c21-2 (779630)
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General IndexAccelerated convergence of seriesschedule 445thermodynamic analogy 444f.traveling salesman problem 445ff.ANSI C standard 2f., 14, 25, 930, 941ANSI macro 17, 930Antonov-Saleev variant of Sobol’ sequence310ff.Apple xviiMacintosh 894Approximate inverse of matrix 57Approximation of functions 105f.by Chebyshev polynomials 191f., 519Padé approximant 200ff.by rational functions 204ff.by wavelets 601f., 791see also FittingArguments, conversion of data types 24f.,930Arithmeticarbitrary precision 889, 915ff.complex 23f., 948ff.floating point 889IEEE standard 285, 890f.rounding 890Arithmetic coding 889, 910ff.Arithmetic-geometric mean (AGM) method915Arraycentered subarray of 119how to allocate 19index range 18one-dimensional 18relation to C pointer 18three-dimensional 23two-dimensional 20f.unit-offset 18, 940f.variable dimension 20zero-offset 18Artificial viscosity 840, 846Ascending transformation, elliptic integrals262ASCII character set 5, 896, 903, 910Assembly language 278Associated Legendre polynomials 252f., 773recurrence relation for 253relation to Legendre polynomials 252Association, measures of 610, 628ff.Asymptotic series 167exponential integral 224Attenuation factors 590166ff.Accuracy 28f.achievable in minimization 398, 404, 410achievable in root finding 353contrasted with fidelity 841, 849CPU different from memory 186vs.
stability 710, 736, 839, 853Acknowledgments xiiAdams-Bashford-Moulton method 749Adams’ stopping criterion 373Adaptive integration 129, 141, 709, 714ff.,725ff., 733f., 737, 744, 749f., 797Adaptive Monte Carlo integration 316ff.,319ff.Addition, multiple precision 916Addition theorem, elliptic integrals 262ADI (alternating direction implicit) method856, 870f., 915Adjoint operator 876Adobe Illustrator xiii, xviiAdvective equation 835AGM (arithmetic geometric mean) 915Airy function 210, 240, 250routine for 250f.Aitken’s delta squared process 166Aitken’s interpolation algorithm 108Algorithms, non-numerical 889ff.Aliasing 501, 576see also Fourier transformAll-poles model 573see also Maximum entropy method (MEM)All-zeros model 573see also PeriodogramAllocation of storage 19, 21f., 940ff.Alternating-direction implicit method (ADI)856, 870f., 915Alternating series 166f.Alternative extended Simpson’s rule 134Amoeba 410see also Simplex, method of Nelder andMeadAmplification factor 837, 839, 841, 849, 854f.Amplitude error 840Analog-to-digital converter 821, 894Analyticity 201Analyze/factorize/operate package 71f., 833Anderson-Darling statistic 626f.Andrew’s sine 702Annealing, method of simulated 394f., 444ff.assessment 454f.for continuous variables 444, 451f.965966Autocorrelationin linear prediction 565use of FFT 545Wiener-Khinchin theorem 498, 574AUTODIN-II polynomial 898Autonomous differential equations 735f.Autoregressive model (AR) see Maximum entropy method (MEM)Average deviation of distribution 611Averaging kernel, in Backus-Gilbert method816B acksubstitution42, 47, 50, 98in band diagonal matrix 54in Cholesky decomposition 97complex equations 49direct for computing A−1 · B 48relaxation solution of boundary value problems 764in singular value decomposition 64Backtracking 427in quasi-Newton methods 384Backus-Gilbert method 815ff.Backward deflation 370Bader-Deuflhard method 737, 742f.Bairstow’s method 371, 376f.Balancing 483Band diagonal matrix 50, 51ff.backsubstitution 54LU decomposition 53f.multiply by vector 52f.storage 52Band-pass filter 558, 562wavelets 592, 599f.Bandwidth limited function 501Bank accounts, checksum for 902Bar codes, checksum for 902Bartlett window 554Base of representation 28, 890BASIC, Numerical Recipes in xv, 1Basis functions in general linear least squares671Bayes’ Theorem 819Bayesianapproach to inverse problems 808, 820,825f.contrasted with frequentist 819vs.
historic maximum entropy method825f.views on straight line fitting 670Bays’ shuffle 280Bernoulli number 138Bessel functions 230ff., 240ff.asymptotic form 230, 236complex 210continued fraction 240f., 246f.double precision 230fractional order 230, 240ff.Miller’s algorithm 181, 234modified 236ff.modified, fractional order 246ff.modified, normalization formula 239, 246modified, routines for 237ff.normalization formula 181recurrence relation 178, 231, 239, 241f.Indexreflection formulas 242reflection formulas, modified functions247routines for 232ff., 243ff.routines for modified functions 248f.series for 166, 230series for Kν 247series for Yν 242spherical 240, 251turning point 241Wronskian 240, 246Best-fit parameters 656, 662, 666, 703see also FittingBeta function 213incomplete see Incomplete beta functionBFGS algorithm see Broyden-Fletcher-GoldfarbShanno algorithmBias, of exponent 28Bias, removal in linear prediction 570Biconjugacy 84Biconjugate gradient methodelliptic partial differential equations 833preconditioning 85f., 833for sparse system 84f., 606Bicubic interpolation 125f.Bicubic spline 127f.Big-endian 302Bilinear interpolation 123f.Binomial coefficients 213recurrences for 215Binomial probability function 215cumulative 229deviates from 290, 295f.Binormal distribution 637, 695Biorthogonality 84Bisection 117, 366compared to minimum bracketing 397f.,399f.minimum finding with derivatives 406root finding 350, 353f., 359ff., 397, 476BISYNCH 898Bit 28reversal in fast Fourier transform (FFT)505f., 532Bitwise logical functions 296ff., 898f.Block-by-block method 797Block of statements 6Bode’s rule 132Boltzmann probability distribution 445Boltzmann’s constant 445Bootstrap method 691f.Bordering method for Toeplitz matrix 92f.Borwein and Borwein method for π 915Boundary 161f., 432f., 753Boundary conditionsfor differential equations 707f.initial value problems 708in multigrid method 877f.partial differential equations 514, 828ff.,857ff.for spheroidal harmonics 774two-point boundary value problems 708,753ff.Boundary value problems see Differentialequations; Elliptic partial differentialIndexequations; Two-point boundary valueproblemsBox-Muller algorithm for normal deviate 289Bracketingof function minimum 350, 397ff., 409of roots 348, 350ff., 360, 369, 371, 376,397Branch cut, for hypergeometric function 209f.Branching 8Break iteration 12f.Brenner, N.M.
506, 522Brent’s methodminimization 395f., 402ff., 666minimization, using derivative 396, 406root finding 348, 356, 666Broyden-Fletcher-Goldfarb-Shanno algorithm397, 426ff.Broyden’s method 380, 389ff., 393singular Jacobian 393Bubble sort 330Bugsin compilers xiiihow to report iv, xviiiBulirsch-Stoeralgorithm for rational function interpolation111f.method (differential equations) 209, 272,708f., 712, 722, 724ff., 733, 747method (differential equations), stepsizecontrol 725, 733f.for second order equations 733Burg’s LP algorithm 568Byte 28C ++ 7, 24C (programming language) 11ANSI 2f., 14, 25, 930, 941C++ 7, 24compilers 3control structures 5deficiencies 16, 24f., 26f.external functions 25features 15f.function declaration 17function definition 17header (.h) file 17implicit conversions 24f., 930Kernighan and Ritchie 2, 16, 24, 930nature of 15f.Numerical Recipes in xv, 1operator associativity 25f.operator precedence 25f.prototypes 2, 25, 930vectors in 18Calendar algorithms 1f., 11ff.Calibration 659Cards, sorting a hand of 330Carlson’s elliptic integrals 261f.Cash-Karp parameters 716f.Cauchy probability distribution see Lorentzianprobability distributionCauchy problem for partial differential equations 827f.Cayley’s representation of exp(−iHt) 853967CCITT (Comité Consultatif International Télégraphique et Téléphonique) 897f., 909CCITT polynomial 897f.Center of mass 305ff.Central limit theorem 658f.Central tendency, measures of 610ff.Change of variablein integration 144ff., 797in Monte Carlo integration 307f.in probability distribution 287ff.Characteristic polynomialdigital filter 561eigensystems 456, 475f.linear prediction 567matrix with a specified 375of recurrence relation 180Characteristics of partial differential equations827Chebyshev acceleration in successive overrelaxation (SOR) 868f.Chebyshev approximation 91, 130, 189, 190ff.Clenshaw-Curtis quadrature 196Clenshaw’s recurrence formula 193coefficients for 191contrasted with Padé approximation 201derivative of approximated function 189,195economization of series 198ff., 201for error function 220f.even function 194and fast cosine transform 519gamma functions 242integral of approximated function 195odd function 194polynomial fits derived from 197rational function 204ff.Remes exchange algorithm for filter 560Chebyshev polynomials 190ff.continuous orthonormality 190f.discrete orthonormality 191explicit formulas for 190formula for xk in terms of 199Check digit 901Checksum 889, 896cyclic redundancy (CRC) 896ff.Cherry, sundae without a 818Chi-by-eye 657Chi-square fitting see Fitting; Least squaresfittingChi-square probability function 216, 221,621, 660, 806as boundary of confidence region 693f.related to incomplete gamma function 221Chi-square test 620f.for binned data 620f.chi-by-eye 657and confidence limit estimation 693f.for contingency table 630ff.degrees of freedom 621f.for inverse problems 806least squares fitting 659ff.nonlinear models 681ff.rule of thumb 661for straight line fitting 661ff.968for straight line fitting, errors in both coordinates 666for two binned data sets 622unequal size samples 623Chip rate 300Chirp signal 563Cholesky decomposition 96ff., 430, 462backsubstitution 97operation count 97pivoting 97solution of normal equations 674Circulant 592Class, data type 7Clenshaw-Curtis quadrature 130, 196, 518,519Clenshaw’s recurrence formula 181ff., 196for Chebyshev polynomials 193stability 181ff.Clocking errors 899cn function 269Coarse-to-fine operator 873Coarse-grid correction 873f.Codingarithmetic 910ff.checksums 896decoding a Huffman-encoded message905Huffman 903ff.run-length 909variable length code 903Ziv-Lempel 903see also Arithmetic coding; Huffman codingCoefficientsbinomial 215for Gaussian quadrature 147ff.for Gaussian quadrature, nonclassical weightfunction 157ff., 797for quadrature formulas 131ff., 797Column degeneracy 32Column operations on matrix 37, 40f.Column totals 630Combinatorial minimization see AnnealingComité Consultatif International Télégraphiqueet Téléphonique (CCITT) 897f., 909Communication theory, use in adaptive integration 727Communications protocol 896Comparison function for rejection method290f.Complementary error functionsee Error functionComplete elliptic integral see Elliptic integralsComplex arithmetic 23f., 176ff., 948ff.avoidance of in path integration 209cubic equations 185for linear equations 49f.quadratic equations 184Complex error function 259Complex planefractal structure for Newton’s rule 367f.path integration for function evaluation208ff., 271poles in 111, 166, 208f., 213, 561, 573,724f.IndexComplex systems of linear equations 49f.complex.c utility functions 23f., 948ff.Compression of data 603, 889, 903ff., 910ff.Concordant pair for Kendall’s tau 642f.Condition number 61, 85Confidence level 692f., 696ff.Confidence limitsbootstrap method 692f.and chi-square 693f.confidence region, confidence interval 692on estimated model parameters 689ff.by Monte Carlo simulation 689ff.from singular value decomposition (SVD)698Confluent hypergeometric function 210, 246Conjugate directions 414f., 421ff.Conjugate gradient methodbiconjugate 84compared to variable metric method 425f.elliptic partial differential equations 833for minimization 396f., 420ff., 812f., 824minimum residual method 85preconditioner 85f.for sparse system 83ff., 606and wavelets 606Conservative differential equations 732f.Constrained linear inversion method 808ff.Constrained linear optimization see Linear programmingConstrained optimization 394Constraints, deterministic 813ff.Constraints, linear 431Contingency coefficient C 631Contingency table 628ff., 644statistics based on chi-square 630ff.statistics based on entropy 632ff.continue construction 14Continued fraction 169ff.Bessel functions 240f.convergence criterion 171equivalence transformation 172evaluation 169ff.evaluation along with normalization condition 247even and odd parts 172, 217, 222even part 255, 257exponential integral 222Fresnel integral 255incomplete beta function 227incomplete gamma function 217Lentz’s method 171, 219modified Lentz’s method 171Pincherle’s theorem 181ratio of Bessel functions 246rational function approximation 170, 217,227recurrence for evaluating 170f.and recurrence relation 181sine and cosine integrals 257Steed’s method 170f.tangent function 169typography for 169Continuous variable (statistics) 628Control structures 6, 8ff.bad 14969IndexConventions in C programs 25ff.Convergenceaccelerated, for series 166ff.of algorithm for π 915criteria for 353, 398f., 410, 489f., 495,684f., 767eigenvalues accelerated by shifting 477f.golden ratio 354, 406of golden section search 398f.of Levenberg-Marquardt method 684f.linear 353, 400of QL method 477f.quadratic 57, 358, 364f., 415f., 427, 915rate 353, 359f., 364f.recurrence relation 181of Ridders’ method 358series vs.
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