976 Index Gauss-Jacobi integration 151 Globally convergent abscissas and weights 154 minimization 425ff Gauss-Jordan elimination 36ff,41,71 root finding380,383f.,390,757f,761 operation count 42,48 GMRES (generalized minimum residual method) solution of normal equations 673 85 storage requirements 38f. GNU Emacs xiii Gauss-Kronrod quadrature 160 Godunov's method 846 Gauss-Laguerre integration 151.798 Golden mean (golden ratio)30,354,399,406 Gauss-Legendre integration 151 Golden section search 348,396,397ff.,403 see also Gaussian integration Golub-Welsch algorithm,for Gaussian quadra- Gauss-Lobatto quadrature 160,196,518 ture 156f. Gauss-Radau quadrature 160 Goodness-of-fit 656,660.663f,668.695 Gauss-Seidel method (relaxation)864,866. goto statements,danger of 8 873,874f. Gram-Schmidt nonlinear 884 biorthogonalization 421f. Gauss transformation 262 orthogonalization 100,457,458 Gaussian (normal)distribution 275,658,807 SVD as alternative to 66 central limit theorem 658f. Graphics,function plotting 349f. deviates from 288f.,578 Gravitational potential 525 kurtosis of 612 Gray code 311,889,894ff multivariate 695 Greenbaum,A.86 Gregorian calendar 12.15 semi-invariants of 614 tails compared to Poisson 659 Grid square 123f. two-dimensional (binormal)637 Group,dihedral 902 Guard digits 890 variance of skewness of 612 Gaussian elimination 41f,59,63 fill-in 53.71 Half weighting 876 integral equations 795 Halton's quasi-random sequence 309f. operation count 42 Hamming window 554 in reduction to Hessenberg form 485 Hamming's motto 348 relaxation solution of boundary value prob- Hann window 554 lems 762ff.,785 Harmonic analysis see Fourier transform Gaussian function Hashing 303 Hardy's theorem on Fourier transforms HDLC checksum 898 607 Header (.h)files 16f. see also Gaussian (normal)distribution Heap (data structure)336f.,344,905 Gaussian integration 133,147ff.,798 Heapsort329,336f.,344 calculation of abscissas and weights 150ff Helmholtz equation 861 error estimate in solution 793 Hermite polynomials 151,153 extensions of 160 Hermitian matrix 457ff.,481f. Golub-Welsch algorithm for weights and Hertz (unit of frequency)496 abscissas 156f. Hessenberg matrix 100,460,477,482,494 for integral equations 790,792 see also Matrix from known recurrence relation 156f. Hessian matrix 389,414,422,427,681ff., nonclassical weight function 157ff.,797 812,824 is inverse of covariance matrix 673.685 and orthogonal polynomials 148 preassigned nodes 160 second derivatives in 683 weight function log z 159 Hexadecimal constants 285,303 weight functions 147ff.,797 Hierarchically band diagonal matrix 606 Hierarchy of program structure 5ff. Gear's method (stiff ODEs)737 High-order not same as high-accuracy 106f, Geiger counter 274 130,396,406,711,715,748f Generalized eigenvalue problems 462 High-pass filter 558 Generalized minimum residual method (GM- Hilbert matrix 90 RES)85 Historic maximum entropy method 825f Geophysics,use of Backus-Gilbert method Homogeneous linear equations 61 818 Hook step methods 393 Gerchberg-Saxton algorithm 814f. Hotelling's method for matrix inverse 57,606 Gilbert and Sullivan 720 Householder transformation 60,460,469ff., Givens reduction 469f.,480 476.480.481.484f.488正 fast 470 operation count 474 operation count 470 in QR decomposition 99 Glassman,A.J.185 Huffman coding 571,889,903f,910 Global optimization 394f,444ff.,656 Hyperbolic functions,explicit formulas for continuous variables 451f. inverse 184976 Index Gauss-Jacobi integration 151 abscissas and weights 154 Gauss-Jordan elimination 36ff., 41, 71 operation count 42, 48 solution of normal equations 673 storage requirements 38f. Gauss-Kronrod quadrature 160 Gauss-Laguerre integration 151, 798 Gauss-Legendre integration 151 see also Gaussian integration Gauss-Lobatto quadrature 160, 196, 518 Gauss-Radau quadrature 160 Gauss-Seidel method (relaxation) 864, 866, 873, 874f. nonlinear 884 Gauss transformation 262 Gaussian (normal) distribution 275, 658, 807 central limit theorem 658f. deviates from 288f., 578 kurtosis of 612 multivariate 695 semi-invariants of 614 tails compared to Poisson 659 two-dimensional (binormal) 637 variance of skewness of 612 Gaussian elimination 41f., 59, 63 fill-in 53, 71 integral equations 795 operation count 42 in reduction to Hessenberg form 485 relaxation solution of boundary value prob￾lems 762ff., 785 Gaussian function Hardy’s theorem on Fourier transforms 607 see also Gaussian (normal) distribution Gaussian integration 133, 147ff., 798 calculation of abscissas and weights 150ff. error estimate in solution 793 extensions of 160 Golub-Welsch algorithm for weights and abscissas 156f. for integral equations 790, 792 from known recurrence relation 156f. nonclassical weight function 157ff., 797 and orthogonal polynomials 148 preassigned nodes 160 weight function log x 159 weight functions 147ff., 797 Gear’s method (stiff ODEs) 737 Geiger counter 274 Generalized eigenvalue problems 462 Generalized minimum residual method (GM￾RES) 85 Geophysics, use of Backus-Gilbert method 818 Gerchberg-Saxton algorithm 814f. Gilbert and Sullivan 720 Givens reduction 469f., 480 fast 470 operation count 470 Glassman, A.J. 185 Global optimization 394f., 444ff., 656 continuous variables 451f. Globally convergent minimization 425ff. root finding 380, 383ff., 390, 757f., 761 GMRES (generalized minimum residual method) 85 GNU Emacs xiii Godunov’s method 846 Golden mean (golden ratio) 30, 354, 399, 406 Golden section search 348, 396, 397ff., 403 Golub-Welsch algorithm, for Gaussian quadra￾ture 156f. Goodness-of-fit 656, 660, 663f., 668, 695 goto statements, danger of 8 Gram-Schmidt biorthogonalization 421f. orthogonalization 100, 457, 458 SVD as alternative to 66 Graphics, function plotting 349f. Gravitational potential 525 Gray code 311, 889, 894ff. Greenbaum, A. 86 Gregorian calendar 12, 15 Grid square 123f. Group, dihedral 902 Guard digits 890 Half weighting 876 Halton’s quasi-random sequence 309f. Hamming window 554 Hamming’s motto 348 Hann window 554 Harmonic analysis see Fourier transform Hashing 303 HDLC checksum 898 Header (.h) files 16f. Heap (data structure) 336f., 344, 905 Heapsort 329, 336f., 344 Helmholtz equation 861 Hermite polynomials 151, 153 Hermitian matrix 457ff., 481f. Hertz (unit of frequency) 496 Hessenberg matrix 100, 460, 477, 482, 494 see also Matrix Hessian matrix 389, 414, 422, 427, 681ff., 812, 824 is inverse of covariance matrix 673, 685 second derivatives in 683 Hexadecimal constants 285, 303 Hierarchically band diagonal matrix 606 Hierarchy of program structure 5ff. High-order not same as high-accuracy 106f., 130, 396, 406, 711, 715, 748f. High-pass filter 558 Hilbert matrix 90 Historic maximum entropy method 825f. Homogeneous linear equations 61 Hook step methods 393 Hotelling’s method for matrix inverse 57, 606 Householder transformation 60, 460, 469ff., 476, 480, 481, 484f., 488ff. operation count 474 in QR decomposition 99 Huffman coding 571, 889, 903ff., 910 Hyperbolic functions, explicit formulas for inverse 184