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implicit differencing 836FACR method 863finite difference method 830ff.finite element methods 833f.flux-conservative initial value problems834ff.forward Euler differencing 835f.Forward Time Centered Space (FTCS)836ff., 847ff., 852, 864Fourier analysis and cyclic reduction (FACR)857ff., 863Gauss-Seidel method (relaxation) 864,873ff., 884Godunov’s method 846Helmholtz equation 861hyperbolic 827, 834f.implicit differencing 848incomplete Cholesky conjugate gradientmethod (ICCG) 833inhomogeneous boundary conditions 859f.initial value problems 827f.initial value problems, recommendations on847ff.Jacobi’s method (relaxation) 864f., 873Laplace’s equation 827Lax method 837ff., 845, 854f.Lax method (multidimensional) 854f.matrix methods 833mesh-drift instability 843f.Monte Carlo methods 833multidimensional initial value problems853ff.multigrid method 833, 871ff.Neumann boundary conditions 829, 849,860, 867nonlinear diffusion equation 851nonlinear instability 840numerical dissipation or viscosity 839operator splitting 832, 856f., 870outgoing wave boundary conditions 829parabolic 827, 847ff.periodic boundary conditions 859, 867piecewise parabolic method (PPM) 846Poisson equation 827, 861rapid (Fourier) methods 514ff., 833, 857ff.relaxation methods 832, 863ff.985Schrödinger equation 851ff.second-order accuracy 842ff., 848f.shock 840, 846sparse matrices from 71spectral methods 833f.spectral radius 865ff., 871stability vs.
accuracy 839stability vs. efficiency 830staggered grids 519, 861staggered leapfrog method 842f.strongly implicit procedure 833successive over-relaxation (SOR) 866ff.,871, 875time splitting 856f., 870two-step Lax-Wendroff method 844ff.upwind differencing 841f., 846variational methods 833varieties of error 840ff.von Neumann stability analysis 836f.,839, 842, 849wave equation 827, 834f.see also Elliptic partial differential equations; Finite difference equations (FDEs)Partial pivoting 38Partition-exchange 332, 341Partitioned matrix, inverse of 77f.Party tricks 102ff., 174f.Parzen window 554Pascal 16, 18, 20Pascal, Numerical Recipes in xv, 1Path integration, for function evaluation 208ff.,271Pattern multiply of sparse matrices 81f.PBCG (preconditioned biconjugate gradientmethod) 85f., 833PC methods see Predictor-corrector methodsPCGPACK 78PDEs see Partial differential equationsPearson’s r 636ff.PECE method 749Pentagon, symmetries of 902Percentile 329Period of linear congruential generator 276Periodic boundary conditions 859, 867Periodogram 550ff., 574Lomb’s normalized 576f., 581f.variance of 552Perl (programming language) xiiiPerron’s theorems, for convergence of recurrence relations 180f.Perturbation methods for matrix inversion73ff.Peter Principle 337Phase error 840Phase-locked loop 705Phi statistic 631Phillips-Twomey method 808ff.Pi, computation of 915ff.Piecewise parabolic method (PPM) 846Pincherle’s theorem 181Pivot element 38, 41, 764in linear programming 435f.Pivoting 36, 38ff., 54, 73, 97full 38implicit 38, 46986in LU decomposition 45f.partial 38, 41, 46and QR decomposition 99in reduction to Hessenberg form 485in relaxation method 764for tridiagonal systems 51Pixel 525, 603, 812, 820Planck’s constant 851Plane rotation see Givens reduction; Jacobitransformation (or rotation)Platykurtic distribution 612Plotting of functions 349f.POCS (method of projection onto convex sets)814Poetry 5Pointerto array 18use for matrices 20, 33f., 940ff.Poisson equation 525, 827, 861Poisson probability functioncumulative 221deviates from 290, 293ff., 579semi-invariants of 614tails compared to Gaussian 659Poisson process 287, 291, 293Polak-Ribiere algorithm 396f., 422f.Poles see Complex plane, poles inPolishing of roots 365, 370f., 376f.Polymorphism 7Polynomial interpolation 105, 108ff.Aitken’s algorithm 108in Bulirsch-Stoer method 728, 730f.coefficients for 120ff.Lagrange’s formula 91, 108f.multidimensional 123ff.Neville’s algorithm 108f., 111, 140, 188pathology in determining coefficients for120in predictor-corrector method 748smoothing filters 650f.see also InterpolationPolynomials 173ff.algebraic manipulations 175approximating modified Bessel functions236approximation from Chebyshev coefficients197AUTODIN-II 898CCITT 897f.characteristic 375characteristic, for digital filters 561, 567characteristic, for eigenvalues of matrix456, 475f.Chebyshev 190ff.CRC-16 898deflation 369ff., 377derivatives of 173f.division 91, 175, 369, 377evaluation of 173evaluation of derivatives 173f.extrapolation in Bulirsch-Stoer method728, 730f.extrapolation in Romberg integration 140fitting 90, 120, 197, 650f., 671, 679f.generator for CRC 897f.Indexill-conditioned 369matrix method for roots 375minimax 192, 204monic 149multiplication 175operation count for 174f.orthonormal 149, 190f.primitive modulo 2 296ff., 311f., 897roots of 183ff., 369ff., 375shifting of 198f.stopping criterion in root finding 373Port, serial data 899Portability 2f., 16Portable random number generator see Random number generatorPositive definite matrix, testing for 97Positivity constraints 431Postal Service (U.S.), barcode 902PostScript xiii, xviiPowell’s method 396, 408, 412ff.Power (in a signal) 498f.Power series 165ff., 173f., 201economization of 198ff.Padé approximant of 200ff.Power spectral density see Fourier transform;Spectral densityPower spectrum estimation see Fourier transform; Spectral densityPPM (piecewise parabolic method) 846Precedence of operators, in C 25f.Precision, floating point 890Precision, multiple 915ff.Preconditioned biconjugate gradient method(PBCG) 85f.Preconditioning, in conjugate gradient methods833Predictor-corrector methods 708, 737, 747ff.Adams-Bashforth-Moulton schemes 749adaptive order methods 751compared to other methods 747f.fallacy of multiple correction 748f.with fixed number of iterations 749functional iteration vs.
Newton’s rule 749multivalue compared with multistep 749f.starting and stopping 750, 751stepsize control 749f.Prime numbers 924f.Primitive polynomials modulo 2 296ff., 311f.,897Principal directions 414f.Principal solution, of inverse problem 806Prize, $1000 offered 281Probability see Random number generator;Statistical testsProbability density, change of variables in287ff.Process loss 554Product Nystrom method 797Program(s)as black boxes xiv, 5, 35, 60, 212, 348,413dependencies 951ff.encapsulation 6f.interfaces 7modularization 6f.987Indexorganization 5ff.recipes by chapter and section xixtypography of 11validation 2f.Projection onto convex sets (POCS) 814Projection operator, nonexpansive 814Prolongation operator 873Protocol, for communications 896Prototypes in C 16f., 25, 930PSD (power spectral density) see Fouriertransform; Spectral densityPseudo-random numbers 274ff.Puns, particularly bad 173, 752, 755Pyramidal algorithm 594Pythagoreans 399Q L see EigensystemsQR see EigensystemsQR decomposition 98f., 389, 393backsubstitution 98and least squares 674operation count 98pivoting 99updating 100, 389use for orthonormal basis 66, 100Quadraticconvergence 57, 262, 358, 364f., 415f.,427, 915equations 29, 183ff., 398, 464interpolation 360, 371programming 443Quadrature 129ff.adaptive 129, 196, 797alternative extended Simpson’s rule 134arbitrary weight function 157ff., 797automatic 160Bode’s rule 132change of variable in 144ff., 797by Chebyshev fitting 130, 195classical formulas for 130ff.Clenshaw-Curtis 130, 196, 518f.closed formulas 131, 133f.and computer science 889by cubic splines 130error estimate in solution 793extended midpoint rule 135, 141f.extended rules 133ff., 140, 795, 797, 799extended Simpson’s rule 134Fourier integrals 584ff.Fourier integrals, infinite range 590f.Gauss-Chebyshev 151, 518f.Gauss-Hermite 151, 798Gauss-Jacobi 151Gauss-Kronrod 160Gauss-Laguerre 151, 798Gauss-Legendre 151, 792, 797Gauss-Lobatto 160, 196, 518Gauss-Radau 160Gaussian integration 133, 147ff., 790,792, 797Gaussian integration, nonclassical weightfunction 157ff., 797for improper integrals 141ff., 797f.for integral equations 790f., 795Monte Carlo 130, 162, 304ff., 316ff.multidimensional 130, 161ff.Newton-Cotes formulas 131ff., 147Newton-Cotes open formulas 132open formulas 131, 132f., 135f., 141related to differential equations 129related to predictor-corrector methods 747f.Romberg integration 130, 140f., 143, 188,723, 797semi-open formulas 135f.Simpson’s rule 132, 139, 143, 590, 791f.,797, 799Simpson’s three-eighths rule 132, 797,799singularity removal 144ff., 797f.singularity removal, worked example 801trapezoidal rule 131, 133, 136ff., 140,586, 590, 791f., 795using FFTs 130weight function log x 159see also Integration of functionsQuadrature mirror filter 592, 600Quantum mechanics, Uncertainty Principle607Quartile value 329Quasi-Newton methods for minimization 397,425ff.Quasi-random sequence 309ff., 327, 889, 896Halton’s 309f.for Monte Carlo integration 313ff., 319,327Sobol’s 311see also Random number generatorQuicksort 329, 332ff., 338, 341Quotient-difference algorithm 170R -estimates699f.Radioactive decay 287Radix base for floating point arithmetic 483,890, 916, 922Radix conversion 910, 914, 922Ramanujan’s identity for π 924RAND_MAX macro 275f., 277Random bits, generation of 296ff.Random deviates 274ff.binomial 295f.exponential 287f.gamma distribution 290ff.Gaussian 275, 288f., 578, 807normal 275, 288f., 578Poisson 293ff., 579quasi-random sequences 309ff., 889, 896uniform 275ff.uniform integer 280, 283ff.Random number generator 274ff.bitwise operations 296ff.Box-Muller algorithm 289Data Encryption Standard 300ff.good choices for modulus, multiplier andincrement 284f.for integer-valued probability distribution293integer vs.
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