IMSL, 10Indefinite systems of linear equations.
241Independence,linear, 20Index array, 82Index numbers, 23, 77Infeasible problems, 221Infinity norm, 104Information loss, 67Initial values for parameters, 146Inner product, 28, 245Insurance premium calculation. 165Interchange,implicit, 81row and column, 95Internal rate of return, 145International Mathematical and StatisticalLibraries, 10Interpolating parabola, 152Interpolation,formulae for differentiation, 218linear, 161Interpreter for computer programming language,91Interval,closed, 17for linear search, 148for root-finding, 160open, 17Inverse,generalised, 44of a matrix, 24of a symmetric positive definite matrix, 97of triangular matrices, 74Inverse interpolation, 151Inverse iteration, 104, 140behaviour of, 108by conjugate gradients, 241,249Inverse linear interpolation, 161Inverse matrix, 95Iteration limit, 109Iteration matrix, 188initialisation, 191Iterative improvement of linear-equationsolutions, 81Jacobi, C.G.
J., 126. 127, 131jacobi (ALGOL procedure), 128, 133Jacobi algorithm, 126, 136,250cyclic, 127organisation of, 128Jacobi rotations,effect of, 126Jacobian, 211, 217, 232matrix, 209Jaffrelot, J. J., 204Jeeves, 185Jenkins, M. A., 143, 148Jones, A., 215Kahan, W., 234Kaiser, H. F., 134Karney, D.
L., 117Kendall, M. G., 40, 180Kernighan, B. W., 12Kowalik, J., 85, 142, 186Kronecker delta, 3 173, 119LLTdecomposition, 84Lagrange multipliers, 221Lanczos method for eigenvalue problems. 234Lawson, C. L., 64Least-squares, 23, 50, 54, 77linear, 21via normal equations, 92via singular-value decomposition, 40, 42Least-squares computations,example, 45Least-squares solution, 22Lefkovitch, L. P., 56, 63, 70Levenberg, K., 211Leverrier, 131Linear algebra, 19Linear approximation of nonlinear function. 187Linear combination, 29Linear dependence, 34Linear equations, 19, 20, 72, 77, 93, 234, 235as a least-squares problem, 23complex, 82consistent, 87Linear independence, 20, 25Linear least-squares, 21, 77, 207, 234, 235Linear relationship, 23Linear search, 143, 146, 148, 156, 159, 188, 189,192, 198, 199, 235, 244acceptable point strategy, 190List of algorithms.
255List of examples, 256Local maxima, 143. 146, 149Local minima, 146, 208Logistic growth function, 144, 216IndexLoss of information in least-squarescomputations, 23, 67Lottery,optimal operation of, 144, 228LU decomposition, 74Machine arithmetic, 6Machine precision, 6, 46, 70, 105, 219Magnetic roots, 232Magnetic zeros, 147Malcolm, M. A., 6Mantissa, 6Market equilibrium,nonlinear equations, 231Marquardt, D. W., 211, 212Marquardt algorithm, 209, 223, 228, 232, 233Mass-spectrograph calibration, 20Mathematical programming, 3, 13Mathematical software, 11Matrix, 19coefficient, 20.23complex, 110cross-products, 66dense, 20, 23diagonal, 26, 3 1elementary, 73Frank, 100generalised inverse of, 24Hermitian, 110inverse, 24, 95Moler, 100non-negative definite, 22, 86non-symmetric, 110null, 52orthogonal, 26, 31, 50positive definite, 22rank of, 20real symmetric, 31, 119rectangular, 24, 44semidefinite, 22singular, 20sparse, 20, 21, 23special, 83symmetric, 23, 28symmetric positive definite, 83, 84, 93triangular, 26, 50, 52, 72, 74unit, 29, 32unitary, 27Matrix decomposition,triangular, 74Matrix eigenvalue problem, 28, 135generalised, 104, 148Matrix eigenvalues for polynomial roots, 148Matrix form of linear equations, 19Matrix inverse for linear equations, 24275Matrix iteration methods for functionminimisation, 187Matrix product count, 250Matrix transpose, 22Maxima, 143Maximal and minimal eigensolutions, 243McKeown, J.
J., 207Mead, R., 168, 170Mean of two numbers, 8Measure of work in function minimisation, 227Method of substitution, 93Minima of functions, 142Minimum-length least-squares solution, 22, 25Model,linear, 23nonlinear, 207of regional hog supply, 204Modular programming, 12Moler, C., 250, 253Moler matrix, 127, 250, 253Choleski decomposition of, 91Moments of inertia, 125Moore-Penrose inverse, 26, 44Mostow, G.
D., 74Multiplicity of eigenvalues, 120Murray. W., 221, 225, 228NAG, 10, 215Nash, J. C., 33, 56, 63, 70, 110, 134, 137, 196, 211,215, 226, 235Nash, S. G., 82, 148, 235Negative definite matrix, 238Nelder, J. A., 168, 170NelderMead search, 168, 197, 223, 228, 230, 233modifications, 172Neptune (planet), 131Newing, R. A., 138, 141Newton-Raphson iteration, 210Newton’s method, 161, 188, 210for more than one parameter, 187Non-diagonal character,measure of, 126Nonlinear equations, 142, 143, 144, 186, 231Nonlinear least-squares, 142, 144, 207, 231Nonlinear model of demand equations, 223Non-negative definite matrix, 22Non-singular matrix, 20Norm, 17, 21, 66, 243Euclidean, 22of vector, 104Normal equations, 22, 25, 41, 50, 55, 66, 92, 239as consistent set, 88Normalisation, 28, 52of eigenvectors, 108, 119of vector to prevent overflow, 104to prevent overflow, 103276Compact numerical methods for computersNormalising constant, 139Notation, 17NOVA, 5, 46, 69, 79, 90, 91, 93, 100, 108, 109,117, 122, 123, 125, 127, 141, 153, 156, 164,199, 206, 208, 220, 225,226, 229, 230,232,241,250Null vector, 20Numerical Algorithms Group, 10Numerical approximation of derivatives, 2 17,218, 223, 228Numerical differentiation, 218Objective function, 205, 207Oliver, F.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
