General Index Accelerated convergence of series 166ff. schedule 445 Accuracy 28f. thermodynamic analogy 444f. achievable in minimization 398,404,410 traveling salesman problem 445ff. achievable in root finding 353 ANSI C standard 2f.,14,25,930,941 contrasted with fidelity 841,849 ANSI macro 17,930 CPU different from memory 186 Antonov-Saleev variant of Sobol'sequence vs.stability 710,736,839,853 310E Acknowledgments xii Apple xvii Adams-Bashford-Moulton method 749 Macintosh 894 Adams'stopping criterion 373 Approximate inverse of matrix 57 Adaptive integration 129,141.709.714ff.. Approximation of functions 105f. 725ff,733f,737,744,749f,797 by Chebyshev polynomials 191f,519 Adaptive Monte Carlo integration 316ff., Pade approximant 200ff. 319f by rational functions 204ff Addition,multiple precision 916 by wavelets 601f..791 Addition theorem,elliptic integrals 262 see also Fitting ADI (altemating direction implicit)method Arguments,conversion of data types 24f.,930 856.870f.915 Arithmetic Adjoint operator 876 arbitrary precision 889.915ff. Adobe Illustrator xiii,xvii complex 23f.,948ff. Advective equation 835 floating point 889 AGM (arithmetic geometric mean)915 IEEE standard 285,890f Airy function 210,240.250 rounding 890 routine for 250f. Arithmetic coding 889,910ff. Aitken's delta squared process 166 Arithmetic-geometric mean (AGM)method Aitken's interpolation algorithm 108 915 Algorithms,non-numerical 889ff. Array Aliasing 501,576 centered subarray of 119 see also Fourier transform how to allocate 19 All-poles model 573 index range 18 see also Maximum entropy method(MEM) one-dimensional 18 All-zeros model 573 relation to C pointer 18 see also Periodogram three-dimensional 23 Allocation of storage 19,21f.,940ff. two-dimensional 20f. Alternating-direction implicit method (ADI) unit-offset 18,940f. 856.870f,915 variable dimension 20 Alternating series 166f zero-offset 18 Alternative extended Simpson's rule 134 Artificial viscosity 840,846 Amoeba 410 Ascending transformation,elliptic integrals see also Simplex,method of Nelder and 262 Mead ASCII character set 5,896,903,910 Amplification factor 837.839.841,849,854f Assembly language 278 Amplitude error 840 Associated Legendre polynomials 252f.,773 Analog-to-digital converter 821,894 recurrence relation for 253 Analyticity 201 relation to Legendre polynomials 252 Analyze/factorize/operate package 71f.,833 Association,measures of 610,628ff. Anderson-Darling statistic 626f. Asymptotic series 167 Andrew's sine 702 exponential integral 224 Annealing,method of simulated 394f.,444ff. Attenuation factors 590 assessment 454f. Autocorrelation for continuous variables 444,451f. in linear prediction 565 965
General Index Accelerated convergence of series 166ff. Accuracy 28f. achievable in minimization 398, 404, 410 achievable in root finding 353 contrasted with fidelity 841, 849 CPU different from memory 186 vs. stability 710, 736, 839, 853 Acknowledgments xii Adams-Bashford-Moulton method 749 Adams’ stopping criterion 373 Adaptive integration 129, 141, 709, 714ff., 725ff., 733f., 737, 744, 749f., 797 Adaptive Monte Carlo integration 316ff., 319ff. Addition, multiple precision 916 Addition theorem, elliptic integrals 262 ADI (alternating direction implicit) method 856, 870f., 915 Adjoint operator 876 Adobe Illustrator xiii, xvii Advective equation 835 AGM (arithmetic geometric mean) 915 Airy function 210, 240, 250 routine for 250f. Aitken’s delta squared process 166 Aitken’s interpolation algorithm 108 Algorithms, non-numerical 889ff. Aliasing 501, 576 see also Fourier transform All-poles model 573 see also Maximum entropy method (MEM) All-zeros model 573 see also Periodogram Allocation of storage 19, 21f., 940ff. Alternating-direction implicit method (ADI) 856, 870f., 915 Alternating series 166f. Alternative extended Simpson’s rule 134 Amoeba 410 see also Simplex, method of Nelder and Mead Amplification factor 837, 839, 841, 849, 854f. Amplitude error 840 Analog-to-digital converter 821, 894 Analyticity 201 Analyze/factorize/operate package 71f., 833 Anderson-Darling statistic 626f. Andrew’s sine 702 Annealing, method of simulated 394f., 444ff. assessment 454f. for continuous variables 444, 451f. schedule 445 thermodynamic analogy 444f. traveling salesman problem 445ff. ANSI C standard 2f., 14, 25, 930, 941 ANSI macro 17, 930 Antonov-Saleev variant of Sobol’ sequence 310ff. Apple xvii Macintosh 894 Approximate inverse of matrix 57 Approximation of functions 105f. by Chebyshev polynomials 191f., 519 Pade approximant 200ff. ´ by rational functions 204ff. by wavelets 601f., 791 see also Fitting Arguments, conversion of data types 24f., 930 Arithmetic arbitrary precision 889, 915ff. complex 23f., 948ff. floating point 889 IEEE standard 285, 890f. rounding 890 Arithmetic coding 889, 910ff. Arithmetic-geometric mean (AGM) method 915 Array centered subarray of 119 how to allocate 19 index range 18 one-dimensional 18 relation to C pointer 18 three-dimensional 23 two-dimensional 20f. unit-offset 18, 940f. variable dimension 20 zero-offset 18 Artificial viscosity 840, 846 Ascending transformation, elliptic integrals 262 ASCII character set 5, 896, 903, 910 Assembly language 278 Associated Legendre polynomials 252f., 773 recurrence relation for 253 relation to Legendre polynomials 252 Association, measures of 610, 628ff. Asymptotic series 167 exponential integral 224 Attenuation factors 590 Autocorrelation in linear prediction 565 965
966 Index use of FFT 545 reflection formulas,modified functions Wiener-Khinchin theorem 498,574 247 AUTODIN-II polynomial 898 routines for 232ff.,243ff Autonomous differential equations 735f. routines for modified functions 248f. Autoregressive model (AR)see Maximum en- series for 166,230 tropy method(MEM) series for K 247 Average deviation of distribution 611 series for Y,242 Averaging kernel,in Backus-Gilbert method spherical 240,251 816 turning point 241 Wronskian 240.246 Best-fit parameters 656,662,666,703 Backsubstitution 42,47,50,98 see also Fitting in band diagonal matrix 54 Beta function 213 in Cholesky decomposition 97 incomplete see Incomplete beta function complex equations 49 BFGS algorithm see Broyden-Fletcher-Goldfarb- direct for computing A-1.B 48 Shanno algorithm relaxation solution of boundary value prob- Bias,of exponent 28 lems 764 Bias,removal in linear prediction 570 in singular value decomposition 64 Biconjugacy 84 Backtracking 427 Biconjugate gradient method in quasi-Newton methods 384 elliptic partial differential equations 833 Backus-Gilbert method 815ff preconditioning 85f.,833 Backward deflation 370 for sparse system 84f.,606 Bader-Deufthard method 737.742f. Bicubic interpolation 125f. Bairstow's method 371,376f. Bicubic spline 127f. Balancing 483 Band diagonal matrix 50,51ff Big-endian 302 Bilinear interpolation 123f backsubstitution 54 LU decomposition 53f. Binomial coefficients 213 multiply by vector 52f recurrences for 215 storage 52 Binomial probability function 215 Band-pass filter 558,562 cumulative 229 wavelets 592,599f. deviates from 290,295f. Bandwidth limited function 501 Binormal distribution 637,695 Bank accounts,checksum for 902 Biorthogonality 84 Bar codes,checksum for 902 Bisection 117,366 Bartlett window 554 compared to minimum bracketing 397f., 300f Base of representation 28,890 BASIC,Numerical Recipes in xv,1 minimum finding with derivatives 406 Basis functions in general linear least squares root finding350.353f.,359f.397.476 671 BISYNCH 898 Bayes'Theorem 819 Bit 28 Bayesian reversal in fast Fourier transform(FFT) approach to inverse problems 808,820, 505f,532 825f Bitwise logical functions 296ff.,898f. contrasted with frequentist 819 Block-by-block method 797 vs.historic maximum entropy method Block of statements 6 825f Bode's rule 132 views on straight line fitting 670 Boltzmann probability distribution 445 Bays'shuffle 280 Boltzmann's constant 445 Beroulli number 138 Bootstrap method 691f. Bessel functions 230ff.,240ff. Bordering method for Toeplitz matrix 92f. asymptotic form 230,236 Borwein and Borwein method for 915 complex 210 Boundary 161f..432f..753 continued fraction 240f,246f Boundary conditions double precision 230 for differential equations 707f. fractional order 230,240ff. initial value problems 708 Miller's algorithm 181,234 in multigrid method 877f. modified 236ff. partial differential equations 514,828ff., modified,fractional order 246ff. 857f modified,normalization formula 239,246 for spheroidal harmonics 774 modified,routines for 237ff two-point boundary value problems 708, normalization formula 181 753狂 recurrence relation 178,231,239,241f. Boundary value problems see Differential reflection formulas 242 equations;Elliptic partial differential
966 Index use of FFT 545 Wiener-Khinchin theorem 498, 574 AUTODIN-II polynomial 898 Autonomous differential equations 735f. Autoregressive model (AR) see Maximum entropy method (MEM) Average deviation of distribution 611 Averaging kernel, in Backus-Gilbert method 816 Backsubstitution 42, 47, 50, 98 in band diagonal matrix 54 in Cholesky decomposition 97 complex equations 49 direct for computing A−1 · B 48 relaxation solution of boundary value problems 764 in singular value decomposition 64 Backtracking 427 in quasi-Newton methods 384 Backus-Gilbert method 815ff. Backward deflation 370 Bader-Deuflhard method 737, 742f. Bairstow’s method 371, 376f. Balancing 483 Band diagonal matrix 50, 51ff. backsubstitution 54 LU decomposition 53f. multiply by vector 52f. storage 52 Band-pass filter 558, 562 wavelets 592, 599f. Bandwidth limited function 501 Bank accounts, checksum for 902 Bar codes, checksum for 902 Bartlett window 554 Base of representation 28, 890 BASIC, Numerical Recipes in xv, 1 Basis functions in general linear least squares 671 Bayes’ Theorem 819 Bayesian approach to inverse problems 808, 820, 825f. contrasted with frequentist 819 vs. historic maximum entropy method 825f. views on straight line fitting 670 Bays’ shuffle 280 Bernoulli number 138 Bessel functions 230ff., 240ff. asymptotic form 230, 236 complex 210 continued fraction 240f., 246f. double precision 230 fractional order 230, 240ff. Miller’s algorithm 181, 234 modified 236ff. modified, fractional order 246ff. modified, normalization formula 239, 246 modified, routines for 237ff. normalization formula 181 recurrence relation 178, 231, 239, 241f. reflection formulas 242 reflection formulas, modified functions 247 routines for 232ff., 243ff. routines for modified functions 248f. series for 166, 230 series for Kν 247 series for Yν 242 spherical 240, 251 turning point 241 Wronskian 240, 246 Best-fit parameters 656, 662, 666, 703 see also Fitting Beta function 213 incomplete see Incomplete beta function BFGS algorithm see Broyden-Fletcher-GoldfarbShanno algorithm Bias, of exponent 28 Bias, removal in linear prediction 570 Biconjugacy 84 Biconjugate gradient method elliptic partial differential equations 833 preconditioning 85f., 833 for sparse system 84f., 606 Bicubic interpolation 125f. Bicubic spline 127f. Big-endian 302 Bilinear interpolation 123f. Binomial coefficients 213 recurrences for 215 Binomial probability function 215 cumulative 229 deviates from 290, 295f. Binormal distribution 637, 695 Biorthogonality 84 Bisection 117, 366 compared to minimum bracketing 397f., 399f. minimum finding with derivatives 406 root finding 350, 353f., 359ff., 397, 476 BISYNCH 898 Bit 28 reversal in fast Fourier transform (FFT) 505f., 532 Bitwise logical functions 296ff., 898f. Block-by-block method 797 Block of statements 6 Bode’s rule 132 Boltzmann probability distribution 445 Boltzmann’s constant 445 Bootstrap method 691f. Bordering method for Toeplitz matrix 92f. Borwein and Borwein method for π 915 Boundary 161f., 432f., 753 Boundary conditions for differential equations 707f. initial value problems 708 in multigrid method 877f. partial differential equations 514, 828ff., 857ff. for spheroidal harmonics 774 two-point boundary value problems 708, 753ff. Boundary value problems see Differential equations; Elliptic partial differential
Index 967 equations;Two-point boundary value CCITT (Comite Consultatif International Tele- problems graphique et Telephonique)897f.909 Box-Muller algorithm for normal deviate 289 CCITT polynomial 897f. Bracketing Center of mass 305ff. of function minimum 350,397ff.,409 Central limit theorem 658f. of roots348.350f,360.369,371,376 Central tendency,measures of 610ff. 397 Change of variable Branch cut,for hypergeometric function 209f. in integration 144ff.,797 Branching 8 in Monte Carlo integration 307f. Break iteration 12f. in probability distribution 287ff. Brenner,N.M.506,522 Characteristic polynomial Brent's method digital filter 561 minimization 395f.,402ff.,666 minimization,using derivative 396,406 eigensystems 456,475f. linear prediction 567 root finding 348,356,666 matrix with a specified 375 Broyden-Fletcher-Goldfarb-Shanno algorithm of recurrence relation 180 397,426ft. Broyden's method 380,389ff.,393 Characteristics of partial differential equations 827 singular Jacobian 393 Chebyshev acceleration in successive over- Bubble sort 330 relaxation (SOR)868f. Bugs in compilers xiii Chebyshev approximation 91,130.189,190ff. Clenshaw-Curtis quadrature 196 how to report iv,xviii Clenshaw's recurrence formula 193 Bulirsch-Stoer coefficients for 191 algorithm for rational function interpolation 111f. contrasted with Pade approximation 201 method (differential equations)209.272. derivative of approximated function 189. 708f,712,722,724f,733,747 195 method (differential equations),stepsize economization of series 198ff.,201 control 725,733f. for error function 220f. for second order equations 733 even function 194 Burg's LP algorithm 568 and fast cosine transform 519 Byte 28 gamma functions 242 integral of approximated function 195 odd function 194 C+7,24 polynomial fits derived from 197 C(programming language)11 rational function 204ff. ANSI2f,14,25,930,941 Remes exchange algorithm for filter 560 C++7.24 Chebyshev polynomials 190ff. compilers 3 continuous orthonormality 190f. discrete orthonormality 191 control structures 5 deficiencies 16,24f.,26f. explicit formulas for 190 exteral functions 25 formula for x in terms of 199 Check digit 901 features 15f. function declaration 17 Checksum 889,896 function definition 17 cyclic redundancy (CRC)896ff. header (.h)file 17 Cherry,sundae without a 818 implicit conversions 24f,930 Chi-by-eye 657 Kernighan and Ritchie 2,16,24,930 Chi-square fitting see Fitting;Least squares nature of 15f. fitting Numerical Recipes in xv,I Chi-square probability function 216,221, operator associativity 25f. 621,660,806 operator precedence 25f. as boundary of confidence region 693f. prototypes 2,25,930 related to incomplete gamma function 221 vectors in 18 Chi-square test 620f. Calendar algorithms If.,11ff. for binned data 620f. Calibration 659 chi-by-eye 657 Cards,sorting a hand of 330 and confidence limit estimation 693f. Carlson's elliptic integrals 261f. for contingency table 630ff. Cash-Karp parameters 716f. degrees of freedom 621f. Cauchy probability distribution see Lorentzian for inverse problems 806 probability distribution least squares fitting 659ff. Cauchy problem for partial differential equa- nonlinear models 681ff. tions 827f. rule of thumb 661 Cayley's representation of exp(-iHt)853 for straight line fitting 661ff
Index 967 equations; Two-point boundary value problems Box-Muller algorithm for normal deviate 289 Bracketing of function minimum 350, 397ff., 409 of roots 348, 350ff., 360, 369, 371, 376, 397 Branch cut, for hypergeometric function 209f. Branching 8 Break iteration 12f. Brenner, N.M. 506, 522 Brent’s method minimization 395f., 402ff., 666 minimization, using derivative 396, 406 root finding 348, 356, 666 Broyden-Fletcher-Goldfarb-Shanno algorithm 397, 426ff. Broyden’s method 380, 389ff., 393 singular Jacobian 393 Bubble sort 330 Bugs in compilers xiii how to report iv, xviii Bulirsch-Stoer algorithm for rational function interpolation 111f. method (differential equations) 209, 272, 708f., 712, 722, 724ff., 733, 747 method (differential equations), stepsize control 725, 733f. for second order equations 733 Burg’s LP algorithm 568 Byte 28 C++ 7, 24 C (programming language) 11 ANSI 2f., 14, 25, 930, 941 C++ 7, 24 compilers 3 control structures 5 deficiencies 16, 24f., 26f. external functions 25 features 15f. function declaration 17 function definition 17 header (.h) file 17 implicit conversions 24f., 930 Kernighan and Ritchie 2, 16, 24, 930 nature of 15f. Numerical Recipes in xv, 1 operator associativity 25f. operator precedence 25f. prototypes 2, 25, 930 vectors in 18 Calendar algorithms 1f., 11ff. Calibration 659 Cards, sorting a hand of 330 Carlson’s elliptic integrals 261f. Cash-Karp parameters 716f. Cauchy probability distribution see Lorentzian probability distribution Cauchy problem for partial differential equations 827f. Cayley’s representation of exp(−iHt) 853 CCITT (Comite Consultatif International T ´ el´ e-´ graphique et Tel´ ephonique) 897f., 909 ´ CCITT polynomial 897f. Center of mass 305ff. Central limit theorem 658f. Central tendency, measures of 610ff. Change of variable in integration 144ff., 797 in Monte Carlo integration 307f. in probability distribution 287ff. Characteristic polynomial digital filter 561 eigensystems 456, 475f. linear prediction 567 matrix with a specified 375 of recurrence relation 180 Characteristics of partial differential equations 827 Chebyshev acceleration in successive overrelaxation (SOR) 868f. Chebyshev approximation 91, 130, 189, 190ff. Clenshaw-Curtis quadrature 196 Clenshaw’s recurrence formula 193 coefficients for 191 contrasted with Pade approximation 201 ´ derivative of approximated function 189, 195 economization of series 198ff., 201 for error function 220f. even function 194 and fast cosine transform 519 gamma functions 242 integral of approximated function 195 odd function 194 polynomial fits derived from 197 rational function 204ff. Remes exchange algorithm for filter 560 Chebyshev polynomials 190ff. continuous orthonormality 190f. discrete orthonormality 191 explicit formulas for 190 formula for xk in terms of 199 Check digit 901 Checksum 889, 896 cyclic redundancy (CRC) 896ff. Cherry, sundae without a 818 Chi-by-eye 657 Chi-square fitting see Fitting; Least squares fitting Chi-square probability function 216, 221, 621, 660, 806 as boundary of confidence region 693f. related to incomplete gamma function 221 Chi-square test 620f. for binned data 620f. chi-by-eye 657 and confidence limit estimation 693f. for contingency table 630ff. degrees of freedom 621f. for inverse problems 806 least squares fitting 659ff. nonlinear models 681ff. rule of thumb 661 for straight line fitting 661ff
968 Index for straight line fitting,errors in both coor- Complex systems of linear equations 49f. dinates 666 complex.c utility functions 23f..948ff. for two binned data sets 622 Compression of data 603,889,903ff.,910ff unequal size samples 623 Concordant pair for Kendall's tau 642f. Chip rate 300 Condition number 61,85 Chirp signal 563 Confidence level 692f.,696ff. Cholesky decomposition 96ff.,430.462 Confidence limits backsubstitution 97 bootstrap method 692f operation count 97 and chi-square 693f. pivoting 97 confidence region,confidence interval 692 solution of normal equations 674 on estimated model parameters 689ff. Circulant 592 by Monte Carlo simulation 689ff. Class,data type 7 from singular value decomposition (SVD) Clenshaw-Curtis quadrature 130,196,518 698 519 Confluent hypergeometric function 210,246 Clenshaw's recurrence formula 181ff.,196 Conjugate directions 414f.,421ff. for Chebyshev polynomials 193 Conjugate gradient method stability 181ff. biconjugate 84 Clocking errors 899 compared to variable metric method 425f. cn function 269 elliptic partial differential equations 833 Coarse-to-fine operator 873 for minimization 396f.,420ff.,812f,824 Coarse-grid correction 873f. minimum residual method 85 Coding preconditioner 85f. arithmetic 910ff. for sparse system 83ff.,606 checksums 896 and wavelets 606 decoding a Huffman-encoded message Conservative differential equations 732f 905 Constrained linear inversion method 808ff. Huffman 903ff. Constrained linear optimization see Linear pro- run-length 909 gramming variable length code 903 Constrained optimization 394 Ziv-Lempel 903 Constraints,deterministic 813ff. see also Arithmetic coding;Huffman cod- Constraints,linear 431 ing Contingency coefficient C 631 Coefficients Contingency table 628ff.,644 binomial 215 statistics based on chi-square 630f. for Gaussian quadrature 147ff. statistics based on entropy 632ff. for Gaussian quadrature,nonclassical weight continue construction 14 function 157ff.,797 Continued fraction 169ff. for quadrature formulas 131f,797 Bessel functions 240f. Column degeneracy 32 convergence criterion 171 Column operations on matrix 37,40f. equivalence transformation 172 Column totals 630 evaluation 169ff. Combinatorial minimization see Annealing evaluation along with normalization condi- Comite Consultatif Intemational Telegraphique tion 247 et Telephonique (CCITT)897f.,909 even and odd parts 172,217,222 Communication theory,use in adaptive integra- even part 255,257 tion 727 exponential integral 222 Communications protocol 896 Fresnel integral 255 Comparison function for rejection method incomplete beta function 227 290f incomplete gamma function 217 Complementary error function Lentz's method 171,219 see Error function modified Lentz's method 171 Complete elliptic integral see Elliptic integrals Pincherle's theorem 181 Complex arithmetic 23f.,176ff.,948ff. ratio of Bessel functions 246 avoidance of in path integration 209 rational function approximation 170,217, cubic equations 185 227 for linear equations 49f. recurrence for evaluating 170f. quadratic equations 184 and recurrence relation 181 Complex error function 259 sine and cosine integrals 257 Complex plane Steed's method 170f. fractal structure for Newton's rule 367f. tangent function 169 path integration for function evaluation typography for 169 208ff,271 Continuous variable (statistics)628 poles in11l,166,208f,213,561,573, Control structures 6,8ff. 724f bad 14
968 Index for straight line fitting, errors in both coordinates 666 for two binned data sets 622 unequal size samples 623 Chip rate 300 Chirp signal 563 Cholesky decomposition 96ff., 430, 462 backsubstitution 97 operation count 97 pivoting 97 solution of normal equations 674 Circulant 592 Class, data type 7 Clenshaw-Curtis quadrature 130, 196, 518, 519 Clenshaw’s recurrence formula 181ff., 196 for Chebyshev polynomials 193 stability 181ff. Clocking errors 899 cn function 269 Coarse-to-fine operator 873 Coarse-grid correction 873f. Coding arithmetic 910ff. checksums 896 decoding a Huffman-encoded message 905 Huffman 903ff. run-length 909 variable length code 903 Ziv-Lempel 903 see also Arithmetic coding; Huffman coding Coefficients binomial 215 for Gaussian quadrature 147ff. for Gaussian quadrature, nonclassical weight function 157ff., 797 for quadrature formulas 131ff., 797 Column degeneracy 32 Column operations on matrix 37, 40f. Column totals 630 Combinatorial minimization see Annealing Comite Consultatif International T ´ el´ egraphique ´ et Tel´ ephonique (CCITT) 897f., 909 ´ Communication theory, use in adaptive integration 727 Communications protocol 896 Comparison function for rejection method 290f. Complementary error function see Error function Complete elliptic integral see Elliptic integrals Complex arithmetic 23f., 176ff., 948ff. avoidance of in path integration 209 cubic equations 185 for linear equations 49f. quadratic equations 184 Complex error function 259 Complex plane fractal structure for Newton’s rule 367f. path integration for function evaluation 208ff., 271 poles in 111, 166, 208f., 213, 561, 573, 724f. Complex 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, 85 Confidence level 692f., 696ff. Confidence limits bootstrap method 692f. and chi-square 693f. confidence region, confidence interval 692 on estimated model parameters 689ff. by Monte Carlo simulation 689ff. from singular value decomposition (SVD) 698 Confluent hypergeometric function 210, 246 Conjugate directions 414f., 421ff. Conjugate gradient method biconjugate 84 compared to variable metric method 425f. elliptic partial differential equations 833 for minimization 396f., 420ff., 812f., 824 minimum residual method 85 preconditioner 85f. for sparse system 83ff., 606 and wavelets 606 Conservative differential equations 732f. Constrained linear inversion method 808ff. Constrained linear optimization see Linear programming Constrained optimization 394 Constraints, deterministic 813ff. Constraints, linear 431 Contingency coefficient C 631 Contingency table 628ff., 644 statistics based on chi-square 630ff. statistics based on entropy 632ff. continue construction 14 Continued fraction 169ff. Bessel functions 240f. convergence criterion 171 equivalence transformation 172 evaluation 169ff. evaluation along with normalization condition 247 even and odd parts 172, 217, 222 even part 255, 257 exponential integral 222 Fresnel integral 255 incomplete beta function 227 incomplete gamma function 217 Lentz’s method 171, 219 modified Lentz’s method 171 Pincherle’s theorem 181 ratio of Bessel functions 246 rational function approximation 170, 217, 227 recurrence for evaluating 170f. and recurrence relation 181 sine and cosine integrals 257 Steed’s method 170f. tangent function 169 typography for 169 Continuous variable (statistics) 628 Control structures 6, 8ff. bad 14
Index 969 Conventions in C programs 25ff. routine for 258f Convergence series 257 accelerated,for series 166ff Cosine transform see Fast Fourier transform of algorithm forπ915 (FFT);Fourier transform criteria for353,398f,410,489f.,495 Coulomb wave function 210,240 684f,767 Courant condition 838,841,843,845 eigenvalues accelerated by shifting 477f multidimensional 855 golden ratio 354,406 Courant-Friedrichs-Lewy stability criterion see of golden section search 398f. Courant condition of Levenberg-Marquardt method 684f Covariance linear 353,400 a priori 705 of QL method 477f. in general linear least squares 673,677 quadratic57,358,364f.,415f,427,915 matrix,by Cholesky decomposition 98, ate353,359f,364f. 673 recurrence relation 181 matrix,of errors 805,817 of Ridders'method 358 matrix,is inverse of Hessian matrix 685 series vs.continued fraction 169f. matrix,when it is meaningful 695f. and spectral radius 865ff.,871 in nonlinear models 685,687 convert_matrix()utility 945 relation to chi-square 695ff. Convex sets,use in inverse problems 813f. from singular value decomposition (SVD) Convolution 698 denoted by asterisk 498 in straight line fitting 663 finite impulse response(FIR)538 CR method see Cyclic reduction (CR) of functions 498.509 Cramer's V 631 of large data sets 543f. Crank-Nicolson method 848,853,855 for multiple precision arithmetic 918 CRC (cyclic redundancy check)896ff. multiplication as 918 CRC-12898 necessity for optimal filtering 542 CRC-16 polynomial 898 overlap-add method 544 CRC-CCITT 898 overlap-save method 543f. Creativity,essay on 8 and polynomial interpolation 120 Critical (Nyquist)sampling 500,550 relation to wavelet transform 1592 Cross(denotes matrix outer product)73 theorem 498,538ff.553 Crosstabulation analysis 629 theorem,discrete 538ff. see also Contingency table treatment of end effects 540 Crout's algorithm 44ff.,53 use of FFT 530,538ff. Cubic equations 183ff.,367 wraparound problem 540 Cubic spline interpolation 113ff. Cooley-Tukey FFT algorithm 509 see also Spline Co-processor,floating point 894 Cumulative binomial distribution 226,229 Copyright rules xvi Cumulative Poisson function 221 Cornwell-Evans algorithm 825 related to incomplete gamma function 221 Corporate promotion ladder 336f. Curvature matrix see Hessian matrix Corrected two-pass algorithm 613 cvector()utility 943 Correction,in multigrid method 872 Cycle,in multigrid method 874 Correlation coefficient (linear)636ff. Cyclic Jacobi method 466 Correlation function 498 Cyclic reduction (CR)857f.,861f. autocorrelation 498,546,565 Cyclic redundancy check(CRC)896ff. and Fourier transforms 498 Cyclic tridiagonal systems 74f. theorem 498.545 treatment of end effects 546 using FFT 545f. Danielson-Lanczos lemma 504f..532 Wiener-Khinchin theorem 498.574 Data Correlation,statistical 609f.,628 assigning keys to 897 Kendall's tau 640,642ff. continuous vs.binned 620 linear correlation coefficient 636ff.,664 entropy 632ff,903f. linear related to least square fitting 636 essay on 609 664 fitting 656ff. nonparametric or rank statistical 639ff. fraudulent 661 among parameters in a fit 663,673,676 glitches in 659 in random number generators 277 iid (independent and identically distributed) Spearman rank-order coefficient 640f. 691 sum squared difference of ranks 640 modeling 656ff. Cosine function,recurrence 178 serial port 899 Cosine integral 255,257ff. smoothing 610,650ff. continued fraction 257 statistical tests 609ff
Index 969 Conventions in C programs 25ff. Convergence accelerated, for series 166ff. of algorithm for π 915 criteria for 353, 398f., 410, 489f., 495, 684f., 767 eigenvalues accelerated by shifting 477f. golden ratio 354, 406 of golden section search 398f. of Levenberg-Marquardt method 684f. linear 353, 400 of QL method 477f. quadratic 57, 358, 364f., 415f., 427, 915 rate 353, 359f., 364f. recurrence relation 181 of Ridders’ method 358 series vs. continued fraction 169f. and spectral radius 865ff., 871 convert_matrix() utility 945 Convex sets, use in inverse problems 813f. Convolution denoted by asterisk 498 finite impulse response (FIR) 538 of functions 498, 509 of large data sets 543f. for multiple precision arithmetic 918 multiplication as 918 necessity for optimal filtering 542 overlap-add method 544 overlap-save method 543f. and polynomial interpolation 120 relation to wavelet transform 592 theorem 498, 538ff., 553 theorem, discrete 538ff. treatment of end effects 540 use of FFT 530, 538ff. wraparound problem 540 Cooley-Tukey FFT algorithm 509 Co-processor, floating point 894 Copyright rules xvi Cornwell-Evans algorithm 825 Corporate promotion ladder 336f. Corrected two-pass algorithm 613 Correction, in multigrid method 872 Correlation coefficient (linear) 636ff. Correlation function 498 autocorrelation 498, 546, 565 and Fourier transforms 498 theorem 498, 545 treatment of end effects 546 using FFT 545f. Wiener-Khinchin theorem 498, 574 Correlation, statistical 609f., 628 Kendall’s tau 640, 642ff. linear correlation coefficient 636ff., 664 linear related to least square fitting 636, 664 nonparametric or rank statistical 639ff. among parameters in a fit 663, 673, 676 in random number generators 277 Spearman rank-order coefficient 640f. sum squared difference of ranks 640 Cosine function, recurrence 178 Cosine integral 255, 257ff. continued fraction 257 routine for 258f. series 257 Cosine transform see Fast Fourier transform (FFT); Fourier transform Coulomb wave function 210, 240 Courant condition 838, 841, 843, 845 multidimensional 855 Courant-Friedrichs-Lewy stability criterion see Courant condition Covariance a priori 705 in general linear least squares 673, 677 matrix, by Cholesky decomposition 98, 673 matrix, of errors 805, 817 matrix, is inverse of Hessian matrix 685 matrix, when it is meaningful 695ff. in nonlinear models 685, 687 relation to chi-square 695ff. from singular value decomposition (SVD) 698 in straight line fitting 663 CR method see Cyclic reduction (CR) Cramer’s V 631 Crank-Nicolson method 848, 853, 855 CRC (cyclic redundancy check) 896ff. CRC-12 898 CRC-16 polynomial 898 CRC-CCITT 898 Creativity, essay on 8 Critical (Nyquist) sampling 500, 550 Cross (denotes matrix outer product) 73 Crosstabulation analysis 629 see also Contingency table Crout’s algorithm 44ff., 53 Cubic equations 183ff., 367 Cubic spline interpolation 113ff. see also Spline Cumulative binomial distribution 226, 229 Cumulative Poisson function 221 related to incomplete gamma function 221 Curvature matrix see Hessian matrix cvector() utility 943 Cycle, in multigrid method 874 Cyclic Jacobi method 466 Cyclic reduction (CR) 857f., 861f. Cyclic redundancy check (CRC) 896ff. Cyclic tridiagonal systems 74f. Danielson-Lanczos lemma 504f., 532 Data assigning keys to 897 continuous vs. binned 620 entropy 632ff., 903f. essay on 609 fitting 656ff. fraudulent 661 glitches in 659 iid (independent and identically distributed) 691 modeling 656ff. serial port 899 smoothing 610, 650ff. statistical tests 609ff
970 Index unevenly or irregularly sampled 576, Determinant 34,49 581f,654 Deviates,random see Random deviates use of CRCs in manipulating 897 DFP algorithm see Davidon-Fletcher-Powell windowing 553ff. algorithm see also Statistical tests Diagonal dominance 51,684,789,865 Data compression 603,889 Difference equations,finite see Finite differ- arithmetic coding 910ff ence equations (FDEs) cosine transform 519 Difference operator 167 Huffman coding 903ff.,910 Differential equations 707ff. linear predictive coding (LPC)571f. accuracy vs.stability 710,736 lossless 903 Adams-Bashforth-Moulton schemes 749 Data Encryption Standard(DES)300ff. adaptive stepsize control 709.714ff..725. Data type 28 733f,738.744.749f,751 DAUB4592ff.595,598f.601 algebraically difficult sets 772 DAUB6 593 backward Euler's method 735 DAUB12 605 Bader-Deuflhard method for stiff 737, DAUB20 598f. 742f Daubechies wavelet coefficients 592ff.,596, boundary conditions 707f.,753ff.,757, 598f.,601,605 760.779f Davidon-Fletcher-Powell algorithm 397,426f. Bulirsch-Stoer method 209,272,708f., Dawson's integral 259f.,606 712,722.724f.747 approximation for 259 Bulirsch-Stoer method for conservative routine for 260 equations 733 D.C.(direct current)498 comparison of methods 708f.,747,751 Debugging 7 conservative 732f. DEC (Digital Equipment Corp.)xvii,3,894 danger of too small stepsize 720 Declarations 930ff. eigenvalue problem 756,772ff.,779f., Decomposition see Cholesky decomposition; 781 LU decomposition;QR decomposition; embedded Runge-Kutta method 715f.,738 Singular value decomposition(SVD) equivalence of multistep and multivalue Deconvolution 542,549 methods 751 see also Convolution;Fast Fourier trans- Euler's method 708.710.735 form (FFT);Fourier transform forward Euler's method 735 Defect,in multigrid method 872 free boundary problem 756,785 Deferred approach to the limit see Richard- high-order implicit methods 737ff. son's deferred approach to the limit implicit differencing 735f.,749 Deflation initial value problems 708 of matrix 478 internal boundary conditions 784ff of polynomials 369ff.,377,378 internal singular points 784ff. Degeneracy of linear algebraic equations 32, interpolation on right-hand sides 117 61,65,676 Kaps-Rentrop method for stiff 737 Degenerate kernel 794 local extrapolation 715 Degenerate minimization principle 804 modified midpoint method 722f.,726 Degrees of freedom 621f.660.695f. multistep methods 747ff. Demonstration programs 3 multivalue methods 747 Derivatives order of method 710f.,725 computation via Chebyshev approximation path integration for function evaluation 189,195 208f,271 computation via Savitzky-Golay filters predictor-corrector methods 708,737, 189,651 747 matrix of first partial see Jacobian determi- reduction to first-order sets 707,753 nant relaxation method 754f,762ff. matrix of second partial see Hessian ma- relaxation method,example of 772ff. trix r.h.s.independent of c 735 numerical computation 186ff,386,651, Rosenbrock methods for stiff 737 738.758.779 Runge-Kutta method 708f.,710ff.,714ff., of polynomial 173f. 738.747 use in optimization 395f.,406 Runge-Kutta method,high-order 711 DES see Data Encryption Standard Runge-Kutta-Fehlberg method 715f. Descending transformation,elliptic integrals scaling stepsize to required accuracy 716f. 262 second order 732f. Descent direction 383,390,427 semi-implicit differencing 737 Descriptive statistics 609ff. semi-implicit Euler method 737,743 see also Statistical tests semi-implicit extrapolation method 737, Design matrix 651,671,804,809 743
970 Index unevenly or irregularly sampled 576, 581f., 654 use of CRCs in manipulating 897 windowing 553ff. see also Statistical tests Data compression 603, 889 arithmetic coding 910ff. cosine transform 519 Huffman coding 903ff., 910 linear predictive coding (LPC) 571f. lossless 903 Data Encryption Standard (DES) 300ff. Data type 28 DAUB4 592ff., 595, 598f., 601 DAUB6 593 DAUB12 605 DAUB20 598f. Daubechies wavelet coefficients 592ff., 596, 598f., 601, 605 Davidon-Fletcher-Powell algorithm 397, 426f. Dawson’s integral 259f., 606 approximation for 259 routine for 260 D.C. (direct current) 498 Debugging 7 DEC (Digital Equipment Corp.) xvii, 3, 894 Declarations 930ff. Decomposition see Cholesky decomposition; LU decomposition; QR decomposition; Singular value decomposition (SVD) Deconvolution 542, 549 see also Convolution; Fast Fourier transform (FFT); Fourier transform Defect, in multigrid method 872 Deferred approach to the limit see Richardson’s deferred approach to the limit Deflation of matrix 478 of polynomials 369ff., 377, 378 Degeneracy of linear algebraic equations 32, 61, 65, 676 Degenerate kernel 794 Degenerate minimization principle 804 Degrees of freedom 621f., 660, 695f. Demonstration programs 3 Derivatives computation via Chebyshev approximation 189, 195 computation via Savitzky-Golay filters 189, 651 matrix of first partial see Jacobian determinant matrix of second partial see Hessian matrix numerical computation 186ff., 386, 651, 738, 758, 779 of polynomial 173f. use in optimization 395f., 406 DES see Data Encryption Standard Descending transformation, elliptic integrals 262 Descent direction 383, 390, 427 Descriptive statistics 609ff. see also Statistical tests Design matrix 651, 671, 804, 809 Determinant 34, 49 Deviates, random see Random deviates DFP algorithm see Davidon-Fletcher-Powell algorithm Diagonal dominance 51, 684, 789, 865 Difference equations, finite see Finite difference equations (FDEs) Difference operator 167 Differential equations 707ff. accuracy vs. stability 710, 736 Adams-Bashforth-Moulton schemes 749 adaptive stepsize control 709, 714ff., 725, 733f., 738, 744, 749f., 751 algebraically difficult sets 772 backward Euler’s method 735 Bader-Deuflhard method for stiff 737, 742f. boundary conditions 707f., 753ff., 757, 760, 779f. Bulirsch-Stoer method 209, 272, 708f., 712, 722, 724ff., 747 Bulirsch-Stoer method for conservative equations 733 comparison of methods 708f., 747, 751 conservative 732f. danger of too small stepsize 720 eigenvalue problem 756, 772ff., 779f., 781 embedded Runge-Kutta method 715f., 738 equivalence of multistep and multivalue methods 751 Euler’s method 708, 710, 735 forward Euler’s method 735 free boundary problem 756, 785 high-order implicit methods 737ff. implicit differencing 735f., 749 initial value problems 708 internal boundary conditions 784ff. internal singular points 784ff. interpolation on right-hand sides 117 Kaps-Rentrop method for stiff 737 local extrapolation 715 modified midpoint method 722f., 726 multistep methods 747ff. multivalue methods 747 order of method 710f., 725 path integration for function evaluation 208ff., 271 predictor-corrector methods 708, 737, 747ff. reduction to first-order sets 707, 753 relaxation method 754f., 762ff. relaxation method, example of 772ff. r.h.s. independent of x 735 Rosenbrock methods for stiff 737 Runge-Kutta method 708f., 710ff., 714ff., 738, 747 Runge-Kutta method, high-order 711 Runge-Kutta-Fehlberg method 715f. scaling stepsize to required accuracy 716f. second order 732f. semi-implicit differencing 737 semi-implicit Euler method 737, 743 semi-implicit extrapolation method 737, 743
Index 971 semi-implicit midpoint rule 743 Downhill simplex method see Simplex,method shooting method 754,757E of Nelder and Mead shooting method,example 779f.,781 Driver programs 3 similarity to Volterra integral equations Dual viewpoint,in multigrid method 883 794f Duplication theorem,elliptic integrals 262f. singular points 724f,760,784ff. dvector()utility 943 step doubling 715 DWT (discrete wavelet transform)see Wavelet stepsize control 709,714ff.,724,733f., transform 738.744.749f.751 stiff 709.734 stiff methods compared 747 Eardley,D.M.346 Stoermer's rule 732f. EBCDIC 898 see also Partial differential equations:Two- Economization of power series 198ff.,201 point boundary value problems Eigensystems 456ff. Diffusion equation 827,847ff.,864 balancing matrix 483 Crank-Nicolson method 848,853,855 bounds on eigenvalues 58 Forward Time Centered Space (FTCS) calculation of few eigenvectors or eigenval- 847E.850f.864 ues461,494 implicit differencing 848 canned routines 461 multidimensional 855f. characteristic polynomial 456,475f Digamma function 222 completeness 457 defective 457.482.494 Digital filtering see Filter deflation 478 Dihedral group Ds 902 degenerate eigenvalues 456,458 Dimensions (units)683f. elimination method 460,485 Diminishing increment sort 331 factorization method 460 Dirac delta function 293,789 fast Givens reduction 470 Direct method see Periodogram Direct methods for linear algebraic equations generalized eigenproblem 462 35 Givens reduction 469f. Hermitian matrix 481f. Direct product see Outer product of matrices Hessenberg matrix 460,477,482ff.,494 Direction of largest decrease 416f. Householder transformation 460,469ff., Direction numbers,Sobol's sequence 311 476,480,481.484f Direction-set methods for minimization 396, ill-conditioned eigenvalues 483 412ff Dirichlet boundary conditions 829,849,859, implicit shifts 478f and integral equations 788ff.,794 865.867 invariance under similarity transform 459 Disclaimer of warranty xvi inverse iteration 462,476,483,493ff. Discordant pair for Kendall's tau 643 Jacobi transformation 460,463ff.,469 Discrete convolution theorem 538ff. 481,495 Discrete Fourier transform (DFT)500ff. left eigenvalues 458 as approximate continuous transform 503 list of tasks 461 see also Fast Fourier transform (FFT) Discrete optimization 444ff. multiple eigenvalues 495 nonlinear 462 Discriminant 184.464 nonsymmetric matrix 482ff. Diskettes,how to order xvi,996f. operation count of balancing 483 Dispersion 840 operation count of Givens reduction 470 DISPO see Savitzky-Golay filters operation count of Householder reduction Dissipation.numerical 839 474 Divergent series 167 operation count of inverse iteration 494 Division operation count of Jacobi method 467 complex 177 operation count of QL method 477,480 multiple precision 919f. operation count of QR method for Hessen- of polynomials 175,369,377 berg matrices 490 dmatrix()utility 944 operation count of reduction to Hessenberg dn function 269 form 485 Do-while iteration 12 orthogonality 457 Dogleg step methods 393 polynomial roots and 375 Domain of integration 161f. QL method 476ff,481,494f. Dominant solution of recurrence relation 179 QL method with implicit shifts 478ff. Dot (denotes matrix multiplication)33 QR method60,460,463,476f. Double exponential error distribution 701 QR method for Hessenberg matrices 486ff. Double precision real,symmetric matrix 156,474,794 as refuge of scoundrels 890 reduction to Hessenberg form 484f. use in iterative improvement 56 right eigenvalues 458 Double root 348 shifting eigenvalues 456,477f.,486f
Index 971 semi-implicit midpoint rule 743 shooting method 754, 757ff. shooting method, example 779f., 781 similarity to Volterra integral equations 794f. singular points 724f., 760, 784ff. step doubling 715 stepsize control 709, 714ff., 724, 733f., 738, 744, 749f., 751 stiff 709, 734ff. stiff methods compared 747 Stoermer’s rule 732f. see also Partial differential equations; Twopoint boundary value problems Diffusion equation 827, 847ff., 864 Crank-Nicolson method 848, 853, 855 Forward Time Centered Space (FTCS) 847ff., 850f., 864 implicit differencing 848 multidimensional 855f. Digamma function 222 Digital filtering see Filter Dihedral group D5 902 Dimensions (units) 683f. Diminishing increment sort 331 Dirac delta function 293, 789 Direct method see Periodogram Direct methods for linear algebraic equations 35 Direct product see Outer product of matrices Direction of largest decrease 416f. Direction numbers, Sobol’s sequence 311 Direction-set methods for minimization 396, 412ff. Dirichlet boundary conditions 829, 849, 859, 865, 867 Disclaimer of warranty xvi Discordant pair for Kendall’s tau 643 Discrete convolution theorem 538ff. Discrete Fourier transform (DFT) 500ff. as approximate continuous transform 503 see also Fast Fourier transform (FFT) Discrete optimization 444ff. Discriminant 184, 464 Diskettes, how to order xvi, 996f. Dispersion 840 DISPO see Savitzky-Golay filters Dissipation, numerical 839 Divergent series 167 Division complex 177 multiple precision 919f. of polynomials 175, 369, 377 dmatrix() utility 944 dn function 269 Do-while iteration 12 Dogleg step methods 393 Domain of integration 161f. Dominant solution of recurrence relation 179 Dot (denotes matrix multiplication) 33 Double exponential error distribution 701 Double precision as refuge of scoundrels 890 use in iterative improvement 56 Double root 348 Downhill simplex method see Simplex, method of Nelder and Mead Driver programs 3 Dual viewpoint, in multigrid method 883 Duplication theorem, elliptic integrals 262f. dvector() utility 943 DWT (discrete wavelet transform) see Wavelet transform Eardley, D.M. 346 EBCDIC 898 Economization of power series 198ff., 201 Eigensystems 456ff. balancing matrix 483 bounds on eigenvalues 58 calculation of few eigenvectors or eigenvalues 461, 494 canned routines 461 characteristic polynomial 456, 475f. completeness 457 defective 457, 482, 494 deflation 478 degenerate eigenvalues 456, 458 elimination method 460, 485 factorization method 460 fast Givens reduction 470 generalized eigenproblem 462 Givens reduction 469f. Hermitian matrix 481f. Hessenberg matrix 460, 477, 482ff., 494 Householder transformation 460, 469ff., 476, 480, 481, 484f. ill-conditioned eigenvalues 483 implicit shifts 478ff. and integral equations 788ff., 794 invariance under similarity transform 459 inverse iteration 462, 476, 483, 493ff. Jacobi transformation 460, 463ff., 469, 481, 495 left eigenvalues 458 list of tasks 461 multiple eigenvalues 495 nonlinear 462 nonsymmetric matrix 482ff. operation count of balancing 483 operation count of Givens reduction 470 operation count of Householder reduction 474 operation count of inverse iteration 494 operation count of Jacobi method 467 operation count of QL method 477, 480 operation count of QR method for Hessenberg matrices 490 operation count of reduction to Hessenberg form 485 orthogonality 457 polynomial roots and 375 QL 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, 794 reduction to Hessenberg form 484f. right eigenvalues 458 shifting eigenvalues 456, 477f., 486f
972 Index special matrices 461 nonnormal 659.695,699ff. termination criterion 489f.495 relative truncation 883 tridiagonal matrix 460,475ff.,494 roundoff 185,889f. Eigenvalue and eigenvector,defined 456 series,advantage of an even 138f.,723 Eigenvalue problem for differential equations systematic vs.statistical 659 756.772f,779,781 truncation30,186,406,715,889f Eigenvalues and polynomial root finding 375 varieties found by check digits 902 EISPACK 461,482 varieties of,in PDEs 840ff. Electromagnetic potential 525 see also Roundoff error Elimination see Gaussian elimination Error function 220f.,607 Ellipse in confidence limit estimation 693 approximation via sampling theorem 6071 Elliptic integrals 261ff.,915 Chebyshev approximation 220f. addition theorem 262 complex 259 Carlson's forms and algorithms 261ff. for Fisher's z-transformation 638 Cauchy principal value 263 relation to Dawson's integral 259 duplication theorem 262f. relation to Fresnel integrals 255 Legendre 261ff.,267ff. relation to incomplete gamma function routines for 264ff. 220 symmetric form 261f routine for 220f. Weierstrass 262 for significance of correlation 636 Elliptic partial differential equations 827 for sum squared difference of ranks 641 alternating-direction implicit method (ADI) Error handling in programs 2,940 870f.915 Estimation of parameters see Fitting;Maxi- analyze/factorize/operate package 833 mum likelihood estimate biconjugate gradient method 833 Estimation of power spectrum 549ff.,572ff. boundary conditions 828f. Euler equation (fluid flow)840 comparison of rapid methods 863 Euler-Maclaurin summation formula 138,142 conjugate gradient method 833 Euler's constant 223f..257 cyclic reduction 857f.,861f. Euler's method for differential equations 708, Fourier analysis and cyclic reduction (FACR) 710.735 857E,863 Euler's transformation 166ff. Gauss-Seidel method 864.873.874f..884 generalized form 168f. incomplete Cholesky conjugate gradient Evaluation of functions see Function method (ICCG)833 Even and odd parts,of continued fraction Jacobi's method 864f.,873 172,217,222 matrix methods 833 Even parity 896 multigrid method 833.871ff. Exception handling in programs 2,940 rapid (Fourier)method 833,857ff exit()function 2 relaxation method 832.863ff. Explicit differencing 836 strongly implicit procedure 833 Exponent in floating point format 28.890 successive over-relaxation (SOR)866ff., Exponential deviate 287f. 871,875 Exponential integral 222f Emacs.GNU xiii asymptotic expansion 224 Embedded Runge-Kutta method 715f.,738 continued fraction 222 Encapsulation,in programs 6f. recurrence relation 178 Encryption 300 related to incomplete gamma function 222 Entropy 903f. relation to cosine integral 257 of data 632ff.,820 routine for En()223f. EOM(end of message)910 routine for Ei(r）225 Equality constraints 431 series 222 Equations Exponential probability distribution 577 cubic 183ff.,367 Extended midpoint rule 130f,135,141f. normal (fitting)651,672ff.,809f. Extended Simpson's rule 134,796,799 quadratic 29,183ff. Extended Simpson's three-eighths rule 797 see also Differential equations;Partial dif- Extended trapezoidal rule 131f.,133,136ff., ferential equations;Root finding 141,795 Equivalence classes 345f. roundoff error 138 Equivalence transformation 172 extern storage class 25 Error Extirpolation (so-called)581f. checksums for preventing 899 Extrapolation 105ff. clocking 899 in Bulirsch-Stoer method 724ff.,731 double exponential distribution 701 differential equations 708 local truncation 883 by linear prediction 564ff. Lorentzian distribution 701f. local 715 in multigrid method 872 maximum entropy method as type of 574
972 Index special matrices 461 termination criterion 489f., 495 tridiagonal matrix 460, 475ff., 494 Eigenvalue and eigenvector, defined 456 Eigenvalue problem for differential equations 756, 772ff., 779, 781 Eigenvalues and polynomial root finding 375 EISPACK 461, 482 Electromagnetic potential 525 Elimination see Gaussian elimination Ellipse in confidence limit estimation 693 Elliptic integrals 261ff., 915 addition theorem 262 Carlson’s forms and algorithms 261ff. Cauchy principal value 263 duplication theorem 262f. Legendre 261ff., 267ff. routines for 264ff. symmetric form 261f. Weierstrass 262 Elliptic partial differential equations 827 alternating-direction implicit method (ADI) 870f., 915 analyze/factorize/operate package 833 biconjugate gradient method 833 boundary conditions 828f. comparison of rapid methods 863 conjugate gradient method 833 cyclic reduction 857f., 861f. Fourier analysis and cyclic reduction (FACR) 857ff., 863 Gauss-Seidel method 864, 873, 874f., 884 incomplete Cholesky conjugate gradient method (ICCG) 833 Jacobi’s method 864f., 873 matrix methods 833 multigrid method 833, 871ff. rapid (Fourier) method 833, 857ff. relaxation method 832, 863ff. strongly implicit procedure 833 successive over-relaxation (SOR) 866ff., 871, 875 Emacs, GNU xiii Embedded Runge-Kutta method 715f., 738 Encapsulation, in programs 6f. Encryption 300 Entropy 903f. of data 632ff., 820 EOM (end of message) 910 Equality constraints 431 Equations cubic 183ff., 367 normal (fitting) 651, 672ff., 809f. quadratic 29, 183ff. see also Differential equations; Partial differential equations; Root finding Equivalence classes 345f. Equivalence transformation 172 Error checksums for preventing 899 clocking 899 double exponential distribution 701 local truncation 883 Lorentzian distribution 701f. in multigrid method 872 nonnormal 659, 695, 699ff. relative truncation 883 roundoff 185, 889f. series, advantage of an even 138f., 723 systematic vs. statistical 659 truncation 30, 186, 406, 715, 889f. varieties found by check digits 902 varieties of, in PDEs 840ff. see also Roundoff error Error function 220f., 607 approximation via sampling theorem 607f. Chebyshev approximation 220f. complex 259 for Fisher’s z-transformation 638 relation to Dawson’s integral 259 relation to Fresnel integrals 255 relation to incomplete gamma function 220 routine for 220f. for significance of correlation 636 for sum squared difference of ranks 641 Error handling in programs 2, 940 Estimation of parameters see Fitting; Maximum likelihood estimate Estimation of power spectrum 549ff., 572ff. Euler equation (fluid flow) 840 Euler-Maclaurin summation formula 138, 142 Euler’s constant 223f., 257 Euler’s method for differential equations 708, 710, 735 Euler’s transformation 166ff. generalized form 168f. Evaluation of functions see Function Even and odd parts, of continued fraction 172, 217, 222 Even parity 896 Exception handling in programs 2, 940 exit() function 2 Explicit differencing 836 Exponent in floating point format 28, 890 Exponential deviate 287f. Exponential integral 222ff. asymptotic expansion 224 continued fraction 222 recurrence relation 178 related to incomplete gamma function 222 relation to cosine integral 257 routine for En(x) 223f. routine for Ei(x) 225 series 222 Exponential probability distribution 577 Extended midpoint rule 130f., 135, 141f. Extended Simpson’s rule 134, 796, 799 Extended Simpson’s three-eighths rule 797 Extended trapezoidal rule 131f., 133, 136ff., 141, 795 roundoff error 138 extern storage class 25 Extirpolation (so-called) 581f. Extrapolation 105ff. in Bulirsch-Stoer method 724ff., 731 differential equations 708 by linear prediction 564ff. local 715 maximum entropy method as type of 574
Index 973 polynomial 728,730f.,748 for multiple precision multiplication 918 rational function 724f,731 number-theoretic transforms 509f relation to interpolation 105 operation count 504 for Romberg integration 140 optimal (Wiener)filtering 547f,565f see also Interpolation order of storage in 507 Extremization see Minimization partial differential equations 833,857ff. Parzen window 554 F-distribution probability function 226,229 periodicity of 503 periodogram 550ff.,574 F-test for differences of variances 617,619 power spectrum estimation 549ff FACR see Fourier analysis and cyclic reduc- tion (FACR) for quadrature 130 of real data in 2D and 3D 525ff. Facsimile standard 909 of real functions 510ff.,525ff. Factorial double (denoted "!!"253 related algorithms 509f. evaluation of 165 Sande-Tukey algorithm 509 sine transform 514ff..859 relation to gamma function 213 routine for 214f. Singleton's algorithm 532 False position 354ff. square window 553 Family tree 345 treatment of end effects in convolution 540 FAS(full approximation storage algorithm) 882任. treatment of end effects in correlation 546 Fast Fourier transform (FFT)504ff.,889 Tukey's trick for frequency doubling 582 alternative algorithms 509f. use in smoothing data 650 applications 537f used for Lomb periodogram 581f. as approximation to continuous transform variance of power spectrum estimate 552, 503 556 Bartlett window 554 virtual memory machine 535f. bit reversal 505f.,532 Welch window 554 and Clenshaw-Curtis quadrature 196 Winograd algorithms 509 convolution 509,530,538ff.,918 see also Discrete Fourier transform (DFT); convolution of large data sets 543f Fourier transform;Spectral density Cooley-Tukey algorithm 509 Faure sequence 310 correlation 545f Fax (facsimile)Group 3 standard 909 cosine transform 196,517f,860f fcomplex (data type) 24,948 cosine transform,second form 519,861 Feasible vector 431 Danielson-Lanczos lemma 504f.532 FFT see Fast Fourier transform (FFT) data sets not a power of 2 509 Field,in data record 338 data smoothing 650 Figure-of-merit function 656 data windowing 553ff. Filon's method 590 decimation-in-frequency algorithm 509 Filter 558ff. decimation-in-time algorithm 509 acausal 559 discrete autocorrelation 546 bilinear transformation method 561 discrete convolution theorem 538ff. causal 559,650 discrete correlation theorem 545 characteristic polynomial 561 at double frequency 582 data smoothing 650 endpoint corrections 585f digital 558ff. extemal storage 532 DISPO 650 figures of merit for data windows 554 by fast Fourier transform(FFT)530, filtering 558ff. 558ff FIR filter 559f finite impulse response (FIR)538,559f. Fourier integrals 584ff. homogeneous modes of 561 Fourier integrals,infinite range 590f. infinite impulse response (IIR)559,573 Hamming window 554 Kalman 705 Hann window 554 linear 559ff. history 504 low-pass for smoothing 650 IIR filter 559 nonrecursive 559f. image processing 812,814 optimal (Wiener)542,547ff.,565f.,650 integrals using 130 quadrature mirror 592.600 inverse of cosine transform 518f. realizable 559,561 inverse of sine transform 517 recursive 559,573 large data sets 532 Remes exchange algorithm 560 leakage 551 Savitzky-Golay 189,650ff memory-local algorithm 535f stability of 561 multidimensional 521ff. in the time domain 558ff. for multiple precision arithmetic 915 Fine-to-coarse operator 873
Index 973 polynomial 728, 730f., 748 rational function 724ff., 731 relation to interpolation 105 for Romberg integration 140 see also Interpolation Extremization see Minimization F-distribution probability function 226, 229 F-test for differences of variances 617, 619 FACR see Fourier analysis and cyclic reduction (FACR) Facsimile standard 909 Factorial double (denoted “!!") 253 evaluation of 165 relation to gamma function 213 routine for 214f. False position 354ff. Family tree 345 FAS (full approximation storage algorithm) 882ff. Fast Fourier transform (FFT) 504ff., 889 alternative algorithms 509f. applications 537ff. as approximation to continuous transform 503 Bartlett window 554 bit reversal 505f., 532 and Clenshaw-Curtis quadrature 196 convolution 509, 530, 538ff., 918 convolution of large data sets 543f. Cooley-Tukey algorithm 509 correlation 545f. cosine transform 196, 517ff., 860f. cosine transform, second form 519, 861 Danielson-Lanczos lemma 504f., 532 data sets not a power of 2 509 data smoothing 650 data windowing 553ff. decimation-in-frequency algorithm 509 decimation-in-time algorithm 509 discrete autocorrelation 546 discrete convolution theorem 538ff. discrete correlation theorem 545 at double frequency 582 endpoint corrections 585f. external storage 532 figures of merit for data windows 554 filtering 558ff. FIR filter 559f. Fourier integrals 584ff. Fourier integrals, infinite range 590f. Hamming window 554 Hann window 554 history 504 IIR filter 559 image processing 812, 814 integrals using 130 inverse of cosine transform 518f. inverse of sine transform 517 large data sets 532 leakage 551 memory-local algorithm 535f. multidimensional 521ff. for multiple precision arithmetic 915 for multiple precision multiplication 918 number-theoretic transforms 509f. operation count 504 optimal (Wiener) filtering 547ff., 565f. order of storage in 507 partial differential equations 833, 857ff. Parzen window 554 periodicity of 503 periodogram 550ff., 574 power spectrum estimation 549ff. for quadrature 130 of real data in 2D and 3D 525ff. of real functions 510ff., 525ff. related algorithms 509f. Sande-Tukey algorithm 509 sine transform 514ff., 859 Singleton’s algorithm 532 square window 553 treatment of end effects in convolution 540 treatment of end effects in correlation 546 Tukey’s trick for frequency doubling 582 use in smoothing data 650 used for Lomb periodogram 581f. variance of power spectrum estimate 552, 556 virtual memory machine 535f. Welch window 554 Winograd algorithms 509 see also Discrete Fourier transform (DFT); Fourier transform; Spectral density Faure sequence 310 Fax (facsimile) Group 3 standard 909 fcomplex (data type) 24, 948 Feasible vector 431 FFT see Fast Fourier transform (FFT) Field, in data record 338 Figure-of-merit function 656 Filon’s method 590 Filter 558ff. acausal 559 bilinear transformation method 561 causal 559, 650 characteristic polynomial 561 data smoothing 650 digital 558ff. DISPO 650 by fast Fourier transform (FFT) 530, 558ff. finite impulse response (FIR) 538, 559f. homogeneous modes of 561 infinite impulse response (IIR) 559, 573 Kalman 705 linear 559ff. low-pass for smoothing 650 nonrecursive 559f. optimal (Wiener) 542, 547ff., 565f., 650 quadrature mirror 592, 600 realizable 559, 561 recursive 559, 573 Remes exchange algorithm 560 Savitzky-Golay 189, 650ff. stability of 561 in the time domain 558ff. Fine-to-coarse operator 873
974 Index Finite difference equations(FDEs)762,772, standard (probable)errors on fitted parame- 783 ters663,667E,673,677.689ǖ alternating-direction implicit method (ADI) straight line 661ff.,673f,703 856.870f. straight line,errors in both coordinates art,not science 838 666正 Cayley's form for unitary operator 853 see also Error,Least squares fitting;Max- Courant condition 838,841,845 imum likelihood estimate;Robust esti- Courant condition (multidimensional)855 mation Crank-Nicolson method 848.853.855 Five-point difference star 876 eigenmodes of 836f. Fixed point format 28 explicit vs.implicit schemes 836 Fletcher-Powell algorithm see Davidon-Fletcher- forward Euler 835f. Powell algorithm Forward Time Centered Space (FTCS) Fletcher-Reeves algorithm 396f.,421ff. 836f,847f,852,864 float to double conversion 24f. implicit scheme 848 Floating point co-processor 894 Lax method 837ff..845 Floating point format 28,890 Lax method (multidimensional)854f. care in numerical derivatives 186 mesh drifting instability 843f IEEE 285.890f. numerical derivatives 186 Flux-conservative initial value problems 834ff. partial differential equations 830ff. FMG (full multigrid method)872,877f. in relaxation methods 762ff for iteration 8,11 staggered leapfrog method 842f Formats of numbers 28,890 two-step Lax-Wendroff method 844ff FORTRAN 16.20 upwind differencing 841f..846 Numerical Recipes in xv,I see also Partial differential equations Forward deflation 370 Finite element methods,partial differential Forward difference operator 167 equations 833f. Forward Euler differencing 835f Finite impulse response (FIR)538 Forward Time Centered Space see FTCS Finkelstein,S.xii Fourier analysis and cyclic reduction (FACR) FIR (finite impulse response)filter 559f. 858,863 Fisher's z-transformation 637f. Fourier and spectral applications 537ff Fitting 656ff. Fourier integrals basis functions 671 attenuation factors 590 by Chebyshev approximation 191f. endpoint corrections 585f. chi-square 659ff. tail integration by parts 591 confidence levels related to chi-square val- use of fast Fourier transform (FFT)584ff. ues 696ff. Fourier transform 105,496ff confidence levels from singular value de aliasing 501,576 composition (SVD)698 approximation of Dawson's integral 259 confidence limits on fitted parameters 689ff. autocorrelation 498 covariance matrix not always meaningful basis functions compared 514f. 657,695 contrasted with wavelet transform 591f., degeneracy of parameters 679 601 an exponential 679 convolution 498.509.538ff.,918 freezing parameters in 674,705 correlation 498.545f. Gaussians,a sum of 687f. cosine transform 196,517E,860f general linear least squares 671ff. cosine transform,second form 519,861 Kalman filter 705 critical sampling 500,550,552 K-S test,caution regarding 627 definition 496 least squares 657ff. discrete Fourier transform (DFT)190, Legendre polynomials 680 500ff. Levenberg-Marquardt method 683ff.,825 Gaussian function 607 linear regression 661ff. image processing 812,814 maximum likelihood estimation 658, infinite range 590f. 699f inverse of discrete Fourier transform 503 Monte Carlo simulation 627,660,689ff. method for partial differential equations multidimensional 680 857E nonlinear models 681ff missing data 576 nonlinear models,advanced methods 688 missing data,fast algorithm 581f. nonlinear problems that are linear 679 Nyquist frequency 500ff,526,550,552, nonnormal errors 662,695,699ff. 576.579 polynomial90,120,197,650f,671,679f optimal(Wiener)filtering 547ff,565f. by rational Chebyshey approximation 204ff Parseval's theorem 498,504,551 robust methods 699ff. power spectral density (PSD)498f. of sharp spectral features 573 power spectrum estimation by FFT 549ff
974 Index Finite difference equations (FDEs) 762, 772, 783 alternating-direction implicit method (ADI) 856, 870f. art, not science 838 Cayley’s form for unitary operator 853 Courant condition 838, 841, 845 Courant condition (multidimensional) 855 Crank-Nicolson method 848, 853, 855 eigenmodes of 836f. explicit vs. implicit schemes 836 forward Euler 835f. Forward Time Centered Space (FTCS) 836ff., 847ff., 852, 864 implicit scheme 848 Lax method 837ff., 845 Lax method (multidimensional) 854f. mesh drifting instability 843f. numerical derivatives 186 partial differential equations 830ff. in relaxation methods 762ff. staggered leapfrog method 842f. two-step Lax-Wendroff method 844ff. upwind differencing 841f., 846 see also Partial differential equations Finite element methods, partial differential equations 833f. Finite impulse response (FIR) 538 Finkelstein, S. xii FIR (finite impulse response) filter 559f. Fisher’s z-transformation 637f. Fitting 656ff. basis functions 671 by Chebyshev approximation 191f. chi-square 659ff. confidence levels related to chi-square values 696ff. confidence levels from singular value decomposition (SVD) 698 confidence limits on fitted parameters 689ff. covariance matrix not always meaningful 657, 695 degeneracy of parameters 679 an exponential 679 freezing parameters in 674, 705 Gaussians, a sum of 687f. general linear least squares 671ff. Kalman filter 705 K–S test, caution regarding 627 least squares 657ff. Legendre polynomials 680 Levenberg-Marquardt method 683ff., 825 linear regression 661ff. maximum likelihood estimation 658, 699ff. Monte Carlo simulation 627, 660, 689ff. multidimensional 680 nonlinear models 681ff. nonlinear models, advanced methods 688 nonlinear problems that are linear 679 nonnormal errors 662, 695, 699ff. polynomial 90, 120, 197, 650f., 671, 679f. by rational Chebyshev approximation 204ff. robust methods 699ff. of sharp spectral features 573 standard (probable) errors on fitted parameters 663, 667f., 673, 677, 689ff. straight line 661ff., 673f., 703 straight line, errors in both coordinates 666ff. see also Error; Least squares fitting; Maximum likelihood estimate; Robust estimation Five-point difference star 876 Fixed point format 28 Fletcher-Powell algorithm see Davidon-FletcherPowell algorithm Fletcher-Reeves algorithm 396f., 421ff. float to double conversion 24f. Floating point co-processor 894 Floating point format 28, 890 care in numerical derivatives 186 IEEE 285, 890f. Flux-conservative initial value problems 834ff. FMG (full multigrid method) 872, 877f. for iteration 8, 11 Formats of numbers 28, 890 FORTRAN 16, 20 Numerical Recipes in xv, 1 Forward deflation 370 Forward difference operator 167 Forward Euler differencing 835f. Forward Time Centered Space see FTCS Fourier analysis and cyclic reduction (FACR) 858, 863 Fourier and spectral applications 537ff. Fourier integrals attenuation factors 590 endpoint corrections 585f. tail integration by parts 591 use of fast Fourier transform (FFT) 584ff. Fourier transform 105, 496ff. aliasing 501, 576 approximation of Dawson’s integral 259 autocorrelation 498 basis functions compared 514f. contrasted with wavelet transform 591f., 601 convolution 498, 509, 538ff., 918 correlation 498, 545f. cosine transform 196, 517ff., 860f. cosine transform, second form 519, 861 critical sampling 500, 550, 552 definition 496 discrete Fourier transform (DFT) 190, 500ff. Gaussian function 607 image processing 812, 814 infinite range 590f. inverse of discrete Fourier transform 503 method for partial differential equations 857ff. missing data 576 missing data, fast algorithm 581f. Nyquist frequency 500ff., 526, 550, 552, 576, 579 optimal (Wiener) filtering 547ff., 565f. Parseval’s theorem 498, 504, 551 power spectral density (PSD) 498f. power spectrum estimation by FFT 549ff