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continued fraction 169f.and spectral radius 865ff., 871convert_matrix() utility 945Convex sets, use in inverse problems 813f.Convolutiondenoted by asterisk 498finite impulse response (FIR) 538of functions 498, 509of large data sets 543f.for multiple precision arithmetic 918multiplication as 918necessity for optimal filtering 542overlap-add method 544overlap-save method 543f.and polynomial interpolation 120relation to wavelet transform 592theorem 498, 538ff., 553theorem, discrete 538ff.treatment of end effects 540use of FFT 530, 538ff.wraparound problem 540Cooley-Tukey FFT algorithm 509Co-processor, floating point 894Copyright rules xviCornwell-Evans algorithm 825Corporate promotion ladder 336f.Corrected two-pass algorithm 613Correction, in multigrid method 872Correlation coefficient (linear) 636ff.Correlation function 498autocorrelation 498, 546, 565and Fourier transforms 498theorem 498, 545treatment of end effects 546using FFT 545f.Wiener-Khinchin theorem 498, 574Correlation, statistical 609f., 628Kendall’s tau 640, 642ff.linear correlation coefficient 636ff., 664linear related to least square fitting 636,664nonparametric or rank statistical 639ff.among parameters in a fit 663, 673, 676in random number generators 277Spearman rank-order coefficient 640f.sum squared difference of ranks 640Cosine function, recurrence 178Cosine integral 255, 257ff.continued fraction 257routine for 258f.series 257Cosine transform see Fast Fourier transform(FFT); Fourier transformCoulomb wave function 210, 240Courant condition 838, 841, 843, 845multidimensional 855Courant-Friedrichs-Lewy stability criterion seeCourant conditionCovariancea priori 705in general linear least squares 673, 677matrix, by Cholesky decomposition 98,673matrix, of errors 805, 817matrix, is inverse of Hessian matrix 685matrix, when it is meaningful 695ff.in nonlinear models 685, 687relation to chi-square 695ff.from singular value decomposition (SVD)698in straight line fitting 663CR method see Cyclic reduction (CR)Cramer’s V 631Crank-Nicholson method 848, 853, 855CRC (cyclic redundancy check) 896ff.CRC-12 898CRC-16 polynomial 898CRC-CCITT 898Creativity, essay on 8Critical (Nyquist) sampling 500, 550Cross (denotes matrix outer product) 73Crosstabulation analysis 629see also Contingency tableCrout’s algorithm 44ff., 53Cubic equations 183ff., 367Cubic spline interpolation 113ff.see also SplineCumulative binomial distribution 226, 229Cumulative Poisson function 221related to incomplete gamma function 221Curvature matrix see Hessian matrixcvector() utility 943Cycle, in multigrid method 874Cyclic Jacobi method 466Cyclic reduction (CR) 857f., 861f.Cyclic redundancy check (CRC) 896ff.Cyclic tridiagonal systems 74f.D anielson-Lanczos lemma504f., 532Dataassigning keys to 897continuous vs.
binned 620entropy 632ff., 903f.essay on 609fitting 656ff.fraudulent 661glitches in 659iid (independent and identically distributed)691modeling 656ff.serial port 899smoothing 610, 650ff.statistical tests 609ff.970unevenly or irregularly sampled 576,581f., 654use of CRCs in manipulating 897windowing 553ff.see also Statistical testsData compression 603, 889arithmetic coding 910ff.cosine transform 519Huffman coding 903ff., 910linear predictive coding (LPC) 571f.lossless 903Data Encryption Standard (DES) 300ff.Data type 28DAUB4 592ff., 595, 598f., 601DAUB6 593DAUB12 605DAUB20 598f.Daubechies wavelet coefficients 592ff., 596,598f., 601, 605Davidon-Fletcher-Powell algorithm 397, 426f.Dawson’s integral 259f., 606approximation for 259routine for 260D.C. (direct current) 498Debugging 7DEC (Digital Equipment Corp.) xvii, 3, 894Declarations 930ff.Decomposition see Cholesky decomposition;LU decomposition; QR decomposition;Singular value decomposition (SVD)Deconvolution 542, 549see also Convolution; Fast Fourier transform (FFT); Fourier transformDefect, in multigrid method 872Deferred approach to the limit see Richardson’s deferred approach to the limitDeflationof matrix 478of polynomials 369ff., 377, 378Degeneracy of linear algebraic equations 32,61, 65, 676Degenerate kernel 794Degenerate minimization principle 804Degrees of freedom 621f., 660, 695f.Demonstration programs 3Derivativescomputation via Chebyshev approximation189, 195computation via Savitzky-Golay filters189, 651matrix of first partial see Jacobian determinantmatrix of second partial see Hessian matrixnumerical computation 186ff., 386, 651,738, 758, 779of polynomial 173f.use in optimization 395f., 406DES see Data Encryption StandardDescending transformation, elliptic integrals262Descent direction 383, 390, 427Descriptive statistics 609ff.see also Statistical testsDesign matrix 651, 671, 804, 809IndexDeterminant 34, 49Deviates, random see Random deviatesDFP algorithm see Davidon-Fletcher-PowellalgorithmDiagonal dominance 51, 684, 789, 865Difference equations, finite see Finite difference equations (FDEs)Difference operator 167Differential equations 707ff.accuracy vs.
stability 710, 736Adams-Bashforth-Moulton schemes 749adaptive stepsize control 709, 714ff., 725,733f., 738, 744, 749f., 751algebraically difficult sets 772backward Euler’s method 735Bader-Deuflhard method for stiff 737,742f.boundary conditions 707f., 753ff., 757,760, 779f.Bulirsch-Stoer method 209, 272, 708f.,712, 722, 724ff., 747Bulirsch-Stoer method for conservativeequations 733comparison of methods 708f., 747, 751conservative 732f.danger of too small stepsize 720eigenvalue problem 756, 772ff., 779f.,781embedded Runge-Kutta method 715f., 738equivalence of multistep and multivaluemethods 751Euler’s method 708, 710, 735forward Euler’s method 735free boundary problem 756, 785high-order implicit methods 737ff.implicit differencing 735f., 749initial value problems 708internal boundary conditions 784ff.internal singular points 784ff.interpolation on right-hand sides 117Kaps-Rentrop method for stiff 737local extrapolation 715modified midpoint method 722f., 726multistep methods 747ff.multivalue methods 747order of method 710f., 725path integration for function evaluation208ff., 271predictor-corrector methods 708, 737,747ff.reduction to first-order sets 707, 753relaxation method 754f., 762ff.relaxation method, example of 772ff.r.h.s.
independent of x 735Rosenbrock methods for stiff 737Runge-Kutta method 708f., 710ff., 714ff.,738, 747Runge-Kutta method, high-order 711Runge-Kutta-Fehlberg method 715f.scaling stepsize to required accuracy 716f.second order 732f.semi-implicit differencing 737semi-implicit Euler method 737, 743semi-implicit extrapolation method 737,743971Indexsemi-implicit midpoint rule 743shooting method 754, 757ff.shooting method, example 779f., 781similarity to Volterra integral equations794f.singular points 724f., 760, 784ff.step doubling 715stepsize control 709, 714ff., 724, 733f.,738, 744, 749f., 751stiff 709, 734ff.stiff methods compared 747Stoermer’s rule 732f.see also Partial differential equations; Twopoint boundary value problemsDiffusion equation 827, 847ff., 864Crank-Nicholson method 848, 853, 855Forward Time Centered Space (FTCS)847ff., 850f., 864implicit differencing 848multidimensional 855f.Digamma function 222Digital filtering see FilterDihedral group D5 902Dimensions (units) 683f.Diminishing increment sort 331Dirac delta function 293, 789Direct method see PeriodogramDirect methods for linear algebraic equations35Direct product see Outer product of matricesDirection of largest decrease 416f.Direction numbers, Sobol’s sequence 311Direction-set methods for minimization 396,412ff.Dirichlet boundary conditions 829, 849, 859,865, 867Disclaimer of warranty xviDiscordant pair for Kendall’s tau 643Discrete convolution theorem 538ff.Discrete Fourier transform (DFT) 500ff.as approximate continuous transform 503see also Fast Fourier transform (FFT)Discrete optimization 444ff.Discriminant 184, 464Diskettes, how to order xvi, 996f.Dispersion 840DISPO see Savitzky-Golay filtersDissipation, numerical 839Divergent series 167Divisioncomplex 177multiple precision 919f.of polynomials 175, 369, 377dmatrix() utility 944dn function 269Do-while iteration 12Dogleg step methods 393Domain of integration 161f.Dominant solution of recurrence relation 179Dot (denotes matrix multiplication) 33Double exponential error distribution 701Double precisionas refuge of scoundrels 890use in iterative improvement 56Double root 348Downhill simplex method see Simplex, methodof Nelder and MeadDriver programs 3Dual viewpoint, in multigrid method 883Duplication theorem, elliptic integrals 262f.dvector() utility 943DWT (discrete wavelet transform) see WavelettransformE ardley, D.M.346EBCDIC 898Economization of power series 198ff., 201Eigensystems 456ff.balancing matrix 483bounds on eigenvalues 58calculation of few eigenvectors or eigenvalues 461, 494canned routines 461characteristic polynomial 456, 475f.completeness 457defective 457, 482, 494deflation 478degenerate eigenvalues 456, 458elimination method 460, 485factorization method 460fast Givens reduction 470generalized eigenproblem 462Givens reduction 469f.Hermitian matrix 481f.Hessenberg matrix 460, 477, 482ff., 494Householder transformation 460, 469ff.,476, 480, 481, 484f.ill-conditioned eigenvalues 483implicit shifts 478ff.and integral equations 788ff., 794invariance under similarity transform 459inverse iteration 462, 476, 483, 493ff.Jacobi transformation 460, 463ff., 469,481, 495left eigenvalues 458list of tasks 461multiple eigenvalues 495nonlinear 462nonsymmetric matrix 482ff.operation count of balancing 483operation count of Givens reduction 470operation count of Householder reduction474operation count of inverse iteration 494operation count of Jacobi method 467operation count of QL method 477, 480operation count of QR method for Hessenberg matrices 490operation count of reduction to Hessenbergform 485orthogonality 457polynomial roots and 375QL method 476ff., 481, 494f.QL method with implicit shifts 478ff.QR method 60, 460, 463, 476ff.QR method for Hessenberg matrices 486ff.real, symmetric matrix 156, 474, 794reduction to Hessenberg form 484f.right eigenvalues 458shifting eigenvalues 456, 477f., 486f.972special matrices 461termination criterion 489f., 495tridiagonal matrix 460, 475ff., 494Eigenvalue and eigenvector, defined 456Eigenvalue problem for differential equations756, 772ff., 779, 781Eigenvalues and polynomial root finding 375EISPACK 461, 482Electromagnetic potential 525Elimination see Gaussian eliminationEllipse in confidence limit estimation 693Elliptic integrals 261ff., 915addition theorem 262Carlson’s forms and algorithms 261ff.Cauchy principal value 263duplication theorem 262f.Legendre 261ff., 267ff.routines for 264ff.symmetric form 261f.Weierstrass 262Elliptic partial differential equations 827alternating-direction implicit method (ADI)870f., 915analyze/factorize/operate package 833biconjugate gradient method 833boundary conditions 828f.comparison of rapid methods 863conjugate gradient method 833cyclic reduction 857f., 861f.Fourier analysis and cyclic reduction (FACR)857ff., 863Gauss-Seidel method 864, 873, 874f., 884incomplete Cholesky conjugate gradientmethod (ICCG) 833Jacobi’s method 864f., 873matrix methods 833multigrid method 833, 871ff.rapid (Fourier) method 833, 857ff.relaxation method 832, 863ff.strongly implicit procedure 833successive over-relaxation (SOR) 866ff.,871, 875Emacs, GNU xiiiEmbedded Runge-Kutta method 715f., 738Encapsulation, in programs 6f.Encryption 300Entropy 903f.of data 632ff., 820EOM (end of message) 910Equality constraints 431Equationscubic 183ff., 367normal (fitting) 651, 672ff., 809f.quadratic 29, 183ff.see also Differential equations; Partial differential equations; Root findingEquivalence classes 345f.Equivalence transformation 172Errorchecksums for preventing 899clocking 899double exponential distribution 701local truncation 883Lorentzian distribution 701f.in multigrid method 872Indexnonnormal 659, 695, 699ff.relative truncation 883roundoff 185, 889f.series, advantage of an even 138f., 723systematic vs.
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