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185Global optimization 394f., 444ff., 656continuous variables 451f.IndexGlobally convergentminimization 425ff.root finding 380, 383ff., 390, 757f., 761GMRES (generalized minimum residual method)85GNU Emacs xiiiGodunov’s method 846Golden mean (golden ratio) 30, 354, 399, 406Golden section search 348, 396, 397ff., 403Golub-Welsch algorithm, for Gaussian quadrature 156f.Goodness-of-fit 656, 660, 663f., 668, 695goto statements, danger of 8Gram-Schmidtbiorthogonalization 421f.orthogonalization 100, 457, 458SVD as alternative to 66Graphics, function plotting 349f.Gravitational potential 525Gray code 311, 889, 894ff.Greenbaum, A.

86Gregorian calendar 12, 15Grid square 123f.Group, dihedral 902Guard digits 890H alf weighting 876Halton’s quasi-random sequence 309f.Hamming window 554Hamming’s motto 348Hann window 554Harmonic analysis see Fourier transformHashing 303HDLC checksum 898Header (.h) files 16f.Heap (data structure) 336f., 344, 905Heapsort 329, 336f., 344Helmholtz equation 861Hermite polynomials 151, 153Hermitian matrix 457ff., 481f.Hertz (unit of frequency) 496Hessenberg matrix 100, 460, 477, 482, 494see also MatrixHessian matrix 389, 414, 422, 427, 681ff.,812, 824is inverse of covariance matrix 673, 685second derivatives in 683Hexadecimal constants 285, 303Hierarchically band diagonal matrix 606Hierarchy of program structure 5ff.High-order not same as high-accuracy 106f.,130, 396, 406, 711, 715, 748f.High-pass filter 558Hilbert matrix 90Historic maximum entropy method 825f.Homogeneous linear equations 61Hook step methods 393Hotelling’s method for matrix inverse 57, 606Householder transformation 60, 460, 469ff.,476, 480, 481, 484f., 488ff.operation count 474in QR decomposition 99Huffman coding 571, 889, 903ff., 910Hyperbolic functions, explicit formulas forinverse 184IndexHyperbolic partial differential equations 827advective equation 835flux-conservative initial value problems834ff.Hypergeometric function 208ff., 271ff.routine for 272f.Hypothesis, null 609I BMxviibad random number generator 277PC 3, 285, 303, 894radix base for floating point arithmetic483IBM checksum 901f.ICCG (incomplete Cholesky conjugate gradientmethod) 833ICF (intrinsic correlation function) model 826Identity (unit) matrix 34IEEE floating point format 285, 890f.if structure 11warning about nesting 11IIR (infinite impulse response) filter 559, 573Ill-conditioned integral equations 789Image processing 525, 812cosine transform 519fast Fourier transform (FFT) 525, 530,812as an inverse problem 812maximum entropy method (MEM) 818ff.from modulus of Fourier transform 814wavelet transform 603imatrix() utility 944Implicitconversion of data types 24f., 930function theorem 347pivoting 38shifts in QL method 478ff.Implicit differencing 836for diffusion equation 848for stiff equations 735f., 749Importance sampling, in Monte Carlo 316f.Improper integrals 141ff.Impulse response function 538, 549, 559IMSL xvii, 35, 72, 212, 371, 376, 461In-place selection 342Include files 17, 930Incomplete beta function 226ff.for F-test 619routine for 227f.for Student’s t 616, 618Incomplete Cholesky conjugate gradient method(ICCG) 833Incomplete gamma function 216for chi-square 621, 660, 663f.deviates from 290ff.in mode estimation 616routine for 218f.Increment of linear congruential generator276Indentation of blocks 11Index 965ff.this entry 977Index table 329, 338Inequality constraints 431977Inheritance 7Initial value problems 708, 827f.see also Differential equations;Partial differential equationsInjection operator 873Instability see StabilityInteger programming 443Integral equations 788ff.adaptive stepsize control 797block-by-block method 797correspondence with linear algebraic equations 788ff.degenerate kernel 794eigenvalue problems 789, 794error estimate in solution 793Fredholm 788f., 791f.Fredholm alternative 789homogeneous, second kind 793f.ill-conditioned 789infinite range 797f.inverse problems 789, 804ff.kernel 788f.nonlinear 790, 796Nystrom method 791f., 797product Nystrom method 797with singularities 797ff.with singularities, worked example 801subtraction of singularity 798symmetric kernel 794unstable quadrature 796Volterra 789f., 794f.wavelets 791see also Inverse problemsIntegral operator, wavelet approximation of603f., 791Integration of functions 129ff.cosine integrals 257Fourier integrals 584ff.Fourier integrals, infinite range 590f.Fresnel integrals 255Gauss-Hermite 153Gauss-Jacobi 154Gauss-Laguerre 152Gauss-Legendre 151integrals that are elliptic integrals 261path integration 208ff.sine integrals 257see also QuadratureIntegro-differential equations 791Interface, in programs 7Intermediate value theorem 350Internet xviiInterpolation 105ff.Aitken’s algorithm 108avoid 2-stage method 106avoid in Fourier analysis 576bicubic 125f.bilinear 123f.caution on high-order 106f.coefficients of polynomial 106, 120ff.,197, 582for computing Fourier integrals 586error estimates for 106of functions with poles 111ff.inverse quadratic 360, 402ff.978multidimensional 107f., 123ff.in multigrid method 876Neville’s algorithm 108f., 188Nystrom 792offset arrays 110, 119operation count for 106operator 873order of 106and ordinary differential equations 107oscillations of polynomial 106, 120, 396,406parabolic, for minimum finding 402polynomial 105, 108ff., 188rational Chebyshev approximation 204ff.rational function 105, 111ff., 200ff., 231f.,724ff., 731reverse (extirpolation) 581f.spline 106, 113ff., 127f.trigonometric 105see also FittingInterval variable (statistics) 628Intrinsic correlation function (ICF) model 826Inverse hyperbolic function 184, 262Inverse iteration see EigensystemsInverse problems 789, 804ff.Backus-Gilbert method 815ff.Bayesian approach 808, 820, 825f.central idea 808constrained linear inversion method 808ff.data inversion 816deterministic constraints 813ff.in geophysics 818Gerchberg-Saxton algorithm 814f.incomplete Fourier coefficients 822and integral equations 789linear regularization 808ff.maximum entropy method (MEM) 818ff.,824f.MEM demystified 823Phillips-Twomey method 808ff.principal solution 806regularization 805ff.regularizing operator 807stabilizing functional 807Tikhonov-Miller regularization 808ff.trade-off curve 804trade-off curve, Backus-Gilbert method818two-dimensional regularization 812use of conjugate gradient minimization812f., 824use of convex sets 813f.use of Fourier transform 812, 814Van Cittert’s method 813Inverse quadratic interpolation 360, 402ff.Inverse response kernel, in Backus-Gilbertmethod 816f.Inverse trigonometric function 262ISBN (International Standard Book Number)checksum 901Iterated integrals 161Iteration 8functional 748to improve solution of linear algebraicequations 55ff., 201Indexfor linear algebraic equations 35required for two-point boundary valueproblems 753in root finding 347f.Iteration matrix 865ITPACK 78ivector() utility 943J acobi matrix, for Gaussian quadrature 156Jacobi transformation (or rotation) 100, 460,463ff., 469, 481, 495Jacobian determinant 288f., 783Jacobian elliptic functions 261, 269f.Jacobian matrix 381, 383, 386, 389, 738singular in Newton’s rule 393Jacobi’s method (relaxation) 864f., 866, 873Jenkins-Traub method 376Julian Day 1, 12, 14Jump transposition errors 902K -S test see Kolmogorov-Smirnov testKalman filter 705Kaps-Rentrop method 737Kendall’s tau 640, 642ff.Kermit checksum 897Kernel 788f.averaging, in Backus-Gilbert method 816f.degenerate 794finite rank 794inverse response 816f.separable 794singular 797f.symmetric 793f.Kernighan & Ritchie C (K&R C) 2, 16, 24,930Keys used in sorting 338, 897Kolmogorov-Smirnov test 620, 623ff., 699two-dimensional 645ff.variants 626ff., 645ff.Kuiper’s statistic 627Kurtosis 612, 614L-estimate699Labels, statement 8Lag 498, 545f., 560Lagrange multiplier 804Lagrange’s formula for polynomial interpolation 91, 108, 582, 585Laguerre’s method 348, 371ff.Lanczos lemma 504f.Lanczos method for gamma function 213Landen transformation 262LAPACK 35Laplace’s equation 252, 827see also Poisson equationLas Vegas 631Latin square or hypercube 315Laurent series 573Lax method 837ff., 845, 854f.multidimensional 854f.Lax-Wendroff method 844ff.Leakage in power spectrum estimation 551,554f.IndexLeakage width 554f.Leapfrog method 842f.Least squares filters see Savitzky-Golay filtersLeast squares fitting 650f., 657ff., 661ff.,666ff., 671ff.contrasted to general minimization problems 689degeneracies in 677, 679Fourier components 577freezing parameters in 674, 705general linear case 671ff.Levenberg-Marquardt method 683ff., 825Lomb periodogram 577as M-estimate for normal errors 701as maximum likelihood estimator 658as method for smoothing data 650f.multidimensional 680nonlinear 393, 681ff., 825nonlinear, advanced methods 688normal equations 651, 672ff., 809f.normal equations often singular 676, 679optimal (Wiener) filtering 547QR method in 100, 674for rational Chebyshev approximation 205relation to linear correlation 636, 664Savitzky-Golay filter as 650f.singular value decomposition (SVD) 34f.,59ff., 205, 676ff.skewed by outliers 659for spectral analysis 577standard (probable) errors on fitted parameters 673, 677weighted 658see also FittingL’Ecuyer’s long period random generator 280ff.Left eigenvalues or eigenvectors 458Legal matters xviLegendre elliptic integral see Elliptic integralsLegendre polynomials 252f.fitting data to 680recurrence relation 178shifted monic 159see also Associated Legendre polynomials;Spherical harmonicsLehmer-Schur algorithm 376Lemarie’s wavelet 600Lentz’s method for continued fraction 171,219Lepage, P.

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