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Combined back-of-book subject index for the two-volume Numerical Recipes in Fortran, with page numbers for Volume 1 (Fortran 77) and Volume 2 (Fortran 90, pages 935 onward). Entries cover numerical methods such as Bessel functions, Bulirsch-Stoer, Brent's method, and biconjugate gradients, and Fortran 90 array features. This is a reference book index by others, not Phil's own writing; the file appears to be one section of the full index.

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General Index to Volumes 1 and 2 In this index, page numbers 1 through 934 refer to Volume 1, Numerical Recipes in Fortran 77 , while page numbers 935 through 1446 refer to Volume 2, Numerical Recipes in Fortran 90 . Front matter in Volume 1 is indicated by page numbers in the range 1/i through 1/xxxi, while front matter in Volume 2 is indicated 2/i through 2/xx. Abstract data types 2/xiii, 1030 Accelerated convergence of series 160ff., 1070 Accuracy 19f. achievable in minimization 392, 397, 404achievable in root ﬁnding 346f.contrasted with ﬁdelity 832, 840CPU different from memory 181vs. stability 704, 729, 830, 844 Accuracy parameters 1362f.Acknowledgments 1/xvi, 2/ixAda 2/xAdams-Bashford-Moulton method 741Adams’ stopping criterion 366Adaptive integration 123, 135, 703, 708ff., 720, 726, 731f., 737, 742ff., 788, 1298ff.,1303, 1308f. Monte Carlo 306ff., 1161ff. Addition, multiple precision 907, 1353Addition theorem, elliptic integrals 255ADI (alternating direction implicit) method 847, 861f., 906 Adjoint operator 867Adobe Illustrator 1/xvi, 2/xxAdvective equation 826AGM (arithmetic geometric mean) 906Airy function 204, 234, 243f. routine for 244f., 1121 Aitken’s delta squared process 160Aitken’s interpolation algorithm 102Algol 2/x, 2/xivAlgorithms, non-numerical 881ff., 1343ff.Aliasing 495, 569 see alsoFourier transform all() intrinsic function 945, 948All-poles model 566 see alsoMaximum entropy method (MEM) All-zeros model 566 see alsoPeriodogram Allocatable array 938, 941, 952ff., 1197, 1212, 1266, 1293, 1306, 1336 allocate statement 938f., 941, 953f., 1197, 1266, 1293, 1306, 1336 allocated() intrinsic function 938, 952ff., 1197, 1266, 1293 Allocation status 938, 952ff., 961, 1197, 1266, 1293Alpha AXP 2/xix Alternating-direction implicit method (ADI) 847, 861f., 906 Alternating series 160f., 1070Alternative extended Simpson’s rule 128 American National Standards Institute (ANSI) 2/x, 2/xiii Amoeba 403 see alsoSimplex, method of Nelder and Mead Ampliﬁcation factor 828, 830, 832, 840, 845f.Amplitude error 831Analog-to-digital converter 812, 886 Analyticity 195 Analyze/factorize/operate package 64, 824Anderson-Darling statistic 621Andrew’s sine 697 Annealing, method of simulated 387f., 436ff., 1219ff. assessment 447 for continuous variables 437, 443ff., 1222schedule 438thermodynamic analogy 437 traveling salesman problem 438ff., 1219ff. ANSI (American National Standards Institute) 2/x, 2/xiii Antonov-Saleev variant of Sobol’ sequence 300, 1160 any() intrinsic function 945, 948APL (computer language) 2/xiApple 1/xxiii Macintosh 2/xix, 4, 886 Approximate inverse of matrix 49Approximation of functions 99, 1043 by Chebyshev polynomials 185f., 513, 1076ff. Pad´e approximant 194ff., 1080f. by rational functions 197ff., 1081f. by wavelets 594f., 782see alsoFitting Argument keyword 2/xiv, 947f., 1341 optional 2/xiv, 947f., 1092, 1228, 1230, 1256, 1272, 1275, 1340 Argument checking 994f., 1086, 1090, 1092, 1370f. Index to Volumes 1 and2 Arithmetic arbitrary precision 881, 906ff., 1352ff. ﬂoating point 881, 1343IEEE standard 276, 882, 1343 rounding 882, 1343 Arithmetic coding 881, 902ff., 1349ff. Arithmetic-geometric mean (AGM) method 906 Arithmetic-if statement 2/xi Arithmetic progression 971f., 996, 1072, 1127, 1365, 1371f. Array 953ff. allocatable 938, 941, 952ff., 1197, 1212, 1266, 1293, 1306, 1336 allocated with pointer 941allocation 953 array manipulation functions 950 array sections 939, 941, 943ff. of arrays 2/xii, 956, 1336 associated pointer 953f.assumed-shape 942 automatic 938, 954, 1197, 1212, 1336 centered subarray of 113conformable to a scalar 942f., 965, 1094 constructor 2/xii, 968, 971, 1022, 1052, 1055, 1127 copying 991, 1034, 1327f., 1365f. cumulative product 997f., 1072, 1086, 1375 cumulative sum 997, 1280f., 1365, 1375 deallocation 938, 953f., 1197, 1266, 1293 disassociated pointer 953extents 938, 949 in Fortran 90 941 increasing storage for 955, 1070, 1302index loss 967f. index table 1173ff. indices 942inquiry functions 948ff. intrinsic procedures 2/xiii, 948ff. of length 0 944of length 1 949 location of ﬁrst “true” 993, 1041, 1369 location of maximum value 993, 1015, 1017, 1365, 1369 location of minimum value 993, 1369f. manipulation functions 950, 1247masked swapping of elements in two arrays 1368 operations on 942, 949, 964ff., 969, 1026, 1040, 1050, 1200, 1326 outer product 949, 1076 parallel features 941ff., 964ff., 985passing variable number of arguments to function 1022 of pointers forbidden 956, 1337rank 938, 949 reallocation 955, 992, 1070f., 1365, 1368f. reduction functions 948ff.shape 938, 944, 949 size 938 skew sections 945, 985 stride 944 subscript bounds 942subscript triplet 944swapping elements of two arrays 991, 1015, 1365ff. target 938three-dimensional, in Fortran 90 1248 transformational functions 948ff. unary and binary functions 949undeﬁned status 952ff., 961, 1266, 1293zero-length 944 Array section 2/xiii, 943ff., 960 matches by shape 944pointer alias 939, 944f., 1286, 1333 skew 2/xii, 945, 960, 985, 1284 vs. eoshift 1078 array copy() utility function 988, 991, 1034, 1153, 1278, 1328 arth() utility function 972, 974, 988, 996, 1072, 1086, 1127 replaces do-list 968 Artiﬁcial viscosity 831, 837 Ascending transformation, elliptic integrals 256 ASCII character set 6, 888, 896, 902 Assembly language 269 assert() utility function 988, 994, 1086, 1090, 1249 asserteq() utility function 988, 995, 1022 associated() intrinsic function 952f. Associated Legendre polynomials 246ff., 764, 1122f., 1319 recurrence relation for 247 relation to Legendre polynomials 246 Association, measures of 604, 622ff., 1275Assumed-shape array 942 Asymptotic series 161 exponential integral 218 Attenuation factors 583, 1261 Autocorrelation 492 in linear prediction 558use of FFT 538f., 1254Wiener-Khinchin theorem 492, 566f. AUTODIN-II polynomial 890 Automatic array 938, 954, 1197, 1212, 1336 specifying size of 938, 954 Automatic deallocation 2/xv, 961 Autonomous differential equations 729f. Autoregressive model (AR) seeMaximum en- tropy method (MEM) Average deviation of distribution 605, 1269 Averaging kernel, in Backus-Gilbert method 807 Backsubstitution 33ff., 39, 42, 92, 1017 in band diagonal matrix 46, 1021in Cholesky decomposition 90, 1039complex equations 41 direct for computing A −1·B40 with QR decomposition 93, 1040relaxation solution of boundary value prob- lems 755, 1316 in singular value decomposition 56, 1022f. Backtracking 419 in quasi-Newton methods 376f., 1195 Backus-Gilbert method 806ff. Backus, John 2/xBackward deﬂation 363 Indexto Volumes1 and 2 Bader-Deuﬂhard method 730, 735, 1310f. Bairstow’s method 364, 370, 1193 Balancing 476f., 1230f. Band diagonal matrix 42ff., 1019 backsubstitution 46, 1021LU decomposition 45, 1020multiply by vector 44, 1019storage 44, 1019 Band-pass ﬁlter 551, 554f. wavelets 584, 592f. Bandwidth limited function 495Bank accounts, checksum for 894Bar codes, checksum for 894Bartlett window 547, 1254ff.Base case, of recursive procedure 958 Base of representation 19, 882, 1343 BASIC, Numerical Recipes in 1, 2/x, 2/xviiiBasis functions in general linear least squares 665 Bayes’ Theorem 810Bayesian approach to inverse problems 799, 810f., 816f. contrasted with frequentist 810 vs. historic maximum entropy method 816f. views on straight line ﬁtting 664 Bays’ shufﬂe 270Bernoulli number 132Bessel functions 223ff., 234ff., 936, 1101ff. asymptotic form 223f., 229f.complex 204continued fraction 234, 239double precision 223 fractional order 223, 234ff., 1115ff. Miller’s algorithm 175, 228, 1106modiﬁed 229ff.modiﬁed, fractional order 239ff.modiﬁed, normalization formula 232, 240modiﬁed, routines for 230ff., 1109ff.normalization formula 175parallel computation of 1107ff.recurrence relation 172, 224, 232, 234reﬂection formulas 236reﬂection formulas, modiﬁed functions 241 routines for 225ff., 236ff., 1101ff.routines for modiﬁed functions 241ff., 1118 series for 160, 223series for K ν241 series for Yν235 spherical 234, 245, 1121f.turning point 234Wronskian 234, 239 Best-ﬁt parameters 650, 656, 660, 698, 1285ff. see alsoFitting Beta function 206ff., 1089 incomplete seeIncomplete beta function BFGS algorithm seeBroyden-Fletcher-Goldfarb- Shanno algorithm Bias, of exponent 19Bias, removal in linear prediction 563Biconjugacy 77Biconjugate gradient method elliptic partial differential equations 824preconditioning 78f., 824, 1037for sparse system 77, 599, 1034ff. Bicubic interpolation 118f., 1049f. Bicubic spline 120f., 1050f.Big-endian 293Bilinear interpolation 117Binary constant, initialization 959 Binomial coefﬁcients 206ff., 1087f. recurrences for 209 Binomial probability function 208 cumulative 222f. deviates from 281, 285f., 1155 Binormal distribution 631, 690Biorthogonality 77Bisection 111, 359, 1045f. compared to minimum bracketing 390ff. minimum ﬁnding with derivatives 399root ﬁnding 343, 346f., 352f., 390, 469, 1184f. BISYNCH 890Bit 18 manipulation functions seeBitwise logical functions reversal in fast Fourier transform (FFT) 499f., 525 bit size() intrinsic function 951 Bitwise logical functions 2/xiii, 17, 287, 890f., 951 Block-by-block method 788 Block of statements 7Bode’s rule 126Boltzmann probability distribution 437 Boltzmann’s constant 437 Bootstrap method 686f.Bordering method for Toeplitz matrix 85f.Borwein and Borwein method for π906, 1357 Boundary 155f., 425f., 745Boundary conditions for differential equations 701f. initial value problems 702in multigrid method 868f.partial differential equations 508, 819ff., 848ff. for spheroidal harmonics 764 two-point boundary value problems 702, 745ff., 1314ff. Boundary value problems seeDifferential equations; Elliptic partial differential equations; Two-point boundary valueproblems Box-Muller algorithm for normal deviate 279f., 1152 Bracketing of function minimum 343, 390ff., 402, 1201f. of roots 341, 343ff., 353f., 362, 364, 369, 390, 1183f. Branch cut, for hypergeometric function 203Branching 9 Break iteration 14 Brenner, N.M. 500, 517 Index to Volumes 1 and2 Brent’s method minimization 389, 395ff., 660f., 1204ff., 1286 minimization, using derivative 389, 399, 1205 root ﬁnding 341, 349, 660f., 1188f., 1286 Broadcast (parallel capability) 965ff.Broyden-Fletcher-Goldfarb-Shanno algorithm 390, 418ff., 1215 Broyden’s method 373, 382f., 386, 1199f. singular Jacobian 386 btest() intrinsic function 951 Bubble sort 321, 1168Bugs 4 in compilers 1/xvii how to report 1/iv, 2/iv Bulirsch-Stoer algorithm for rational function interpolation 105f., 1043 method (differential equations) 202, 263, 702f., 706, 716, 718ff., 726, 740, 1138,1303ff. method (differential equations), stepsize control 719, 726 for second order equations 726, 1307 Burg’s LP algorithm 561, 1256 Byte 18 C(programming language) 13, 2/viii and case construct 1010 Numerical Recipes in 1, 2/x, 2/xvii C++ 1/xiv, 2/viii, 2/xvi, 7f. class templates 1083, 1106 Calendar algorithms 1f., 13ff., 1010ff. Calibration 653Capital letters in programs 3, 937Cards, sorting a hand of 321 Carlson’s elliptic integrals 255f., 1128ff. case construct 2/xiv, 1010 trapping errors 1036 Cash-Karp parameters 710, 1299f. Cauchy probability distribution seeLorentzian probability distribution Cauchy problem for partial differential equa- tions 818f. Cayley’s representation of exp(−iHt )844 CCITT (Comit ´e Consultatif International T ´el´e- graphique et T ´el´ephonique) 889f., 901 CCITT polynomial 889f. ceiling() intrinsic function 947Center of mass 295ff. Central limit theorem 652f. Central tendency, measures of 604ff., 1269Change of variable in integration 137ff., 788, 1056ff. in Monte Carlo integration 298 in probability distribution 279 Character functions 952 Character variables, in Fortran 90 1183 Characteristic polynomial digital ﬁlter 554eigensystems 449, 469 linear prediction 559 matrix with a speciﬁed 368, 1193of recurrence relation 175Characteristics of partial differential equations 818 Chebyshev acceleration in successive over- relaxation (SOR) 859f., 1332 Chebyshev approximation 84, 124, 183, 184ff., 1076ff. Clenshaw-Curtis quadrature 190Clenshaw’s recurrence formula 187, 1076coefﬁcients for 185f., 1076contrasted with Pad ´e approximation 195 derivative of approximated function 183, 189, 1077f. economization of series 192f., 195, 1080for error function 214, 1095even function 188and fast cosine transform 513gamma functions 236, 1118integral of approximated function 189, 1078 odd function 188polynomial ﬁts derived from 191, 1078rational function 197ff., 1081f.Remes exchange algorithm for ﬁlter 553 Chebyshev polynomials 184ff., 1076ff. continuous orthonormality 184discrete orthonormality 185explicit formulas for 184formula for x kin terms of 193, 1080 Check digit 894, 1345f.Checksum 881, 888 cyclic redundancy (CRC) 888ff., 1344f. Cherry, sundae without a 809Chi-by-eye 651Chi-square ﬁtting seeFitting; Least squares ﬁtting Chi-square probability function 209ff., 215, 615, 654, 798, 1272 as boundary of conﬁdence region 688f.related to incomplete gamma function 215 Chi-square test 614f. for binned data 614f., 1272chi-by-eye 651and conﬁdence limit estimation 688f.for contingency table 623ff., 1275degrees of freedom 615f.for inverse problems 797least squares ﬁtting 653ff., 1285nonlinear models 675ff., 1292rule of thumb 655for straight line ﬁtting 655ff., 1285for straight line ﬁtting, errors in both coor- dinates 660, 1286ff. for two binned data sets 616, 1272unequal size samples 617 Chip rate 290Chirp signal 556Cholesky decomposition 89f., 423, 455, 1038 backsubstitution 90, 1039operation count 90pivoting 90solution of normal equations 668 Circulant 585Class, data type 7Clenshaw-Curtis quadrature 124, 190, 512f. Indexto Volumes1 and 2 Clenshaw’s recurrence formula 176f., 191, 1078 for Chebyshev polynomials 187, 1076stability 176f. Clocking errors 891 CM computers (Thinking Machines Inc.) 964CM Fortran 2/xvcn function 261, 1137f. Coarse-grid correction 864f. Coarse-to-ﬁne operator 864, 1337Coding arithmetic 902ff., 1349ff. checksums 888, 1344decoding a Huffman-encoded message 900, 1349 Huffman 896f., 1346ff.run-length 901variable length code 896, 1346ff. Ziv-Lempel 896 see alsoArithmetic coding; Huffman cod- ing Coefﬁcients binomial 208, 1087f.for Gaussian quadrature 140ff., 1059ff. for Gaussian quadrature, nonclassical weight function 151ff., 788f., 1064 for quadrature formulas 125ff., 789, 1328 Cohen, Malcolm 2/xiv Column degeneracy 22 Column operations on matrix 29, 31f.Column totals 624 Combinatorial minimization seeAnnealing Comit´e Consultatif International T ´el´egraphique et T´el´ephonique (CCITT) 889f., 901 Common block obsolescent 2/xif. superseded by internal subprogram 957, 1067 superseded by module 940, 953, 1298, 1320, 1322, 1324, 1330 Communication costs, in parallel processing 969, 981, 1250 Communication theory, use in adaptive integra- tion 721 Communications protocol 888 Comparison function for rejection method 281 Compilers 964, 1364 CM Fortran 968 DEC (Digital Equipment Corp.) 2/viiiIBM (International Business Machines) 2/viii Microsoft Fortran PowerStation 2/viii NAG (Numerical Algorithms Group) 2/viii, 2/xiv for parallel supercomputers 2/viii Complementary error function 1094f. seeError function Complete elliptic integral seeElliptic integrals Complex arithmetic 171f. avoidance of in path integration 203cubic equations 179f. for linear equations 41 quadratic equations 178 Complex error function 252Complex plane fractal structure for Newton’s rule 360f. path integration for function evaluation 201ff., 263, 1138 poles in 105, 160, 202f., 206, 554, 566, 718f. Complex systems of linear equations 41f. Compression of data 596f. Concordant pair for Kendall’s tau 637, 1281 Condition number 53, 78 Conﬁdence level 687, 691ff.Conﬁdence limits bootstrap method 687f. and chi-square 688f.conﬁdence region, conﬁdence interval 687 on estimated model parameters 684ff. by Monte Carlo simulation 684ff.from singular value decomposition (SVD) 693f. Conﬂuent hypergeometric function 204, 239 Conformable arrays 942f., 1094 Conjugate directions 408f., 414ff., 1210Conjugate gradient method biconjugate 77, 1034 compared to variable metric method 418elliptic partial differential equations 824 for minimization 390, 413ff., 804, 815, 1210, 1214 minimum residual method 78 preconditioner 78f., 1037 for sparse system 77ff., 599, 1034and wavelets 599 Conservative differential equations 726, 1307 Constrained linear inversion method 799ff. Constrained linear optimization seeLinear pro- gramming Constrained optimization 387 Constraints, deterministic 804ff. Constraints, linear 423CONTAINS statement 954, 957, 1067, 1134, 1202 Contingency coefﬁcient C 625, 1275Contingency table 622ff., 638, 1275f. statistics based on chi-square 623ff., 1275 statistics based on entropy 626ff., 1275f. Continued fraction 163ff. Bessel functions 234 convergence criterion 165 equivalence transformation 166 evaluation 163ff.evaluation along with normalization condi- tion 240 even and odd parts 166, 211, 216even part 249, 251 exponential integral 216 Fresnel integral 248f.incomplete beta function 219f., 1099f. incomplete gamma function 211, 1092f. Lentz’s method 165, 212modiﬁed Lentz’s method 165 Pincherle’s theorem 175 ratio of Bessel functions 239 rational function approximation 164, 211, 219f. recurrence for evaluating 164f. Index to Volumes 1 and2 and recurrence relation 175 sine and cosine integrals 250f. Steed’s method 164f.tangent function 164typography for 163 Continuous variable (statistics) 623 Control structures 7ff., 2/xiv bad 15named 959, 1219, 1305 Convergence accelerated, for series 160ff., 1070 of algorithm for pi 906criteria for 347, 392, 404, 483, 488, 679, 759 eigenvalues accelerated by shifting 470f. golden ratio 349, 399of golden section search 392f.of Levenberg-Marquardt method 679linear 346, 393 of QL method 470f. quadratic 49, 351, 356, 409f., 419, 906rate 346f., 353, 356recurrence relation 175of Ridders’ method 351 series vs. continued fraction 163f. and spectral radius 856ff., 862 Conversion intrinsic functions 946f.Convex sets, use in inverse problems 804 Convolution denoted by asterisk 492ﬁnite impulse response (FIR) 531of functions 492, 503f.of large data sets 536f. for multiple precision arithmetic 909, 1354 multiplication as 909, 1354necessity for optimal ﬁltering 535 overlap-add method 537 overlap-save method 536f.and polynomial interpolation 113relation to wavelet transform 585 theorem 492, 531ff., 546 theorem, discrete 531ff. treatment of end effects 533use of FFT 523, 531ff., 1253wraparound problem 533 Cooley-Tukey FFT algorithm 503, 1250 parallel version 1239f. Co-processor, ﬂoating point 886Copyright rules 1/xx, 2/xixCornwell-Evans algorithm 816 Corporate promotion ladder 328 Corrected two-pass algorithm 607, 1269Correction, in multigrid method 863Correlation coefﬁcient (linear) 630ff., 1276 Correlation function 492 autocorrelation 492, 539, 558and Fourier transforms 492theorem 492, 538treatment of end effects 538f. using FFT 538f., 1254 Wiener-Khinchin theorem 492, 566f. Correlation, statistical 603f., 622 Kendall’s tau 634, 637ff., 1279linear correlation coefﬁcient 630ff., 658, 1276 linear related to least square ﬁtting 630, 658 nonparametric or rank statistical 633ff., 1277 among parameters in a ﬁt 657, 667, 670in random number generators 268 Spearman rank-order coefﬁcient 634f., 1277 sum squared difference of ranks 634, 1277 Cosine function, recurrence 172 Cosine integral 248, 250ff., 1125f. continued fraction 250 routine for 251f., 1125 series 250 Cosine transform seeFast Fourier transform (FFT); Fourier transform Coulomb wavefunction 204, 234 count() intrinsic function 948Courant condition 829, 832ff., 836 multidimensional 846 Courant-Friedrichs-Lewy stability criterion see Courant condition Covariance a priori 700 in general linear least squares 667, 671, 1288ff. matrix, by Cholesky decomposition 91, 667 matrix, of errors 796, 808matrix, is inverse of Hessian matrix 679 matrix, when it is meaningful 690ff. in nonlinear models 679, 681, 1292relation to chi-square 690ff.from singular value decomposition (SVD) 693f. in straight line ﬁtting 657 cpu time() intrinsic function (Fortran 95) 961 CR method seeCyclic reduction (CR) Cramer’s V 625, 1275Crank-Nicolson method 840, 844, 846Cray computers 964 CRC (cyclic redundancy check) 888ff., 1344f. CRC-12 890CRC-16 polynomial 890 CRC-CCITT 890 Creativity, essay on 9Critical (Nyquist) sampling 494, 543Cross (denotes matrix outer product) 66 Crosstabulation analysis 623 see alsoContingency table Crout’s algorithm 36ff., 45, 1017cshift() intrinsic function 950 communication bottleneck 969 Cubic equations 178ff., 360Cubic spline interpolation 107ff., 1044f. see alsoSpline cumprod() utility function 974, 988, 997, 1072, 1086 cumsum() utility function 974, 989, 997, 1280, 1305 Cumulant, of a polynomial 977, 999, 1071f., 1192 Indexto Volumes1 and 2 Cumulative binomial distribution 222f. Cumulative Poisson function 214 related to incomplete gamma function 214 Curvature matrix seeHessian matrix cycle statement 959, 1219 Cycle, in multigrid method 865 Cyclic Jacobi method 459, 1225Cyclic reduction (CR) 848f., 852ff. linear recurrences 974tridiagonal systems 976, 1018 Cyclic redundancy check (CRC) 888ff., 1344f. Cyclic tridiagonal systems 67, 1030 D.C. (direct current) 492 Danielson-Lanczos lemma 498f., 525, 1235ff.DAP Fortran 2/xi Data assigning keys to 889continuous vs. binned 614entropy 626ff., 896, 1275essay on 603 ﬁtting 650ff., 1285ff. fraudulent 655glitches in 653iid (independent and identically distributed) 686 modeling 650ff., 1285ff.serial port 892 smoothing 604, 644ff., 1283f. statistical tests 603ff., 1269ff.unevenly or irregularly sampled 569, 574, 648f., 1258ff. use of CRCs in manipulating 889windowing 545ff., 1254see alsoStatistical tests Data compression 596f., 881 arithmetic coding 902ff., 1349ff. cosine transform 513Huffman coding 896f., 902, 1346ff.linear predictive coding (LPC) 563ff.lossless 896 Data Encryption Standard (DES) 290ff., 1144, 1147f., 1156ff. Data hiding 956ff., 1209, 1293, 1296 Data parallelism 941, 964ff., 985DATA statement 959 for binary, octal, hexadecimal constants 959 repeat count feature 959superseded by initialization expression 943, 959, 1127 Data type 18, 936 accuracy parameters 1362f. character 1183 derived 2/xiii, 937, 1030, 1336, 1346derived, for array of arrays 956, 1336derived, initialization 2/xv derived, for Numerical Recipes 1361 derived, storage allocation 955DP (double precision) 1361f.DPC (double precision complex) 1361I1B (1 byte integer) 1361 I2B (2 byte integer) 1361 I4B (4 byte integer) 1361intrinsic 937 LGT (default logical type) 1361 nrtype.f90 1361f.passing complex as real 1140SP (single precision) 1361f. SPC (single precision complex) 1361 user-deﬁned 1346 DAUB4 584ff., 588, 590f., 594, 1264f.DAUB6 586DAUB12 598 DAUB20 590f., 1265 Daubechies wavelet coefﬁcients 584ff., 588, 590f., 594, 598, 1264ff. Davidon-Fletcher-Powell algorithm 390, 418ff., 1215 Dawson’s integral 252ff., 600, 1127f. approximation for 252f.routine for 253f., 1127 dble() intrinsic function (deprecated) 947 deallocate statement 938f., 953f., 1197, 1266, 1293 Deallocation, of allocatable array 938, 953f., 1197, 1266, 1293 Debugging 8 DEC (Digital Equipment Corp.) 1/xxiii, 2/xix, 886 Alpha AXP 2/viii Fortran 90 compiler 2/viii quadruple precision option 1362VAX 4 Decomposition seeCholesky decomposition; LU decomposition; QR decomposition; Singular value decomposition (SVD) Deconvolution 535, 540, 1253 see alsoConvolution; Fast Fourier trans- form (FFT); Fourier transform Defect, in multigrid method 863 Deferred approach to the limit seeRichard- son’s deferred approach to the limit Deﬂation of matrix 471 of polynomials 362ff., 370f., 977 Degeneracy of linear algebraic equations 22, 53, 57, 670 Degenerate kernel 785 Degenerate minimization principle 795Degrees of freedom 615f., 654, 691Dekker, T.J. 353Demonstration programs 3, 936 Deprecated features common block 2/xif., 940, 953, 957, 1067, 1298, 1320, 1322, 1324, 1330 dble() intrinsic function 947 EQUIVALENCE statement 2/xif., 1161, 1286 statement function 1057, 1256 Derivatives computation via Chebyshev approximation 183, 189, 1077f. computation via Savitzky-Golay ﬁlters 183, 645 matrix of ﬁrst partial seeJacobian determi- nant matrix of second partial seeHessian ma- trix Index to Volumes 1 and2 numerical computation 180ff., 379, 645, 732, 750, 771, 1075, 1197, 1309 of polynomial 167, 978, 1071f. use in optimization 388f., 399, 1205ff. Derived data type seeData type, derived DESseeData Encryption Standard Descending transformation, elliptic integrals 256 Descent direction 376, 382, 419Descriptive statistics 603ff., 1269ff. see alsoStatistical tests Design matrix 645, 665, 795, 801, 1082Determinant 25, 41 Deviates, random seeRandom deviates DFP algorithm seeDavidon-Fletcher-Powell algorithm diagadd() utility function 985, 989, 1004 diagmult() utility function 985, 989, 1004, 1294 Diagonal dominance 43, 679, 780, 856 Difference equations, ﬁnite seeFinite differ- ence equations (FDEs) Difference operator 161Differential equations 701ff., 1297ff. accuracy vs. stability 704, 729 Adams-Bashforth-Moulton schemes 741adaptive stepsize control 703, 708ff., 719, 726, 731, 737, 742f., 1298ff., 1303ff.,1308f., 1311ff. algebraically difﬁcult sets 763backward Euler’s method 729 Bader-Deuﬂhard method for stiff 730, 735, 1310f. boundary conditions 701f., 745ff., 749, 751f., 771, 1314ff. Bulirsch-Stoer method 202, 263, 702, 706, 716, 718ff., 740, 1138, 1303 Bulirsch-Stoer method for conservative equations 726, 1307 comparison of methods 702f., 739f., 743conservative 726, 1307 danger of too small stepsize 714 eigenvalue problem 748, 764ff., 770ff., 1319ff. embedded Runge-Kutta method 709f., 731, 1298, 1308 equivalence of multistep and multivalue methods 743 Euler’s method 702, 704, 728f. forward Euler’s method 728 free boundary problem 748, 776high-order implicit methods 730ff., 1308ff. implicit differencing 729, 740, 1308 initial value problems 702internal boundary conditions 775ff. internal singular points 775ff. interpolation on right-hand sides 111Kaps-Rentrop method for stiff 730, 1308 local extrapolation 709 modiﬁed midpoint method 716f., 719, 1302f. multistep methods 740ff.multivalue methods 740 order of method 704f., 719path integration for function evaluation 201ff., 263, 1138 predictor-corrector methods 702, 730, 740ff. reduction to ﬁrst-order sets 701, 745 relaxation method 746f., 753ff., 1316ff.relaxation method, example of 764ff., 1319ff. r.h.s. independent of x729f. Rosenbrock methods for stiff 730, 1308f. Runge-Kutta method 702, 704ff., 708ff., 731, 740, 1297f., 1308 Runge-Kutta method, high-order 705, 1297 Runge-Kutta-Fehlberg method 709ff., 1298 scaling stepsize to required accuracy 709 second order 726, 1307 semi-implicit differencing 730semi-implicit Euler method 730, 735f. semi-implicit extrapolation method 730, 735f., 1311ff. semi-implicit midpoint rule 735f., 1310f. shooting method 746, 749ff., 1314ff.shooting method, example 770ff., 1321ff. similarity to Volterra integral equations 786 singular points 718f., 751, 775ff., 1315f., 1323ff. step doubling 708f.stepsize control 703, 708ff., 719, 726, 731, 737, 742f., 1298, 1303ff., 1308f. stiff 703, 727ff., 1308ff. stiff methods compared 739 Stoermer’s rule 726, 1307see alsoPartial differential equations; Two- point boundary value problems Diffusion equation 818, 838ff., 855 Crank-Nicolson method 840, 844, 846 Forward Time Centered Space (FTCS) 839ff., 855 implicit differencing 840multidimensional 846 Digamma function 216 Digital ﬁltering seeFilter Dihedral group D 5894 dim optional argument 948 Dimensional expansion 965ff.Dimensions (units) 678 Diminishing increment sort 322, 1168 Dirac delta function 284, 780 Direct method seePeriodogram Direct methods for linear algebraic equations 26, 1014 Direct product seeOuter product of matrices Direction of largest decrease 410f. Direction numbers, Sobol’s sequence 300 Direction-set methods for minimization 389, 406f., 1210ff. Dirichlet boundary conditions 820, 840, 850, 856, 858 Disclaimer of warranty 1/xx, 2/xviiDiscordant pair for Kendall’s tau 637, 1281 Discrete convolution theorem 531ff. Indexto Volumes1 and 2 Discrete Fourier transform (DFT) 495ff., 1235ff. as approximate continuous transform 497 see alsoFast Fourier transform (FFT) Discrete optimization 436ff., 1219ff.Discriminant 178, 457Diskettes are ANSI standard 3 how to order 1/xxi, 2/xvii Dispersion 831DISPOseeSavitzky-Golay ﬁlters Dissipation, numerical 830 Divergent series 161 Divide and conquer algorithm 1226, 1229Division complex 171multiple precision 910f., 1356 of polynomials 169, 362, 370, 1072 dn function 261, 1137f.Do-list, implied 968, 971, 1127Do-loop 2/xiv Do-until iteration 14 Do-while iteration 13Dogleg step methods 386Domain of integration 155f.Dominant solution of recurrence relation 174 Dot (denotes matrix multiplication) 23 dot product() intrinsic function 945, 949, 969, 1216 Double exponential error distribution 696Double precision converting to 1362as refuge of scoundrels 882 use in iterative improvement 47, 1022 Double root 341Downhill simplex method seeSimplex, method of Nelder and Mead DP, deﬁned 937Driver programs 3Dual viewpoint, in multigrid method 875 Duplication theorem, elliptic integrals 256 DWT (discrete wavelet transform) seeWavelet transform Dynamical allocation of storage 2/xiii, 869, 938, 941f., 953ff., 1327, 1336 garbage collection 956 increasing 955, 1070, 1302 Eardley, D.M. 338 EBCDIC 890Economization of power series 192f., 195, 1080 Eigensystems 449ff., 1225ff. balancing matrix 476f., 1230f. bounds on eigenvalues 50calculation of few eigenvalues 454, 488canned routines 454f.characteristic polynomial 449, 469 completeness 450 defective 450, 476, 489deﬂation 471degenerate eigenvalues 449ff. elimination method 453, 478, 1231 factorization method 453fast Givens reduction 463 generalized eigenproblem 455Givens reduction 462f.Hermitian matrix 475 Hessenberg matrix 453, 470, 476ff., 488, 1232 Householder transformation 453, 462ff., 469, 473, 475, 478, 1227f., 1231 ill-conditioned eigenvalues 477 implicit shifts 472ff., 1228f.and integral equations 779, 785 invariance under similarity transform 452 inverse iteration 455, 469, 476, 487ff., 1230 Jacobi transformation 453, 456ff., 462, 475, 489, 1225f. left eigenvalues 451list of tasks 454f.multiple eigenvalues 489 nonlinear 455 nonsymmetric matrix 476ff., 1230ff.operation count of balancing 476 operation count of Givens reduction 463 operation count of Householder reduction 467 operation count of inverse iteration 488operation count of Jacobi method 460 operation count of QL method 470, 473 operation count of QR method for Hessen- berg matrices 484 operation count of reduction to Hessenberg form 479 orthogonality 450parallel algorithms 1226, 1229polynomial roots and 368, 1193 QL method 469ff., 475, 488f. QL method with implicit shifts 472ff., 1228f. QR method 52, 453, 456, 469ff., 1228QR method for Hessenberg matrices 480ff., 1232ff. real, symmetric matrix 150, 467, 785, 1225, 1228 reduction to Hessenberg form 478f., 1231 right eigenvalues 451 shifting eigenvalues 449, 470f., 480special matrices 454termination criterion 484, 488 tridiagonal matrix 453, 469ff., 488, 1228 Eigenvalue and eigenvector, deﬁned 449Eigenvalue problem for differential equations 748, 764ff., 770ff., 1319ff. Eigenvalues and polynomial root ﬁnding 368, 1193 EISPACK 454, 475 Electromagnetic potential 519 ELEMENTAL attribute (Fortran 95) 961, 1084 Elemental functions 2/xiii, 2/xv, 940, 942, 946f., 961, 986, 1015, 1083, 1097f. Elimination seeGaussian elimination Ellipse in conﬁdence limit estimation 688Elliptic integrals 254ff., 906 addition theorem 255 Index to Volumes 1 and2 Carlson’s forms and algorithms 255f., 1128ff. Cauchy principal value 256f.duplication theorem 256 Legendre 254ff., 260f., 1135ff. routines for 257ff., 1128ff.symmetric form 255 Weierstrass 255 Elliptic partial differential equations 818, 1332ff. alternating-direction implicit method (ADI) 861f., 906 analyze/factorize/operate package 824 biconjugate gradient method 824 boundary conditions 820comparison of rapid methods 854 conjugate gradient method 824 cyclic reduction 848f., 852ff. Fourier analysis and cyclic reduction (FACR) 848ff., 854 Gauss-Seidel method 855, 864ff., 876, 1338, 1341 incomplete Cholesky conjugate gradient method (ICCG) 824 Jacobi’s method 855f., 864 matrix methods 824 multigrid method 824, 862ff., 1009, 1334ff.rapid (Fourier) method 824, 848ff. relaxation method 823, 854ff., 1332 strongly implicit procedure 824 successive over-relaxation (SOR) 857ff., 862, 866, 1332 elsewhere construct 943 Emacs, GNU 1/xvi Embedded Runge-Kutta method 709f., 731, 1298, 1308 Encapsulation, in programs 7 Encryption 290, 1156 enddo statement 12, 17 Entropy 896 of data 626ff., 811, 1275 EOM (end of message) 902 eoshift() intrinsic function 950 communication bottleneck 969vector shift argument 1019f. vs. array section 1078 epsilon() intrinsic function 951, 1189Equality constraints 423 Equations cubic 178ff., 360normal (ﬁtting) 645, 666ff., 800, 1288 quadratic 20, 178 see alsoDifferential equations; Partial dif- ferential equations; Root ﬁnding Equivalence classes 337f., 1180 EQUIVALENCE statement 2/xif., 1161, 1286Equivalence transformation 166 Error checksums for preventing 891clocking 891 double exponential distribution 696 local truncation 875 Lorentzian distribution 696f. in multigrid method 863nonnormal 653, 690, 694ff.relative truncation 875 roundoff 180f., 881, 1362 series, advantage of an even 132f., 717, 1362 systematic vs. statistical 653, 1362 truncation 20f., 180, 399, 709, 881, 1362varieties found by check digits 895varieties of, in PDEs 831ff. see alsoRoundoff error Error function 213f., 601, 1094f. approximation via sampling theorem 601 Chebyshev approximation 214, 1095 complex 252for Fisher’s z-transformation 632, 1276 relation to Dawson’s integral 252, 1127 relation to Fresnel integrals 248relation to incomplete gamma function 213 routine for 214, 1094 for signiﬁcance of correlation 631, 1276for sum squared difference of ranks 635, 1277 Error handling in programs 2/xii, 2/xvi, 3, 994f., 1036, 1370f. Estimation of parameters seeFitting; Maxi- mum likelihood estimate Estimation of power spectrum 542ff., 565ff., 1254ff., 1258 Euler equation (ﬂuid ﬂow) 831 Euler-Maclaurin summation formula 132, 135Euler’s constant 216ff., 250 Euler’s method for differential equations 702, 704, 728f. Euler’s transformation 160f., 1070 generalized form 162f. Evaluation of functions seeFunction Even and odd parts, of continued fraction 166, 211, 216 Even parity 888 Exception handling in programs seeError han- dling in programs exit statement 959, 1219 Explicit differencing 827 Exponent in ﬂoating point format 19, 882, 1343 exponent intrinsic function 1107 Exponential deviate 278, 1151f.Exponential integral 215ff., 1096f. asymptotic expansion 218 continued fraction 216recurrence relation 172related to incomplete gamma function 215 relation to cosine integral 250 routine for Ei(x)218, 1097 routine for E n(x)217, 1096 series 216 Exponential probability distribution 570Extended midpoint rule 124f., 129f., 135, 1054f. Extended Simpson’s rule 128, 788, 790 Extended Simpson’s three-eighths rule 789Extended trapezoidal rule 125, 127, 130ff., 135, 786, 1052ff., 1326 roundoff error 132 Extirpolation (so-called) 574, 1261 Indexto Volumes1 and 2 Extrapolation 99ff. in Bulirsch-Stoer method 718ff., 726, 1305ff. differential equations 702 by linear prediction 557ff., 1256f. local 709maximum entropy method as type of 567polynomial 724, 726, 740, 1305f. rational function 718ff., 726, 1306f. relation to interpolation 101for Romberg integration 134 see alsoInterpolation Extremization seeMinimization F-distribution probability function 222 F-test for differences of variances 611, 613, 1271 FACRseeFourier analysis and cyclic reduc- tion (FACR) Facsimile standard 901 Factorial double (denoted “!!”) 247evaluation of 159, 1072, 1086 relation to gamma function 206 routine for 207f., 1086ff. False position 347ff., 1185f. Family tree 338 FAS (full approximation storage algorithm) 874, 1339ff. Fast Fourier transform (FFT) 498ff., 881, 981, 1235f. alternative algorithms 503f.as approximation to continuous transform 497 Bartlett window 547, 1254bit reversal 499f., 525and Clenshaw-Curtis quadrature 190 column-parallel algorithm 981, 1237ff. communication bottleneck 969, 981, 1250convolution 503f., 523, 531ff., 909, 1253, 1354 convolution of large data sets 536f.Cooley-Tukey algorithm 503, 1250Cooley-Tukey algorithm, parallel 1239f. correlation 538f., 1254 cosine transform 190, 511ff., 851, 1245f.cosine transform, second form 513, 852, 1246 Danielson-Lanczos lemma 498f., 525 data sets not a power of 2 503data smoothing 645 data windowing 545ff., 1254 decimation-in-frequency algorithm 503decimation-in-time algorithm 503discrete autocorrelation 539, 1254 discrete convolution theorem 531ff. discrete correlation theorem 538at double frequency 575 effect of caching 982 endpoint corrections 578f., 1261ff.external storage 525ﬁgures of merit for data windows 548 ﬁltering 551ff. FIR ﬁlter 553four-step framework 983, 1239Fourier integrals 577ff., 1261 Fourier integrals, inﬁnite range 583 Hamming window 547 Hann window 547 history 498IIR ﬁlter 553ff. image processing 803, 805 integrals using 124inverse of cosine transform 512ff. inverse of sine transform 511 large data sets 525 leakage 544 memory-local algorithm 528multidimensional 515ff., 1236f., 1241, 1246, 1251 for multiple precision arithmetic 906for multiple precision multiplication 909, 1354 number-theoretic transforms 503f. operation count 498 optimal (Wiener) ﬁltering 539ff., 558order of storage in 501 parallel algorithms 981ff., 1235ff. partial differential equations 824, 848ff.Parzen window 547 periodicity of 497 periodogram 543ff., 566 power spectrum estimation 542ff., 1254ff. for quadrature 124of real data in 2D and 3D 519ff., 1248f. of real functions 504ff., 519ff., 1242f., 1248f. related algorithms 503f. row-parallel algorithm 981, 1235f. Sande-Tukey algorithm 503 sine transform 508ff., 850, 1245 Singleton’s algorithm 525six-step framework 983, 1240 square window 546, 1254 timing 982treatment of end effects in convolution 533 treatment of end effects in correlation 538f. Tukey’s trick for frequency doubling 575 use in smoothing data 645 used for Lomb periodogram 574, 1259 variance of power spectrum estimate 544f., 549 virtual memory machine 528 Welch window 547, 1254Winograd algorithms 503 see alsoDiscrete Fourier transform (DFT); Fourier transform; Spectral density Faure sequence 300 Fax (facsimile) Group 3 standard 901 Feasible vector 424 FFTseeFast Fourier transform (FFT) Field, in data record 329Figure-of-merit function 650 Filon’s method 583 Filter 551ff. acausal 552 bilinear transformation method 554 causal 552, 644 Index to Volumes 1 and2 characteristic polynomial 554 data smoothing 644f., 1283f. digital 551ff. DISPO 644 by fast Fourier transform (FFT) 523, 551ff. ﬁnite impulse response (FIR) 531, 552 homogeneous modes of 554inﬁnite impulse response (IIR) 552ff., 566 Kalman 700 linear 552ff.low-pass for smoothing 644ff., 1283f. nonrecursive 552 optimal (Wiener) 535, 539ff., 558, 644quadrature mirror 585, 593 realizable 552, 554f. recursive 552ff., 566 Remes exchange algorithm 553 Savitzky-Golay 183, 644ff., 1283f.stability of 554f. in the time domain 551ff. Fine-to-coarse operator 864, 1337Finite difference equations (FDEs) 753, 763, 774 alternating-direction implicit method (ADI) 847, 861f. art not science 829 Cayley’s form for unitary operator 844Courant condition 829, 832ff., 836 Courant condition (multidimensional) 846 Crank-Nicolson method 840, 844, 846 eigenmodes of 827f. explicit vs. implicit schemes 827forward Euler 826f. Forward Time Centered Space (FTCS) 827ff., 839ff., 843, 855 implicit scheme 840 Lax method 828ff., 836 Lax method (multidimensional) 845f.mesh drifting instability 834f. numerical derivatives 181 partial differential equations 821ff. in relaxation methods 753ff. staggered leapfrog method 833f.two-step Lax-Wendroff method 835ff. upwind differencing 832f., 837 see alsoPartial differential equations Finite element methods, partial differential equations 824 Finite impulse response (FIR) 531Finkelstein, S. 1/xvi, 2/ix FIR (ﬁnite impulse response) ﬁlter 552 Fisher’s z-transformation 631f., 1276Fitting 650ff., 1285ff. basis functions 665 by Chebyshev approximation 185f., 1076 chi-square 653ff., 1285ff. conﬁdence levels related to chi-square val- ues 691ff. conﬁdence levels from singular value de- composition (SVD) 693f. conﬁdence limits on ﬁtted parameters 684ff. covariance matrix not always meaningful 651, 690 degeneracy of parameters 674an exponential 674 freezing parameters in 668, 700 Gaussians, a sum of 682, 1294general linear least squares 665ff., 1288, 1290f. Kalman ﬁlter 700 K–S test, caution regarding 621f.least squares 651ff., 1285 Legendre polynomials 674, 1291f. Levenberg-Marquardt method 678ff., 816, 1292f. linear regression 655ff., 1285ff. maximum likelihood estimation 652f., 694ff. Monte Carlo simulation 622, 654, 684ff.multidimensional 675 nonlinear models 675ff., 1292f. nonlinear models, advanced methods 683nonlinear problems that are linear 674nonnormal errors 656, 690, 694ff. polynomial 83, 114, 191, 645, 665, 674, 1078, 1291 by rational Chebyshev approximation 197ff., 1081f. robust methods 694ff., 1294 of sharp spectral features 566standard (probable) errors on ﬁtted pa- rameters 657f., 661, 667, 671, 684ff., 1285f., 1288, 1290 straight line 655ff., 667f., 698, 1285ff., 1294ff. straight line, errors in both coordinates 660ff., 1286ff. see alsoError; Least squares ﬁtting; Max- imum likelihood estimate; Robust esti- mation Five-point difference star 867Fixed point format 18Fletcher-Powell algorithm seeDavidon-Fletcher- Powell algorithm Fletcher-Reeves algorithm 390, 414ff., 1214Floating point co-processor 886 Floating point format 18ff., 882, 1343 care in numerical derivatives 181IEEE 276, 882, 1343 ﬂoor() intrinsic function 948 Flux-conservative initial value problems 825ff. FMG (full multigrid method) 863, 868, 1334ff.FOR iteration 9f., 12forall statement 2/xii, 2/xv, 960, 964, 986 access to associated index 968 skew array sections 985, 1007 Formats of numbers 18ff., 882, 1343Fortran 9 arithmetic-if statement 2/xi COMMON block 2/xif., 953, 957deprecated features 2/xif., 947, 1057, 1161, 1256, 1286 dynamical allocation of storage 869, 1336 EQUIVALENCE statement 2/xif., 1161, 1286 evolution of 2/xivff. exception handling 2/xii, 2/xvi ﬁlenames 935Fortran 2000 (planned) 2/xvi Indexto Volumes1 and 2 Fortran 95 2/xv, 945, 947, 1084, 1100, 1364 HPF (High-Performance Fortran) 2/xvf.Numerical Recipes in 2/x, 2/xvii, 1 obsolescent features 2/xif. side effects 960see alsoFortran 90 Fortran D 2/xv Fortran 77 1/xix bit manipulation functions 17 hexadecimal constants 17 Fortran 8x 2/xi, 2/xiii Fortran 90 3 abstract data types 2/xiii, 1030all() intrinsic function 945, 948 allocatable array 938, 941, 953ff., 1197, 1212, 1266, 1293, 1306, 1336 allocate statement 938f., 941, 953f., 1197, 1266, 1293, 1306, 1336 allocated() intrinsic function 938, 952ff., 1197, 1266, 1293 any() intrinsic function 945, 948 array allocation and deallocation 953array of arrays 2/xii, 956, 1336 array constructor 2/xii, 968, 971, 1022, 1052, 1055, 1127 array constructor with implied do-list 968, 971 array extents 938, 949 array features 941ff., 953ff. array intrinsic procedures 2/xiii, 948ff. array of length 0 944array of length 1 949 array manipulation functions 950 array parallel operations 964f.array rank 938, 949 array reallocation 955 array section 2/xiif., 2/xiii, 939, 941ff., 960, 1078, 1284, 1286, 1333 array shape 938, 949 array size 938, 942array transpose 981f. array unary and binary functions 949 associated() intrinsic function 952f.associated pointer 953f. assumed-shape array 942 automatic array 938, 954, 1197, 1212, 1336 backwards-compatibility 935, 946 bit manipulation functions 2/xiii, 951bit size() intrinsic function 951 broadcasts 965f. btest() intrinsic function 951case construct 1010, 1036 case insensitive 937 ceiling() intrinsic function 947 character functions 952 character variables 1183cmplx function 1125 communication bottlenecks 969, 981, 1250 compatibility with Fortran 77 935, 946 compilers 2/viii, 2/xiv, 1364 compiling 936conformable arrays 942f., 1094CONTAINS statement 954, 957, 985, 1067, 1134, 1202 control structure 2/xiv, 959, 1219, 1305conversion elemental functions 946 count() intrinsic function 948 cshift() intrinsic function 950, 969cycle statement 959, 1219data hiding 956ff., 1209 data parallelism 964 DATA statement 959data types 937, 1336, 1346, 1361deallocate statement 938f., 953f., 1197, 1266, 1293 deallocating array 938, 953f., 1197, 1266, 1293 deﬁned types 956 deprecated features 947, 1057, 1161, 1256, 1286 derived types 937, 955 dimensional expansion 965ff. do-loop 2/xivdot product() intrinsic function 945, 949, 969, 1216 dynamical allocation of storage 2/xiii, 938, 941f., 953ff., 1327, 1336 elemental functions 940, 942, 946f., 951, 1015, 1083, 1364 elsewhere construct 943eoshift() intrinsic function 950, 969, 1019f., 1078 epsilon() intrinsic function 951, 1189evolution 2/xivff., 959, 987f.example 936 exit statement 959, 1219 exponent() intrinsic function 1107ﬂoor() intrinsic function 948Fortran tip icon 1009 garbage collection 956 gather-scatter operations 2/xiif., 969, 981, 984, 1002, 1032, 1034, 1250 generic interface 2/xiii, 1083 generic procedures 939, 1015, 1083, 1094, 1096, 1364 global variables 955, 957, 1210 history 2/xff. huge() intrinsic function 951iand() intrinsic function 951 ibclr() intrinsic function 951 ibits() intrinsic function 951ibset() intrinsic function 951ieor() intrinsic function 951 IMPLICIT NONE statement 2/xiv, 936 implied do-list 968, 971, 1127index loss 967f.initialization expression 943, 959, 1012, 1127 inquiry functions 948integer model 1144, 1149, 1156 INTENT attribute 1072, 1092 interface 939, 942, 1067, 1084, 1384internal subprogram 2/xii, 2/xiv, 957, 1057, 1067, 1202f., 1256, 1302 interprocessor communication 969, 981, 1250 intrinsic data types 937 Index to Volumes 1 and2 intrinsic procedures 939, 945ff., 987, 1016 ior() intrinsic function 951 ishft() intrinsic function 951ishftc() intrinsic function 951ISO (International Standards Organization) 2/xf., 2/xiiif. keyword argument 2/xiv, 947f., 1341kind() intrinsic function 951KIND parameter 937, 946, 1125, 1144, 1192, 1254, 1261, 1284, 1361 language features 935ff.lbound() intrinsic function 949lexical comparison 952 linear algebra 969f., 1000ff., 1018f., 1026, 1040, 1200, 1326 linear recurrence 971, 988linking 936 literal constant 937, 1361 logo for tips 2/viii, 1009mask 948, 967f., 1006f., 1038, 1102, 1200, 1226, 1305, 1333f., 1368, 1378, 1382 matmul() intrinsic function 945, 949, 969, 1026, 1040, 1050, 1076, 1200, 1216,1290, 1326 maxexponent() intrinsic function 1107 maxloc() intrinsic function 949, 961, 992f., 1015 maxval() intrinsic function 945, 948, 961, 1016, 1273 memory leaks 953, 956, 1327memory management 938, 953ff.merge() intrinsic function 945, 950, 1010, 1094f., 1099f. Metcalf and Reid (M&R) 935minloc() intrinsic function 949, 961, 992f.minval() intrinsic function 948, 961 missing language features 983ff., 987ff. modularization 956f.MODULE facility 2/xiii, 936f., 939f., 953f., 957, 1067, 1298, 1320, 1322, 1324, 1330, 1346 MODULE subprograms 940modulo() intrinsic function 946, 1156named constant 940, 1012, 1361 named control structure 959, 1219, 1305 nearest() intrinsic function 952, 1146nested where construct forbidden 943not() intrinsic function 951 nullify statement 953f., 1070, 1302 numerical representation functions 951ONLY option 941, 957, 1067operator overloading 2/xiif. operator, user-deﬁned 2/xii optional argument 2/xiv, 947f., 1092, 1228, 1230, 1256, 1272, 1275, 1340 outer product 969f.overloading 940, 1083, 1102 pack() intrinsic function 945, 950, 964, 969, 991, 1170, 1176, 1178 pack, for selective evaluation 1087parallel extensions 2/xv, 959ff., 964, 981, 984, 987, 1002, 1032 parallel programming 963ff.PARAMETER attribute 1012pointer 2/xiiif., 938f., 941, 944f., 952ff., 1067, 1070, 1197, 1210, 1212, 1266, 1302, 1327, 1336 pointer to function (missing) 1067 portability 963 present() intrinsic function 952PRIVATE attribute 957, 1067product() intrinsic function 948 programming conventions 937 PUBLIC attribute 957, 1067quick start 936radix() intrinsic function 1231 random number() intrinsic function 1141, 1143 randomseed() intrinsic function 1141 real() intrinsic function 947, 1125 RECURSIVE keyword 958, 1065, 1067recursive procedure 2/xiv, 958, 1065, 1067, 1166 reduction functions 948 reshape() intrinsic function 950, 969, 1247RESULT keyword 958, 1073 SAVE attribute 953f., 958f., 1052, 1070, 1266, 1293 scale() intrinsic function 1107scatter-with-combine (missing function) 984 scope 956ff.scoping units 939select case statement 2/xiv, 1010, 1036 shape() intrinsic function 938, 949 size() intrinsic function 938, 942, 945, 948 skew sections 985 sparse matrix representation 1030speciﬁcation statement 2/xivspread() intrinsic function 945, 950, 966ff., 969, 1000, 1094, 1290f. statement functions deprecated 1057stride (of an array) 944 structure constructor 2/xii subscript triplet 944sum() intrinsic function 945, 948, 966tiny() intrinsic function 952 transformational functions 948 transpose() intrinsic function 950, 960, 969, 981, 1247 tricks 1009, 1072, 1146, 1274, 1278, 1280 truncation elemental functions 946type checking 1140ubound() intrinsic function 949 undeﬁned pointer 953 unpack() intrinsic function 950, 964, 969USE statement 936, 939f., 954, 957, 1067, 1384 utility functions 987ff. vector subscripts 2/xiif., 969, 981, 984, 1002, 1032, 1034, 1250 visibility 956ff., 1209, 1293, 1296 WG5 technical committee 2/xi, 2/xiii, 2/xvf. where construct 943, 985, 1060, 1291 X3J3 Committee 2/viii, 2/xff., 2/xv, 947, 959, 964, 968, 990 zero-length array 944 Indexto Volumes1 and 2 see alsoIntrinsic procedures see alsoFortran Fortran 95 947, 959ff. allocatable variables 961 blocks 960 cputime() intrinsic function 961 elemental functions 2/xiii, 2/xv, 940, 961, 986, 1015, 1083f., 1097f. forall statement 2/xii, 2/xv, 960, 964, 968, 986, 1007 initialization of derived data type 2/xv initialization of pointer 2/xv, 961minor changes from Fortran 90 961 modiﬁed intrinsic functions 961 nested where construct 2/xv, 960, 1100pointer association status 961 pointers 961 PURE attribute 2/xv, 960f., 964, 986SAVE attribute 961 side effects 960 and skew array section 945, 985see alsoFortran Fortran 2000 2/xvi Forward deﬂation 363Forward difference operator 161 Forward Euler differencing 826f. Forward Time Centered Space seeFTCS Four-step framework, for FFT 983, 1239 Fourier analysis and cyclic reduction (FACR) 848f., 854 Fourier integrals attenuation factors 583, 1261 endpoint corrections 578f., 1261tail integration by parts 583 use of fast Fourier transform (FFT) 577ff., 1261ff. Fourier transform 99, 490ff., 1235ff. aliasing 495, 569 approximation of Dawson’s integral 253autocorrelation 492 basis functions compared 508f. contrasted with wavelet transform 584, 594 convolution 492, 503f., 531ff., 909, 1253, 1354 correlation 492, 538f., 1254cosine transform 190, 511ff., 851, 1245f. cosine transform, second form 513, 852, 1246 critical sampling 494, 543, 545 deﬁnition 490 discrete Fourier transform (DFT) 184, 495ff. Gaussian function 600 image processing 803, 805inﬁnite range 583 inverse of discrete Fourier transform 497 method for partial differential equations 848ff. missing data 569 missing data, fast algorithm 574f., 1259Nyquist frequency 494ff., 520, 543, 545, 569, 571 optimal (Wiener) ﬁltering 539ff., 558Parseval’s theorem 492, 498, 544power spectral density (PSD) 492f. power spectrum estimation by FFT 542ff., 1254ff. power spectrum estimation by maximum entropy method 565ff., 1258 properties of 491f.sampling theorem 495, 543, 545, 600 scalings of 491 signiﬁcance of a peak in 570sine transform 508ff., 850, 1245 symmetries of 491 uneven sampling, fast algorithm 574f., 1259 unevenly sampled data 569ff., 574, 1258 and wavelets 592f.Wiener-Khinchin theorem 492, 558, 566f. see alsoFast Fourier transform (FFT); Spectral density Fractal region 360f. Fractional step methods 847f. Fredholm alternative 780 Fredholm equations 779f. eigenvalue problems 780, 785error estimate in solution 784 ﬁrst kind 779 Fredholm alternative 780homogeneous, second kind 785, 1325 homogeneous vs. inhomogeneous 779f. ill-conditioned 780inﬁnite range 789 inverse problems 780, 795ff. kernel 779f.nonlinear 781 Nystrom method 782ff., 789, 1325 product Nystrom method 789, 1328ff.second kind 779f., 782ff., 1325, 1331 with singularities 788, 1328ff. with singularities, worked example 792, 1328ff. subtraction of singularity 789 symmetric kernel 785see alsoInverse problems Frequency domain 490 Frequency spectrum seeFast Fourier transform (FFT) Frequentist, contrasted with Bayesian 810 Fresnel integrals 248ff. asymptotic form 249 continued fraction 248f. routine for 249f., 1123 series 248 Friday the Thirteenth 14f., 1011f.FTCS (forward time centered space) 827ff., 839ff., 843 stability of 827ff., 839ff., 855 Full approximation storage (FAS) algorithm 874, 1339ff. Full moon 14f., 936, 1011f.Full multigrid method (FMG) 863, 868, 1334ff. Full Newton methods, nonlinear least squares 683 Full pivoting 29, 1014 Full weighting 867 Function Airy 204, 243f., 1121 Index to Volumes 1 and2 approximation 99ff., 184ff., 1043, 1076ff. associated Legendre polynomial 246ff., 764, 1122f., 1319 autocorrelation of 492 bandwidth limited 495Bessel 172, 204, 223ff., 234, 1101ff., 1115ff. beta 209, 1089binomial coefﬁcients 208f., 1087f.branch cuts of 202f. chi-square probability 215, 798 complex 202conﬂuent hypergeometric 204, 239convolution of 492correlation of 492cosine integral 250f., 1123f. Coulomb wave204, 234 cumulative binomial probability 222f.cumulative Poisson 209ff.Dawson’s integral 252ff., 600, 1127f.digamma 216 elliptic integrals 254ff., 906, 1128ff. error 213f., 248, 252, 601, 631, 635, 1094f., 1127, 1276f. evaluation 159ff., 1070ff. evaluation by path integration 201ff., 263, 1138 exponential integral 172, 215ff., 250, 1096f. F-distribution probability 222 Fresnel integral 248ff., 1123gamma 206, 1085hypergeometric 202f., 263ff., 1138ff.incomplete beta 219ff., 610, 1098ff., 1269 incomplete gamma 209ff., 615, 654, 657f., 1089ff., 1272, 1285 inverse hyperbolic 178, 255 inverse trigonometric 255Jacobian elliptic 261, 1137f.Kolmogorov-Smirnov probability 618f., 640, 1274, 1281 Legendre polynomial 172, 246, 674, 1122, 1291 logarithm 255modiﬁed Bessel 229ff., 1109ff.modiﬁed Bessel, fractional order 239ff., 1118ff. overloading 1083parallel evaluation 986, 1009, 1084, 1087, 1090, 1102, 1128, 1134 path integration to evaluate 201ff.pathological 99f., 343 Poisson cumulant 214 representations of 490routine for plotting a 342, 1182sine and cosine integrals 248, 250ff., 1125f. sn, dn, cn 261, 1137f.spherical harmonics 246ff., 1122spheroidal harmonic 764ff., 770ff., 1319ff., 1323ff. Student’s probability 221f. variable number of arguments 1022 Weber 204Functional iteration, for implicit equations 740f. FWHM (full width at half maximum) 548f. Gamma deviate 282f., 1153f. Gamma function 206ff., 1085 incomplete seeIncomplete gamma func- tion Garbage collection 956Gather-scatter operations 2/xiif., 984, 1002, 1032, 1034 communication bottleneck 969, 981, 1250many-to-one 984, 1002, 1032, 1034 Gauss-Chebyshev integration 141, 144, 512f. Gauss-Hermite integration 144, 789 abscissas and weights 147, 1062normalization 147 Gauss-Jacobi integration 144 abscissas and weights 148, 1063 Gauss-Jordan elimination 27ff., 33, 64, 1014f. operation count 34, 39 solution of normal equations 667, 1288 storage requirements 30 Gauss-Kronrod quadrature 154Gauss-Laguerre integration 144, 789, 1060 Gauss-Legendre integration 145f., 1059 see alsoGaussian integration Gauss-Lobatto quadrature 154, 190, 512Gauss-Radau quadrature 154 Gauss-Seidel method (relaxation) 855, 857, 864ff., 1338 nonlinear 876, 1341 Gauss transformation 256Gaussian (normal) distribution 267, 652, 798 central limit theorem 652f. deviates from 279f., 571, 1152 kurtosis of 606multivariate 690semi-invariants of 608 tails compared to Poisson 653 two-dimensional (binormal) 631variance of skewness of 606 Gaussian elimination 33f., 51, 55, 1014f. ﬁll-in 45, 64 integral equations 786, 1326operation count 34outer product variant 1017in reduction to Hessenberg form 478, 1231 relaxation solution of boundary value prob- lems 753ff., 777, 1316 Gaussian function Hardy’s theorem on Fourier transforms 600 see alsoGaussian (normal) distribution Gaussian integration 127, 140ff., 789, 1059ff. calculation of abscissas and weights 142ff., 1009, 1059ff. error estimate in solution 784extensions of 153f. Golub-Welsch algorithm for weights and abscissas 150, 1064 for integral equations 781, 783, 1325 from known recurrence relation 150, 1064 Indexto Volumes1 and 2 nonclassical weight function 151ff., 788f., 1064f., 1328f. and orthogonal polynomials 142, 1009, 1061 parallel calculation of formulas 1009, 1061 preassigned nodes 153f.weight function logx153 weight functions 140ff., 788f., 1059ff., 1328f. Gear’s method (stiff ODEs) 730 Geiger counter 266Generalized eigenvalue problems 455Generalized minimum residual method (GM- RES) 78 Generic interface seeInterface, generic Generic procedures 939, 1083, 1094, 1096, 1364 elemental 940, 942, 946f., 1015, 1083 Geometric progression 972, 996f., 1365, 1372ff. geop() utility function 972, 974, 989, 996, 1127 Geophysics, use of Backus-Gilbert method 809 Gerchberg-Saxton algorithm 805 get diag() utility function 985, 989, 1005, 1226 Gilbert and Sullivan 714Givens reduction 462f., 473 fast 463 operation count 463 Glassman, A.J. 180Global optimization 387f., 436ff., 650, 1219ff. continuous variables 443f., 1222 Global variables 940, 953f., 1210 allocatable array method 954, 1197, 1212, 1266, 1287, 1298 communicated via internal subprogram 954, 957f., 1067, 1226 danger of 957, 1209, 1293, 1296 pointer method 954, 1197, 1212, 1266, 1287, 1302 Globally convergent minimization 418ff., 1215root ﬁnding 373, 376ff., 382, 749f., 752, 1196, 1314f. GMRES (generalized minimum residual method) 78 GNU Emacs 1/xvi Godunov’s method 837 Golden mean (golden ratio) 21, 349, 392f., 399 Golden section search 341, 389ff., 395, 1202ff.Golub-Welsch algorithm, for Gaussian quadra- ture 150, 1064 Goodness-of-ﬁt 650, 654, 657f., 662, 690, 1285 GOTO statements, danger of 9, 959Gram-Schmidt biorthogonalization 415f. orthogonalization 94, 450f., 1039SVD as alternative to 58 Graphics, function plotting 342, 1182f.Gravitational potential 519Gray code 300, 881, 886ff., 1344 Greenbaum, A. 79 Gregorian calendar 13, 16, 1011, 1013Grid square 116f.Group, dihedral 894, 1345Guard digits 882, 1343 Half weighting 867, 1337 Halton’s quasi-random sequence 300Hamming window 547Hamming’s motto 341Hann window 547Harmonic analysis seeFourier transform Hashing 293, 1144, 1148, 1156 for random number seeds 1147f. HDLC checksum 890Heap (data structure) 327f., 336, 897, 1179Heapsort 320, 327f., 336, 1171f., 1179Helmholtz equation 852Hermite polynomials 144, 147 approximation of roots 1062 Hermitian matrix 450ff., 475Hertz (unit of frequency) 490Hessenberg matrix 94, 453, 470, 476ff., 488, 1231 see alsoMatrix Hessian matrix 382, 408, 415f., 419f., 676ff., 803, 815 is inverse of covariance matrix 667, 679second derivatives in 676 Hexadecimal constants 17f., 276, 293 initialization 959 Hierarchically band diagonal matrix 598Hierarchy of program structure 6ff.High-order not same as high-accuracy 100f., 124, 389, 399, 705, 709, 741 High-pass ﬁlter 551High-Performance Fortran (HPF) 2/xvf., 964, 981, 984 scatter-with-add 1032 Hilbert matrix 83Home page, Numerical Recipes 1/xx, 2/xviiHomogeneous linear equations 53Hook step methods 386Hotelling’s method for matrix inverse 49, 598Householder transformation 52, 453, 462ff., 469, 473, 475, 478, 481ff., 1227f. operation count 467 in QR decomposition 92, 1039 HPFseeHigh-Performance Fortran Huffman coding 564, 881, 896f., 902, 1346ff.huge() intrinsic function 951Hyperbolic functions, explicit formulas for inverse 178 Hyperbolic partial differential equations 818 advective equation 826 ﬂux-conservative initial value problems 825ff. Hypergeometric function 202f., 263ff. routine for 264f., 1138 Hypothesis, null 603 I2B, deﬁned 937 Index to Volumes 1 and2 I4B, deﬁned 937 iand() intrinsic function 951 ibclr() intrinsic function 951 ibits() intrinsic function 951 IBM 1/xxiii, 2/xix bad random number generator 268 Fortran 90 compiler 2/viii PC 4, 276, 293, 886PC-RT 4 radix base for ﬂoating point arithmetic 476 RS6000 2/viii, 4 IBM checksum 894 ibset() intrinsic function 951ICCG (incomplete Cholesky conjugate gradient method) 824 ICF (intrinsic correlation function) model 817Identity (unit) matrix 25 IEEE ﬂoating point format 276, 882f., 1343 ieor() intrinsic function 951 if statement, arithmetic 2/xi if structure 12f.iﬁrstloc() utility function 989, 993, 1041, 1346 IIR (inﬁnite impulse response) ﬁlter 552ff., 566 Ill-conditioned integral equations 780 Image processing 519, 803 cosine transform 513 fast Fourier transform (FFT) 519, 523, 803 as an inverse problem 803 maximum entropy method (MEM) 809ff. from modulus of Fourier transform 805wavelet transform 596f., 1267f. imaxloc() utility function 989, 993, 1017 iminloc() utility function 989, 993, 1046, 1076 Implicit function theorem 340 pivoting 30, 1014 shifts in QL method 472ff. Implicit differencing 827 for diffusion equation 840 for stiff equations 729, 740, 1308 IMPLICIT NONE statement 2/xiv, 936 Implied do-list 968, 971, 1127 Importance sampling, in Monte Carlo 306f. Improper integrals 135ff., 1055 Impulse response function 531, 540, 552IMSL 1/xxiii, 2/xx, 26, 64, 205, 364, 369, 454 In-place selection 335, 1178f.Included ﬁle, superseded by module 940 Incomplete beta function 219ff., 1098ff. for F-test 613, 1271routine for 220f., 1097 for Student’s t 610, 613, 1269 Incomplete Cholesky conjugate gradient method (ICCG) 824 Incomplete gamma function 209ff., 1089ff. for chi-square 615, 654, 657f., 1272, 1285 deviates from 282f., 1153 in mode estimation 610routine for 211f., 1089Increment of linear congruential generator 268 Indentation of blocks 9Index 934ff., 1446ff. this entry 1464 Index loss 967f., 1038Index table 320, 329f., 1173ff., 1176Inequality constraints 423Inheritance 8 Initial value problems 702, 818f. see alsoDifferential equations; Partial differential equations Initialization of derived data type 2/xvInitialization expression 943, 959, 1012, 1127Injection operator 864, 1337Instability seeStability Integer model, in Fortran 90 1144, 1149, 1156 Integer programming 436Integral equations 779ff. adaptive stepsize control 788block-by-block method 788correspondence with linear algebraic equa- tions 779ff. degenerate kernel 785 eigenvalue problems 780, 785error estimate in solution 784Fredholm 779f., 782ff., 1325, 1331Fredholm alternative 780homogeneous, second kind 785, 1325ill-conditioned 780inﬁnite range 789inverse problems 780, 795ff. kernel 779 nonlinear 781, 787Nystrom method 782ff., 789, 1325product Nystrom method 789, 1328ff.with singularities 788ff., 1328ff.with singularities, worked example 792, 1328ff. subtraction of singularity 789 symmetric kernel 785unstable quadrature 787f.Volterra 780f., 786ff., 1326f.wavelets 782see alsoInverse problems Integral operator, wavelet approximation of 597, 782 Integration of functions 123ff., 1052ff. cosine integrals 250, 1125Fourier integrals 577ff., 1261Fourier integrals, inﬁnite range 583Fresnel integrals 248, 1123Gauss-Hermite 147f., 1062Gauss-Jacobi 148, 1063Gauss-Laguerre 146, 1060Gauss-Legendre 145, 1059 integrals that are elliptic integrals 254 path integration 201ff.sine integrals 250, 1125see alsoQuadrature Integro-differential equations 782INTENT attribute 1072, 1092Interface (Fortran 90) 939, 942, 1067 Indexto Volumes1 and 2 for communication between program parts 957, 1209, 1293, 1296 explicit 939, 942, 1067, 1384generic 2/xiii, 940, 1015, 1083, 1094, 1096 implicit 939for Numerical Recipes 1384ff. Interface block 939, 1084, 1384 Interface, in programs 2, 8 Intermediate value theorem 343 Internal subprogram (Fortran 90) 2/xiv, 954, 957, 1067, 1202f., 1226 nesting of 2/xii resembles C macro 1302supersedes statement function 1057, 1256 International Standards Organization (ISO) 2/xf., 2/xiii Internet, availability of code over 1/xx, 2/xvii Interpolation 99ff. Aitken’s algorithm 102avoid 2-stage method 100 avoid in Fourier analysis 569 bicubic 118f., 1049f. bilinear 117 caution on high-order 100coefﬁcients of polynomial 100, 113ff., 191, 575, 1047f., 1078 for computing Fourier integrals 578error estimates for 100 of functions with poles 104ff., 1043f. inverse quadratic 353, 395ff., 1204multidimensional 101f., 116ff., 1049ff. in multigrid method 866, 1337 Neville’s algorithm 102f., 182, 1043Nystrom 783, 1326 offset arrays 104, 113 operation count for 100 operator 864, 1337 order of 100and ordinary differential equations 101 oscillations of polynomial 100, 116, 389, 399 parabolic, for minimum ﬁnding 395, 1204 polynomial 99, 102ff., 182, 1043 rational Chebyshev approximation 197ff., 1081 rational function 99, 104ff., 194ff., 225, 718ff., 726, 1043f., 1080, 1306 reverse (extirpolation) 574, 1261 spline 100, 107ff., 120f., 1044f., 1050f. trigonometric 99see alsoFitting Interprocessor communication 969, 981 Interval variable (statistics) 623 Intrinsic correlation function (ICF) model 817 Intrinsic data types 937Intrinsic procedures array inquiry 938, 942, 948ff. array manipulation 950array reduction 948 array unary and binary functions 949 backwards-compatibility 946 bit manipulation 2/xiii, 951 character 952cmplx 1254conversion elemental 946 elemental 940, 942, 946f., 951, 1083, 1364 generic 939, 1083f., 1364lexical comparison 952numeric inquiry 2/xiv, 1107, 1231, 1343 numerical 946, 951f. numerical representation 951pack used for sorting 1171random number 1143 real 1254top 10 945truncation 946f.see alsoFortran 90 Inverse hyperbolic function 178, 255Inverse iteration seeEigensystems Inverse problems 779, 795ff. Backus-Gilbert method 806ff.Bayesian approach 799, 810f., 816f.central idea 799constrained linear inversion method 799ff.data inversion 807deterministic constraints 804ff.in geophysics 809Gerchberg-Saxton algorithm 805incomplete Fourier coefﬁcients 813 and integral equations 780 linear regularization 799ff.maximum entropy method (MEM) 810, 815f. MEM demystiﬁed 814Phillips-Twomey method 799ff. principal solution 797 regularization 796ff.regularizing operator 798stabilizing functional 798Tikhonov-Miller regularization 799ff.trade-off curve 795trade-off curve, Backus-Gilbert method 809 two-dimensional regularization 803use of conjugate gradient minimization 804, 815 use of convex sets 804use of Fourier transform 803, 805 Van Cittert’s method 804 Inverse quadratic interpolation 353, 395ff., 1204 Inverse response kernel, in Backus-Gilbert method 807 Inverse trigonometric function 255ior() intrinsic function 951ISBN (International Standard Book Number) checksum 894 ishft() intrinsic function 951ishftc() intrinsic function 951ISO (International Standards Organization) 2/xf., 2/xiii Iterated integrals 155Iteration 9f. functional 740f.to improve solution of linear algebraic equations 47ff., 195, 1022 for linear algebraic equations 26 Index to Volumes 1 and2 required for two-point boundary value problems 745 in root ﬁnding 340f. Iteration matrix 856 ITPACK 71 Iverson, John 2/xi Jacobi matrix, for Gaussian quadrature 150, 1064 Jacobi polynomials, approximation of roots 1064 Jacobi transformation (or rotation) 94, 453, 456ff., 462, 475, 489, 1041, 1225 Jacobian determinant 279, 774Jacobian elliptic functions 261, 1137f.Jacobian matrix 374, 376, 379, 382, 731, 1197f., 1309 singular in Newton’s rule 386 Jacobi’s method (relaxation) 855ff., 864 Jenkins-Traub method 369 Julian Day 1, 13, 16, 936, 1010ff.Jump transposition errors 895 K-S testseeKolmogorov-Smirnov test Kalman ﬁlter 700 Kanji 2/xiiKaps-Rentrop method 730, 1308 Kendall’s tau 634, 637ff., 1279 Kennedy, Ken 2/xvKepler’s equation 1061Kermit checksum 889 Kernel 779 averaging, in Backus-Gilbert method 807degenerate 785 ﬁnite rank 785 inverse response 807separable 785singular 788f., 1328 symmetric 785 Keys used in sorting 329, 889Keyword argument 2/xiv, 947f., 1341 kind() intrinsic function 951 KIND parameter 946, 1261, 1284 and cmplx() intrinsic function 1125, 1192, 1254 default 937 for Numerical Recipes 1361for random numbers 1144and real() intrinsic function 1125 Kolmogorov-Smirnov test 614, 617ff., 694, 1273f. two-dimensional 640, 1281ff. variants 620ff., 640, 1281 Kuiper’s statistic 621Kurtosis 606, 608, 1269 L-estimate 694 Labels, statement 9Lag 492, 538, 553Lagged Fibonacci generator 1142, 1148ff. Lagrange multiplier 795 Lagrange’s formula for polynomial interpola- tion 84, 102f., 575, 578Laguerre polynomials, approximation of roots 1061 Laguerre’s method 341, 365f., 1191f.Lanczos lemma 498f. Lanczos method for gamma function 206, 1085 Landen transformation 256 LAPACK 26, 1230 Laplace’s equation 246, 818 see alsoPoisson equation Las Vegas 625 Latin square or hypercube 305f.Laurent series 566Lax method 828ff., 836, 845f. multidimensional 845f. Lax-Wendroff method 835ff.lbound() intrinsic function 949 Leakage in power spectrum estimation 544, 548 Leakage width 548f. Leapfrog method 833f. Least squares ﬁlters seeSavitzky-Golay ﬁlters Least squares ﬁtting 645, 651ff., 655ff., 660ff., 665ff., 1285f., 1288f. contrasted to general minimization prob- lems 684ff. degeneracies in 671f., 674 Fourier components 570 as M-estimate for normal errors 696as maximum likelihood estimator 652 as method for smoothing data 645, 1283 Fourier components 1258freezing parameters in 668, 700 general linear case 665ff., 1288, 1290f. Levenberg-Marquardt method 678ff., 816, 1292f. Lomb periodogram 570, 1258 multidimensional 675nonlinear 386, 675ff., 816, 1292 nonlinear, advanced methods 683 normal equations 645, 666f., 800, 1288normal equations often singular 670, 674optimal (Wiener) ﬁltering 540f. QR method in 94, 668 for rational Chebyshev approximation 199f., 1081f. relation to linear correlation 630, 658 Savitzky-Golay ﬁlter as 645, 1283singular value decomposition (SVD) 25f., 51ff., 199f., 670ff., 1081, 1290 skewed by outliers 653for spectral analysis 570, 1258 standard (probable) errors on ﬁtted parame- ters 667, 671 weighted 652see alsoFitting L’Ecuyer’s long period random generator 271, 273 Least squares ﬁtting standard (probable) errors on ﬁtted parame- ters 1288, 1290 weighted 1285 Left eigenvalues or eigenvectors 451 Legal matters 1/xx, 2/xviiLegendre elliptic integral seeElliptic integrals Indexto Volumes1 and 2 Legendre polynomials 246, 1122 ﬁtting data to 674, 1291f. recurrence relation 172shifted monic 151 see alsoAssociated Legendre polynomials; Spherical harmonics Lehmer-Schur algorithm 369Lemarie’s wavelet 593 Lentz’s method for continued fraction 165, 212 Lepage, P. 309Leptokurtic distribution 606 Levenberg-Marquardt algorithm 386, 678ff., 816, 1292 advanced implementation 683 Levinson’s method 86, 1038 Lewis, H.W. 275Lexical comparison functions 952LGT, deﬁned 937 License information 1/xx, 2/xviiff. Limbo 356Limit cycle, in Laguerre’s method 365 Line minimization seeMinimization, along a ray Line search seeMinimization, along a ray Linear algebra, intrinsic functions for paral- lelization 969f., 1026, 1040, 1200, 1326 Linear algebraic equations 22ff., 1014 band diagonal 43ff., 1019 biconjugate gradient method 77, 1034ff. Cholesky decomposition 89f., 423, 455, 668, 1038f. complex 41 computing A −1·B40 conjugate gradient method 77ff., 599, 1034 cyclic tridiagonal 67, 1030 direct methods 26, 64, 1014, 1030Fortran 90 vs. library routines 1016 Gauss-Jordan elimination 27ff., 1014 Gaussian elimination 33f., 1014f.Hilbert matrix 83Hotelling’s method 49, 598 and integral equations 779ff., 783, 1325 iterative improvement 47ff., 195, 1022iterative methods 26, 77ff., 1034 large sets of 23 least squares solution 53ff., 57f., 199f., 671, 1081, 1290 LU decomposition 34ff., 195, 386, 732, 783, 786, 801, 1016, 1022, 1325f. nonsingular 23overdetermined 25f., 199, 670, 797partitioned 70 QR decomposition 91f., 382, 386, 668, 1039f., 1199 row vs. column elimination 31f. Schultz’s method 49, 598 Sherman-Morrison formula 65ff., 83singular 22, 53, 58, 199, 670singular value decomposition (SVD) 51ff., 199f., 670ff., 797, 1022, 1081, 1290 sparse 23, 43, 63ff., 732, 804, 1020f., 1030summary of tasks 25f. Toeplitz 82, 85ff., 195, 1038 tridiagonal 26, 42f., 64, 109, 150, 453f., 462ff., 469ff., 488, 839f., 853, 861f., 1018f., 1227ff. Vandermonde 82ff., 114, 1037, 1047 wavelet solution 597ff., 782 Woodbury formula 68ff., 83see alsoEigensystems Linear congruential random number generator 267ff., 1142 choice of constants for 274ff. Linear constraints 423 Linear convergence 346, 393Linear correlation (statistics) 630ff., 1276 Linear dependency constructing orthonormal basis 58, 94 of directions in N-dimensional space 409 in linear algebraic equations 22f. Linear equations seeDifferential equations; In- tegral equations; Linear algebraic equa- tions Linear inversion method, constrained 799ff. Linear prediction 557ff. characteristic polynomial 559coefﬁcients 557ff., 1256 compared to maximum entropy method 558 compared with regularization 801 contrasted to polynomial extrapolation 560 related to optimal ﬁltering 558 removal of bias in 563stability 559f., 1257 Linear predictive coding (LPC) 563ff. Linear programming 387, 423ff., 1216ff. artiﬁcial variables 429 auxiliary objective function 430basic variables 426 composite simplex algorithm 435 constraints 423convergence criteria 432 degenerate feasible vector 429 dual problem 435 equality constraints 423 feasible basis vector 426feasible vector 424 fundamental theorem 426 inequality constraints 423left-hand variables 426 nonbasic variables 426 normal form 426 objective function 424 optimal feasible vector 424pivot element 428f. primal-dual algorithm 435 primal problem 435 reduction to normal form 429ff. restricted normal form 426ff.revised simplex method 435 right-hand variables 426 simplex method 402, 423ff., 431ff., 1216ff.slack variables 429 tableau 427 vertex of simplex 426 Index to Volumes 1 and2 Linear recurrence seeRecurrence relation Linear regression 655ff., 660ff., 1285ff. see alsoFitting Linear regularization 799ff. LINPACK 26 Literal constant 937, 1361Little-endian 293Local extrapolation 709 Local extremum 387f., 437 Localization of roots seeBracketing Logarithmic function 255 Lomb periodogram method of spectral analysis 569f., 1258f. fast algorithm 574f., 1259 Loops 9f. Lorentzian probability distribution 282, 696f. Low-pass ﬁlter 551, 644f., 1283f.Lower subscript 944lower triangle() utility function 989, 1007, 1200 LP coefﬁcients seeLinear prediction LPC (linear predictive coding) 563ff. LU decomposition 34ff., 47f., 51, 55, 64, 97, 374, 667, 732, 1016, 1022 forA−1·B40 backsubstitution 39, 1017 band diagonal matrix 43ff., 1020 complex equations 41f.Crout’s algorithm 36ff., 45, 1017 for integral equations 783, 786, 1325f. for inverse iteration of eigenvectors 488for inverse problems 801for matrix determinant 41 for matrix inverse 40, 1016 for nonlinear sets of equations 374, 386, 1196 operation count 36, 39 outer product Gaussian elimination 1017for Pad´e approximant 195, 1080 pivoting 37f., 1017 repeated backsubstitution 40, 46 solution of linear algebraic equations 40, 1017 solution of normal equations 667 for Toeplitz matrix 87 Lucifer 290 M&R (Metcalf and Reid) 935 M-estimates 694ff. how to compute 697f.local 695ff. see alsoMaximum likelihood estimate Machine accuracy 19f., 881f., 1189, 1343Macintosh, seeApple Macintosh Maehly’s procedure 364, 371 Magic in MEM image restoration 814in Pad´e approximation 195 Mantissa in ﬂoating point format 19, 882, 909, 1343 Marginals 624Marquardt method (least squares ﬁtting) 678ff., 816, 1292f. Marsaglia shift register 1142, 1148ff.Marsaglia, G. 1142, 1149mask 1006f., 1102, 1200, 1226, 1305, 1333f., 1368, 1378, 1382 optional argument 948optional argument, facilitates parallelism 967f., 1038 Mass, center of 295ff.MasterCard checksum 894Mathematical Center (Amsterdam) 353 Mathematical intrinsic functions 946, 951f. matmul() intrinsic function 945, 949, 969, 1026, 1040, 1050, 1076, 1200, 1216, 1290, 1326 Matrix 23ff. add vector to diagonal 1004, 1234, 1366, 1381 approximation of 58f., 598f.band diagonal 42ff., 64, 1019band triangular 64 banded 26, 454 bidiagonal 52block diagonal 64, 754 block triangular 64 block tridiagonal 64bordered 64characteristic polynomial 449, 469 Cholesky decomposition 89f., 423, 455, 668, 1038f. column augmented 28, 1014 complex 41 condition number 53, 78create unit matrix 1006, 1382 curvature 677 cyclic banded 64cyclic tridiagonal 67, 1030defective 450, 476, 489 of derivatives seeHessian matrix; Jacobian determinant design (ﬁtting) 645, 665, 801, 1082 determinant of 25, 41 diagonal of sparse matrix 1033ff.diagonalization 452ff., 1225ff.elementary row and column operations 28f. ﬁnite differencing of partial differential equations 821ff. get diagonal 985, 1005, 1226f., 1366, 1381f. Hermitian 450, 454, 475 Hermitian conjugate 450 Hessenberg 94, 453, 470, 476ff., 488, 1231ff. HessianseeHessian matrix hierarchically band diagonal 598 Hilbert 83identity 25 ill-conditioned 53, 56, 114 indexed storage of 71f., 1030and integral equations 779, 783, 1325inverse 25, 27, 34, 40, 65ff., 70, 95ff., 1014, 1016f. inverse, approximate 49inverse by Hotelling’s method 49, 598 inverse by Schultz’s method 49, 598 inverse multiplied by a matrix 40iteration for inverse 49, 598 Indexto Volumes1 and 2 Jacobi transformation 453, 456ff., 462, 1225f. Jacobian 731, 1309logical dimension 24 lower triangular 34f., 89, 781, 1016 lower triangular mask 1007, 1200, 1382multiplication denoted by dot 23multiplication, intrinsic function 949, 969, 1026, 1040, 1050, 1200, 1326 norm 50normal 450ff.nullity 53 nullspace 25, 53f., 449, 795 orthogonal 91, 450, 463ff., 587orthogonal transformation 452, 463ff., 469, 1227 orthonormal basis 58, 94outer product denoted by cross 66, 420partitioning for determinant 70 partitioning for inverse 70 pattern multiply of sparse 74physical dimension 24 positive deﬁnite 26, 89f., 668, 1038 QR decomposition 91f., 382, 386, 668, 1039, 1199 range 53 rank 53 residual 49row and column indices 23row vs. column operations 31f. self-adjoint 450 set diagonal elements 1005, 1200, 1366, 1382 similarity transform 452ff., 456, 476, 478, 482 singular 53f., 58, 449singular value decomposition 26, 51ff., 797 sparse 23, 63ff., 71, 598, 732, 754, 804, 1030ff. special forms 26 splitting in relaxation method 856f.spread 808square root of 423, 455 symmetric 26, 89, 450, 454, 462ff., 668, 785, 1038, 1225, 1227 threshold multiply of sparse 74, 1031 Toeplitz 82, 85ff., 195, 1038 transpose() intrinsic function 950transpose of sparse 73f., 1033triangular 453 tridiagonal 26, 42f., 64, 109, 150, 453f., 462ff., 469ff., 488, 839f., 853, 861f.,1018f., 1227ff. tridiagonal with fringes 822 unitary 450 updating 94, 382, 386, 1041, 1199upper triangular 34f., 91, 1016 upper triangular mask 1006, 1226, 1305, 1382 Vandermonde 82ff., 114, 1037, 1047see alsoEigensystems Matrix equations seeLinear algebraic equa- tions Matterhorn 606maxexponent() intrinsic function 1107 Maximization seeMinimization Maximum entropy method (MEM) 565ff., 1258 algorithms for image restoration 815f. Bayesian 816f.Cornwell-Evans algorithm 816 demystiﬁed 814 historic vs. Bayesian 816f. image restoration 809ff. intrinsic correlation function (ICF) model 817 for inverse problems 809ff. operation count 567 see alsoLinear prediction Maximum likelihood estimate (M-estimates) 690, 694ff. and Bayes’ Theorem 811 chi-square test 690 deﬁned 652how to compute 697f. mean absolute deviation 696, 698, 1294 relation to least squares 652 maxloc() intrinsic function 949, 992f., 1015 modiﬁed in Fortran 95 961 maxval() intrinsic function 945, 948, 961, 1016, 1273 Maxwell’s equations 825f. Mean(s) of distribution 604f., 608f., 1269 statistical differences between two 609ff., 1269f. Mean absolute deviation of distribution 605, 696, 1294 related to median 698 Measurement errors 650 Median 320 calculating 333of distribution 605, 608f. as L-estimate 694 role in robust straight line ﬁtting 698by selection 698, 1294 Median-of-three, in Quicksort 324 MEMseeMaximum entropy method (MEM) Memory leak 953, 956, 1071, 1327 Memory management 938, 941f., 953ff., 1327, 1336 merge construct 945, 950, 1099f. for conditional scalar expression 1010, 1094f. contrasted with where 1023 parallelization 1011 Merge-with-dummy-values idiom 1090Merit function 650 in general linear least squares 665 for inverse problems 797 nonlinear models 675 for straight line ﬁtting 656, 698for straight line ﬁtting, errors in both coor- dinates 660, 1286 Mesh-drift instability 834f. Mesokurtic distribution 606 Metcalf, Michael 2/viii see alsoM&R Method of regularization 799ff. Index to Volumes 1 and2 Metropolis algorithm 437f., 1219 Microsoft 1/xxii, 2/xix Microsoft Fortran PowerStation 2/viiiMidpoint method seeModiﬁed midpoint method; Semi-implicit midpoint rule Mikado, or Town of Titipu 714 Miller’s algorithm 175, 228, 1106MIMD machines (Multiple Instruction Multiple Data) 964, 985, 1071, 1084 Minimal solution of recurrence relation 174Minimax polynomial 186, 198, 1076Minimax rational function 198 Minimization 387ff. along a ray 77, 376f., 389, 406ff., 412f., 415f., 418, 1195f., 1211, 1213 annealing, method of simulated 387f., 436ff., 1219ff. bracketing of minimum 390ff., 402, 1201f.Brent’s method 389, 395ff., 399, 660f., 1204ff., 1286 Broyden-Fletcher-Goldfarb-Shanno algo- rithm 390, 418ff., 1215 chi-square 653ff., 675ff., 1285, 1292choice of methods 388f. combinatorial 436f., 1219 conjugate gradient method 390, 413ff., 804, 815, 1210, 1214 convergence rate 393, 409 Davidon-Fletcher-Powell algorithm 390, 418ff., 1215 degenerate 795 direction-set methods 389, 406ff., 1210ff. downhill simplex method 389, 402ff., 444, 697f., 1208, 1222ff. ﬁnding best-ﬁt parameters 650 Fletcher-Reeves algorithm 390, 414ff., 1214 functional 795global 387f., 443f., 650, 1219, 1222 globally convergent multidimensional 418, 1215 golden section search 390ff., 395, 1202ff. multidimensional 388f., 402ff., 1208ff., 1214 in nonlinear model ﬁtting 675f., 1292Polak-Ribiere algorithm 389, 414ff., 1214 Powell’s method 389, 402, 406ff., 1210ff. quasi-Newton methods 376, 390, 418ff., 1215 and root ﬁnding 375 scaling of variables 420 by searching smaller subspaces 815steepest descent method 414, 804termination criterion 392, 404 use in ﬁnding double roots 341 use for sparse linear systems 77ff.using derivatives 389f., 399ff., 1205ff.variable metric methods 390, 418ff., 1215 see alsoLinear programming Minimum residual method, for sparse system 78 minloc() intrinsic function 949, 992f. modiﬁed in Fortran 95 961 MINPACK 683minval() intrinsic function 948, 961MIPS 886 Missing data problem 569Mississippi River 438f., 447MMP (massively multiprocessor) machines 965ff., 974, 981, 984, 1016ff., 1021,1045, 1226ff., 1250 Mode of distribution 605, 609Modeling of data seeFitting Model-trust region 386, 683Modes, homogeneous, of recursive ﬁlters 554 Modiﬁed Bessel functions seeBessel func- tions Modiﬁed Lentz’s method, for continued frac- tions 165 Modiﬁed midpoint method 716ff., 720, 1302f. Modiﬁed moments 152 Modula-2 7Modular arithmetic, without overﬂow 269, 271, 275 Modular programming 2/xiii, 7f., 956ff., 1209, 1293, 1296, 1346 MODULE facility 2/xiii, 936f., 939f., 957, 1067, 1298, 1320, 1322, 1324, 1330, 1346 initializing random number generator 1144ff. in nr.f90 936, 941f., 1362, 1384ff. in nrtype.f90 936f., 1361f.in nrutil.f90 936, 1070, 1362, 1364ff.sparse matrix 1031undeﬁned variables on exit 953, 1266 Module subprogram 940 modulo() intrinsic function 946, 1156 Modulus of linear congruential generator 268Moments of distribution 604ff., 1269ﬁlter that preserves 645 modiﬁed problem of 151f. problem of 83and quadrature formulas 791, 1328semi-invariants 608 Monic polynomial 142f. Monotonicity constraint, in upwind differenc- ing 837 Monte Carlo 155ff., 267 adaptive 306ff., 1161ff. bootstrap method 686f.comparison of sampling methods 309exploration of binary tree 290importance sampling 306f. integration 124, 155ff., 295ff., 306ff., 1161 integration, recursive 314ff., 1164ff. integration, using Sobol’ sequence 304 integration, VEGAS algorithm 309ff., 1161 and Kolmogorov-Smirnov statistic 622, 640 partial differential equations 824 quasi-random sequences in 299ff.quick and dirty 686f.recursive 306ff., 314ff., 1161, 1164ff.signiﬁcance of Lomb periodogram 570 simulation of data 654, 684ff., 690 stratiﬁed sampling 308f., 314, 1164 Indexto Volumes1 and 2 Moon, calculate phases of 1f., 14f., 936, 1010f. Mother functions 584Mother Nature 684, 686 Moving average (MA) model 566 Moving window averaging 644Mozart 9 MS 1/xxii, 2/xix Muller’s method 364, 372 Multidimensional conﬁdence levels of ﬁtting 688f.data, use of binning 623 Fourier transform 515ff., 1241, 1246, 1251 Fourier transform, real data 519ff., 1248f. initial value problems 844ff. integrals 124, 155ff., 295ff., 306ff., 1065ff., 1161ff. interpolation 116ff., 1049ff. Kolmogorov-Smirnov test 640, 1281least squares ﬁtting 675 minimization 402ff., 406ff., 413ff., 1208ff., 1214f., 1222ff. Monte Carlo integration 295ff., 306ff., 1161ff. normal (Gaussian) distribution 690 optimization 388f. partial differential equations 844ff. root ﬁnding 340ff., 358, 370, 372ff., 746, 749f., 752, 754, 1194ff., 1314ff. search using quasi-random sequence 300 secant method 373, 382f., 1199f.wavelet transform 595, 1267f. Multigrid method 824, 862ff., 1334ff. avoid SOR 866boundary conditions 868f. choice of operators 868 coarse-to-ﬁne operator 864, 1337coarse-grid correction 864f. cycle 865 dual viewpoint 875ﬁne-to-coarse operator 864, 1337 full approximation storage (FAS) algorithm 874, 1339ff. full multigrid method (FMG) 863, 868, 1334ff. full weighting 867Gauss-Seidel relaxation 865f., 1338 half weighting 867, 1337 importance of adjoint operator 867injection operator 864, 1337 interpolation operator 864, 1337 line relaxation 866local truncation error 875 Newton’s rule 874, 876, 1339, 1341 nonlinear equations 874ff., 1339ff. nonlinear Gauss-Seidel relaxation 876, 1341 odd-even ordering 866, 869, 1338operation count 862 prolongation operator 864, 1337 recursive nature 865, 1009, 1336 relative truncation error 875 relaxation as smoothing operator 865restriction operator 864, 1337speeding up FMG algorithm 873 stopping criterion 875f. straight injection 867symbol of operator 866f. use of Richardson extrapolation 869 V-cycle 865, 1336W-cycle 865, 1336zebra relaxation 866 Multiple precision arithmetic 906ff., 1352ff. Multiple roots 341, 362Multiplication, complex 171 Multiplication, multiple precision 907, 909, 1353f. Multiplier of linear congruential generator 268 Multistep and multivalue methods (ODEs) 740ff. see alsoDifferential Equations; Predictor- corrector methods Multivariate normal distribution 690 Murphy’s Law 407Musical scores 5f. NAG 1/xxiii, 2/xx, 26, 64, 205, 454 Fortran 90 compiler 2/viii, 2/xiv Named constant 940 initialization 1012 for Numerical Recipes 1361 Named control structure 959, 1219, 1305National Science Foundation (U.S.) 1/xvii, 1/xix, 2/ix Natural cubic spline 109, 1044f.Navier-Stokes equation 830f. nearest() intrinsic function 952, 1146 Needle, eye of (minimization) 403Negation, multiple precision 907, 1353f.Negentropy 811, 896 Nelder-Mead minimization method 389, 402, 1208 Nested iteration 868 Neumann boundary conditions 820, 840, 851, 858 Neutrino 640Neville’s algorithm 102f., 105, 134, 182, 1043 Newton-Cotes formulas 125ff., 140Newton-Raphson method seeNewton’s rule Newton’s rule 143f., 180, 341, 355ff., 362, 364, 469, 1059, 1189 with backtracking 376, 1196caution on use of numerical derivatives 356ff. fractal domain of convergence 360f.globally convergent multidimensional 373, 376ff., 382, 749f., 752, 1196, 1199, 1314f. for matrix inverse 49, 598in multidimensions 370, 372ff., 749f., 752, 754, 1194ff., 1314ff. in nonlinear multigrid 874, 876, 1339, 1341 nonlinear Volterra equations 787 for reciprocal of number 911, 1355 safe 359, 1190scaling of variables 381 Index to Volumes 1 and2 singular Jacobian 386 solving stiff ODEs 740 for square root of number 912, 1356 Niederreiter sequence 300 NL2SOL 683Noise bursty 889 effect on maximum entropy method 567equivalent bandwidth 548 ﬁtting data which contains 647f., 650 model, for optimal ﬁltering 541 Nominal variable (statistics) 623 Nonexpansive projection operator 805 Non-interfering directions seeConjugate direc- tions Nonlinear eigenvalue problems 455 Nonlinear elliptic equations, multigrid method 874ff., 1339ff. Nonlinear equations, in MEM inverse problems 813 Nonlinear equations, roots of 340ff. Nonlinear instability 831Nonlinear integral equations 781, 787 Nonlinear programming 436 Nonnegativity constraints 423Nonparametric statistics 633ff., 1277ff. Nonpolynomial complete (NP-complete) 438 Norm, of matrix 50Normal (Gaussian) distribution 267, 652, 682, 798, 1294 central limit theorem 652f.deviates from 279f., 571, 1152 kurtosis of 607 multivariate 690 semi-invariants of 608 tails compared to Poisson 653two-dimensional (binormal) 631 variance of skewness of 606 Normal equations (ﬁtting) 26, 645, 666ff., 795, 800, 1288 often are singular 670 Normalization of Bessel functions 175 of ﬂoating-point representation 19, 882, 1343 of functions 142, 765 of modiﬁed Bessel functions 232 not() intrinsic function 951 Notch ﬁlter 551, 555f. NP-complete problem 438nr.f90 (module ﬁle) 936, 1362, 1384ff. nrerror() utility function 989, 995 nrtype.f90 (module ﬁle) 936f. named constants 1361 nrutil.f90 (module ﬁle) 936, 1070, 1362, 1364ff. table of contents 1364 Null hypothesis 603 nullify statement 953f., 1070, 1302Nullity 53 Nullspace 25, 53f., 449, 795 Number-theoretic transforms 503f. Numeric inquiry functions 2/xiv, 1107, 1231, 1343 Numerical derivatives 180ff., 645, 1075Numerical integration seeQuadrature Numerical intrinsic functions 946, 951f.Numerical Recipes compatibility with First Edition 4 Example Book 3 Fortran 90 types 936f., 1361how to get programs 1/xx, 2/xviihow to report bugs 1/iv, 2/iv interface blocks (Fortran 90) 937, 941f., 1084, 1384ff. no warranty on 1/xx, 2/xvii plan of two-volume edition 1/xiiitable of dependencies 921ff., 1434ff.as trademark 1/xxiii, 2/xxutility functions (Fortran 90) 936f., 945, 968, 970, 972ff., 977, 984, 987ff., 1015,1071f., 1361ff. Numerical Recipes Software 1/xv, 1/xxiiff., 2/xviiff. address and fax number 1/iv, 1/xxii, 2/iv, 2/xix Web home page 1/xx, 2/xvii Nyquist frequency 494ff., 520, 543, 545, 569ff. Nystrom method 782f., 789, 1325 product version 789, 1331 Object extensibility 8 Objective function 424Object-oriented programming 2/xvi, 2, 8 Oblateness parameter 764 Obsolete features seeFortran, Obsolescent fea- tures Octal constant, initialization 959Odd-even ordering allows parallelization 1333 in Gauss-Seidel relaxation 866, 869, 1338 in successive over-relaxation (SOR) 859, 1332 Odd parity 888OEM information 1/xxiiOne-sided power spectral density 492 ONLY option, for USE statement 941, 957, 1067 Operation count balancing 476Bessel function evaluation 228bisection method 346 Cholesky decomposition 90 coefﬁcients of interpolating polynomial 114f. complex multiplication 97cubic spline interpolation 109evaluating polynomial 168 fast Fourier transform (FFT) 498 Gauss-Jordan elimination 34, 39Gaussian elimination 34Givens reduction 463 Householder reduction 467 interpolation 100inverse iteration 488iterative improvement 48 Jacobi transformation 460 Kendall’s tau 637 Indexto Volumes1 and 2 linear congruential generator 268 LU decomposition 36, 39 matrix inversion 97matrix multiplication 96 maximum entropy method 567 multidimensional minimization 413f.multigrid method 862 multiplication 909 polynomial evaluation 97f., 168QL method 470, 473 QR decomposition 92 QR method for Hessenberg matrices 484reduction to Hessenberg form 479 selection by partitioning 333 sorting 320ff.Spearman rank-order coefﬁcient 638 Toeplitz matrix 83 Vandermonde matrix 83 Operator overloading 2/xiif., 7 Operator splitting 823, 847f., 861 Operator, user-deﬁned 2/xiiOptimal feasible vector 424 Optimal (Wiener) ﬁltering 535, 539ff., 558, 644 compared with regularization 801 Optimization seeMinimization Optimization of code 2/xiiiOptional argument 2/xiv, 947f., 1092, 1228, 1230, 1256, 1272, 1275, 1340 dim 948mask 948, 968, 1038 testing for 952 Ordering Numerical Recipes 1/xxf., 2/xviif.Ordinal variable (statistics) 623 Ordinary differential equations seeDifferential equations Orthogonal seeOrthonormal functions; Or- thonormal polynomials Orthogonal transformation 452, 463ff., 469, 584, 1227 Orthonormal basis, constructing 58, 94, 1039 Orthonormal functions 142, 246Orthonormal polynomials Chebyshev 144, 184ff., 1076ff. construct for arbitrary weight 151ff., 1064in Gauss-Hermite integration 147, 1062 and Gaussian quadrature 142, 1009, 1061 Gaussian weights from recurrence 150, 1064 Hermite 144, 1062 Jacobi 144, 1063 Laguerre 144, 1060Legendre 144, 1059 weight function logx153 Orthonormality 51, 142, 463Outer product Gaussian elimination 1017 Outer product of matrices (denoted by cross) 66, 420, 949, 969f., 989, 1000ff., 1017,1026, 1040, 1076, 1200, 1216, 1275 outerand() utility function 989, 1002, 1015 outerdiff() utility function 989, 1001outerdiv() utility function 989, 1001 outerprod() utility function 970, 989, 1000, 1017, 1026, 1040, 1076, 1200, 1216,1275outersum() utility function 989, 1001 Outgoing waveboundary conditions 820 Outlier 605, 653, 656, 694, 697 see alsoRobust estimation Overcorrection 857 Overﬂow 882, 1343 how to avoid in modulo multiplication 269 in complex arithmetic 171 Overlap-add and overlap-save methods 536f. Overloading operator 2/xiif. procedures 940, 1015, 1083, 1094, 1096 Overrelaxation parameter 857, 1332 choice of 858 Pack() intrinsic function 945, 950, 964, 991, 1031 communication bottleneck 969 for index table 1176 for partition-exchange 1170 for selection 1178 for selective evaluation 1087 Pack-unpack idiom 1087, 1134, 1153 Pad´e approximant 194ff., 1080f. Pad´e approximation 105 Parabolic interpolation 395, 1204 Parabolic partial differential equations 818, 838ff. Parallel axis theorem 308 Parallel programming 2/xv, 941, 958ff., 962ff., 965f., 968f., 987 array operations 964f. array ranking 1278f. band diagonal linear equations 1021 Bessel functions 1107ff. broadcasts 965ff. C and C++ 2/viii communication costs 969, 981, 1250counting do-loops 1015 cyclic reduction 974 deﬂation 977ff. design matrix 1082 dimensional expansion 965ff. eigensystems 1226, 1229f. fast Fourier transform (FFT) 981, 1235ff., 1250 in Fortran 90 963ff. Fortran 90 tricks 1009, 1274, 1278, 1280 function evaluation 986, 1009, 1084f., 1087, 1090, 1102, 1128, 1134 Gaussian quadrature 1009, 1061 geometric progressions 972 index loss 967f., 1038 index table 1176f. interprocessor communication 981 Kendall’s tau 1280 linear algebra 969f., 1000ff., 1018f., 1026, 1040, 1200, 1326 linear recurrence 973f., 1073ff. logo 2/viii, 1009masks 967f., 1006f., 1038, 1102, 1200, 1226, 1305, 1333f., 1368, 1378, 1382 merge statement 1010 Index to Volumes 1 and2 MIMD (multiple instruction, multiple data) 964, 985f., 1084 MMP (massively multiprocessor) machines 965ff., 974, 984, 1016ff., 1226ff., 1250 nrutil.f90 (module ﬁle) 1364ff. odd-even ordering 1333one-dimensional FFT 982f. parallel note icon 1009 partial differential equations 1333in-place selection 1178f. polynomial coefﬁcients from roots 980 polynomial evaluation 972f., 977, 998random numbers 1009, 1141ff. recursive doubling 973f., 976f., 979, 988, 999, 1071ff. scatter-with-combine 984, 1002f., 1032f. second order recurrence 974f., 1074 SIMD (Single Instruction Multiple Data) 964, 985f., 1009, 1084f. singular value decomposition (SVD) 1026 sorting 1167ff., 1171, 1176f.special functions 1009 SSP (small-scale parallel) machines 965ff., 984, 1010ff., 1016ff., 1059f., 1226ff.,1250 subvector scaling 972, 974, 996, 1000 successive over-relaxation (SOR) 1333 supercomputers 2/viii, 962SVD algorithm 1026 synthetic division 977ff., 999, 1048, 1071f., 1079, 1192 tridiagonal systems 975f., 1018, 1229f. utilities 1364ff. vector reduction 972f., 977, 998vs. serial programming 965, 987 PARAMETER attribute 1012 Parameters in ﬁtting function 651, 684ff.Parity bit 888 Park and Miller minimal standard random gen- erator 269, 1142 Parkinson’s Law 328 Parseval’s Theorem 492, 544 discrete form 498 Partial differential equations 818ff., 1332ff. advective equation 826 alternating-direction implicit method (ADI) 847, 861f. ampliﬁcation factor 828, 834 analyze/factorize/operate package 824artiﬁcial viscosity 831, 837 biconjugate gradient method 824 boundary conditions 819ff.boundary value problems 819, 848 Cauchy problem 818f. caution on high-order methods 844f.Cayley’s form 844 characteristics 818 Chebyshev acceleration 859f., 1332classiﬁcation of 818f. comparison of rapid methods 854 conjugate gradient method 824Courant condition 829, 832ff., 836 Courant condition (multidimensional) 846 Crank-Nicolson method 840, 842, 844, 846cyclic reduction (CR) method 848f., 852ff. diffusion equation 818, 838ff., 846, 855Dirichlet boundary conditions 508, 820, 840, 850, 856, 858 elliptic, deﬁned 818error, varieties of 831ff. explicit vs. implicit differencing 827 FACR method 854ﬁnite difference method 821ff.ﬁnite element methods 824ﬂux-conservative initial value problems 825ff. forward Euler differencing 826f. Forward Time Centered Space (FTCS) 827ff., 839ff., 843, 855 Fourier analysis and cyclic reduction (FACR) 848ff., 854 Gauss-Seidel method (relaxation) 855, 864ff., 876, 1338, 1341 Godunov’s method 837Helmholtz equation 852hyperbolic 818, 825f.implicit differencing 840incomplete Cholesky conjugate gradient method (ICCG) 824 inhomogeneous boundary conditions 850f.initial value problems 818f.initial value problems, recommendations on 838ff. Jacobi’s method (relaxation) 855ff., 864Laplace’s equation 818Lax method 828ff., 836, 845f.Lax method (multidimensional) 845f.matrix methods 824mesh-drift instability 834f.Monte Carlo methods 824multidimensional initial value problems 844ff. multigrid method 824, 862ff., 1009, 1334ff.Neumann boundary conditions 508, 820, 840, 851, 858 nonlinear diffusion equation 842nonlinear instability 831numerical dissipation or viscosity 830operator splitting 823, 847f., 861outgoing waveboundary conditions 820 parabolic 818, 838ff.parallel computing 1333periodic boundary conditions 850, 858piecewise parabolic method (PPM) 837Poisson equation 818, 852rapid (Fourier) methods 508ff., 824, 848ff.relaxation methods 823, 854ff., 1332f.Schr¨odinger equation 842ff. second-order accuracy 833ff., 840shock 831, 837sparse matrices from 64spectral methods 825 spectral radius 856ff., 862 stability vs. accuracy 830stability vs. efﬁciency 821staggered grids 513, 852staggered leapfrog method 833f.strongly implicit procedure 824 Indexto Volumes1 and 2 successive over-relaxation (SOR) 857ff., 862, 866, 1332f. time splitting 847f., 861two-step Lax-Wendroff method 835ff.upwind differencing 832f., 837variational methods 824varieties of error 831ff.von Neumann stability analysis 827f., 830, 833f., 840 waveequation 818, 825f. see alsoElliptic partial differential equa- tions; Finite difference equations (FDEs) Partial pivoting 29Partition-exchange 323, 333 and pack() intrinsic function 1170 Partitioned matrix, inverse of 70Party tricks 95ff., 168Parzen window 547Pascal, Numerical Recipes in 2/x, 2/xvii, 1Pass-the-buck idiom 1102, 1128Path integration, for function evaluation 201ff., 263, 1138 Pattern multiply of sparse matrices 74PBCG (preconditioned biconjugate gradient method) 78f., 824 PC methods seePredictor-corrector methods PCGPACK 71 PDEsseePartial differential equations Pearson’s r 630ff., 1276PECE method 741Pentagon, symmetries of 895Percentile 320Period of linear congruential generator 268Periodic boundary conditions 850, 858Periodogram 543ff., 566, 1258ff. Lomb’s normalized 569f., 574f., 1258ff.variance of 544f. Perl (programming language) 1/xviPerron’s theorems, for convergence of recur- rence relations 174f. Perturbation methods for matrix inversion 65ff. Phase error 831Phase-locked loop 700Phi statistic 625Phillips-Twomey method 799ff.Pi, computation of 906ff., 1352ff., 1357f.Piecewise parabolic method (PPM) 837Pincherle’s theorem 175Pivot element 29, 33, 757 in linear programming 428f. Pivoting 27, 29ff., 46, 66, 90, 1014 full 29, 1014implicit 30, 38, 1014, 1017in LU decomposition 37f., 1017partial 29, 33, 37f., 1017and QR decomposition 92in reduction to Hessenberg form 478in relaxation method 757as row and column operations 32for tridiagonal systems 43 Pixel 519, 596, 803, 811PL/1 2/xPlanck’s constant 842Plane rotation seeGivens reduction; Jacobi transformation (or rotation) Platykurtic distribution 606Plotting of functions 342, 1182f.POCS (projection onto convex sets) 805 Poetry 5f. Pointer (Fortran 90) 2/xiiif., 938f., 944f., 953ff., 1197, 1212, 1266 as alias 939, 944f., 1286, 1333 allocating an array 941allocating storage for derived type 955for array of arrays 956, 1336 array of, forbidden 956, 1337 associated with target 938f., 944f., 952f., 1197 in Fortran 95 961 to function, forbidden 1067, 1210 initialization to null 2/xv, 961returning array of unknown size 955f., 1184, 1259, 1261, 1327 undeﬁned status 952f., 961, 1070, 1266, 1302 Poisson equation 519, 818, 852Poisson probability function cumulative 214 deviates from 281, 283ff., 571, 1154semi-invariants of 608 tails compared to Gaussian 653 Poisson process 278, 282ff., 1153Polak-Ribiere algorithm 390, 414ff., 1214PolesseeComplex plane, poles in Polishing of roots 356, 363ff., 370f., 1193 poly() utility function 973, 977, 989, 998, 1072, 1096, 1192, 1258, 1284 Polymorphism 8 Polynomial interpolation 99, 102ff., 1043 Aitken’s algorithm 102in Bulirsch-Stoer method 724, 726, 1305coefﬁcients for 113ff., 1047f. Lagrange’s formula 84, 102f. multidimensional 116ff., 1049ff.Neville’s algorithm 102f., 105, 134, 182, 1043 pathology in determining coefﬁcients for 116 in predictor-corrector method 740smoothing ﬁlters 645 see alsoInterpolation Polynomials 167ff. algebraic manipulations 169, 1072approximate roots of Hermite polynomials 1062 approximate roots of Jacobi polynomials 1064 approximate roots of Laguerre polynomials 1061 approximating modiﬁed Bessel functions 230 approximation from Chebyshev coefﬁcients 191, 1078f. AUTODIN-II 890CCITT 889f. characteristic 368, 1193 characteristic, for digital ﬁlters 554, 559, 1257 Index to Volumes 1 and2 characteristic, for eigenvalues of matrix 449, 469 Chebyshev 184ff., 1076ff.coefﬁcients from roots 980 CRC-16 890 cumulants of 977, 999, 1071f., 1192, 1365, 1378f. deﬂation 362ff., 370f., 977 derivatives of 167, 978, 1071 division 84, 169, 362, 370, 977, 1072evaluation of 167, 972, 977, 998f., 1071, 1258, 1365, 1376ff. evaluation of derivatives 167, 978, 1071extrapolation in Bulirsch-Stoer method 724, 726, 1305f. extrapolation in Romberg integration 134ﬁtting 83, 114, 191, 645, 665, 674, 1078f., 1291 generator for CRC 889 ill-conditioned 362masked evaluation of 1378 matrix method for roots 368, 1193 minimax 186, 198, 1076monic 142f.multiplication 169 operation count for 168 orthonormal 142, 184, 1009, 1061parallel operations on 977ff., 998f., 1071f., 1192 primitive modulo 2 287ff., 301f., 889roots of 178ff., 362ff., 368, 1191ff. shifting of 192f., 978, 1079 stopping criterion in root ﬁnding 366 poly term() utility function 974, 977, 989, 999, 1071f., 1192 Port, serial data 892 Portability 3, 963Portable random number generator seeRan- dom number generator Positive deﬁnite matrix, testing for 90Positivity constraints 423Postal Service (U.S.), barcode 894 PostScript 1/xvi, 1/xxiii, 2/xx Powell’s method 389, 402, 406ff., 1210ff.Power (in a signal) 492f. Power series 159ff., 167, 195 economization of 192f., 1061, 1080Pad´e approximant of 194ff., 1080f. Power spectral density seeFourier transform; Spectral density Power spectrum estimation seeFourier trans- form; Spectral density PowerStation, Microsoft Fortran 2/xix PPM (piecewise parabolic method) 837Precision converting to double 1362 ﬂoating point 882, 937, 1343, 1361ff.multiple 906ff., 1352ff., 1362 Preconditioned biconjugate gradient method (PBCG) 78f. Preconditioning, in conjugate gradient methods 824 Predictor-corrector methods 702, 730, 740ff. Adams-Bashforth-Moulton schemes 741adaptive order methods 744compared to other methods 740 fallacy of multiple correction 741with ﬁxed number of iterations 741functional iteration vs. Newton’s rule 742multivalue compared with multistep 742ff. starting and stopping 742, 744 stepsize control 742f. present() intrinsic function 952Prime numbers 915Primitive polynomials modulo 2 287ff., 301f., 889 Principal directions 408f., 1210Principal solution, of inverse problem 797 PRIVATE attribute 957, 1067 Prize, $1000 offered 272, 1141, 1150f. Probability seeRandom number generator; Statistical tests Probability density, change of variables in 278f. Procedure seeProgram(s); Subprogram Process loss 548product() intrinsic function 948Product Nystrom method 789, 1331Program(s) as black boxes 1/xviii, 6, 26, 52, 205, 341, 406 dependencies 921ff., 1434ff. encapsulation 7interfaces 2, 8modularization 7f.organization 5ff. type declarations 2 typography of 2f., 12, 937validation 3f. Programming, serial vs. parallel 965, 987Projection onto convex sets (POCS) 805Projection operator, nonexpansive 805 Prolongation operator 864, 1337 Protocol, for communications 888PSD (power spectral density) seeFourier transform; Spectral density Pseudo-random numbers 266ff., 1141ff. PUBLIC attribute 957, 1067 Puns, particularly bad 167, 744, 747PURE attribute 2/xv, 960f., 964, 986put diag() utility function 985, 990, 1005, 1200 Pyramidal algorithm 586, 1264Pythagoreans 392 QLseeEigensystems QRseeEigensystems QR decomposition 91f., 382, 386, 1039f., 1199 backsubstitution 92, 1040 and least squares 668operation count 92pivoting 92updating 94, 382, 386, 1041, 1199use for orthonormal basis 58, 94 Quadratic convergence 49, 256, 351, 356, 409f., 419, 906 equations 20, 178, 391, 457 Indexto Volumes1 and 2 interpolation 353, 364 programming 436 Quadrature 123ff., 1052ff. adaptive 123, 190, 788 alternative extended Simpson’s rule 128 arbitrary weight function 151ff., 789, 1064, 1328 automatic 154 Bode’s rule 126change of variable in 137ff., 788, 1056ff.by Chebyshev ﬁtting 124, 189, 1078 classical formulas for 124ff. Clenshaw-Curtis 124, 190, 512f.closed formulas 125, 127f.and computer science 881 by cubic splines 124 error estimate in solution 784extended midpoint rule 129f., 135, 1054f.extended rules 127ff., 134f., 786, 788ff., 1326, 1328 extended Simpson’s rule 128 Fourier integrals 577ff., 1261ff. Fourier integrals, inﬁnite range 583Gauss-Chebyshev 144, 512f.Gauss-Hermite 144, 789, 1062Gauss-Jacobi 144, 1063 Gauss-Kronrod 154 Gauss-Laguerre 144, 789, 1060Gauss-Legendre 144, 783, 789, 1059, 1325 Gauss-Lobatto 154, 190, 512Gauss-Radau 154 Gaussian integration 127, 140ff., 781, 783, 788f., 1009, 1059ff., 1325, 1328f. Gaussian integration, nonclassical weight function 151ff., 788f., 1064f., 1328f. for improper integrals 135ff., 789, 1055, 1328 for integral equations 781f., 786, 1325ff. Monte Carlo 124, 155ff., 295ff., 306ff., 1161ff. multidimensional 124, 155ff., 1052, 1065ff. multidimensional, by recursion 1052, 1065 Newton-Cotes formulas 125ff., 140open formulas 125ff., 129f., 135related to differential equations 123 related to predictor-corrector methods 740 Romberg integration 124, 134f., 137, 182, 717, 788, 1054f., 1065, 1067 semi-open formulas 130Simpson’s rule 126, 133, 136f., 583, 782, 788ff., 1053 Simpson’s three-eighths rule 126, 789f.singularity removal 137ff., 788, 1057ff., 1328ff. singularity removal, worked example 792, 1328ff. trapezoidal rule 125, 127, 130ff., 134f., 579, 583, 782, 786, 1052ff., 1326f. using FFTs 124weight function logx153 see alsoIntegration of functions Quadrature mirror ﬁlter 585, 593Quantum mechanics, Uncertainty Principle 600 Quartile value 320 Quasi-Newton methods for minimization 390, 418ff., 1215 Quasi-random sequence 299ff., 318, 881, 888 Halton’s 300for Monte Carlo integration 304, 309, 318Sobol’s 300ff., 1160 see alsoRandom number generator Quicksort 320, 323ff., 330, 333, 1169f.Quotient-difference algorithm 164 R-estimates 694 Radioactive decay 278Radix base for ﬂoating point arithmetic 476, 882, 907, 913, 1231, 1343, 1357 Radix conversion 902, 906, 913, 1357radix() intrinsic function 1231Radix sort 1172 Ramanujan’s identity for π915 Random bits, generation of 287ff., 1159f.Random deviates 266ff., 1141ff. binomial 285f., 1155 exponential 278, 1151f. gamma distribution 282f., 1153Gaussian 267, 279f., 571, 798, 1152f.normal 267, 279f., 571, 1152f. Poisson 283ff., 571, 1154f. quasi-random sequences 299ff., 881, 888, 1160f. uniform 267ff., 1158f., 1166uniform integer 270, 274ff. Random number generator 266ff., 1141ff. bitwise operations 287 Box-Muller algorithm 279, 1152 Data Encryption Standard 290ff., 1144, 1156ff. good choices for modulus, multiplier and increment 274ff. initializing 1144ff.for integer-valued probability distribution 283f., 1154 integer vs. real implementation 273L’Ecuyer’s long period 271f. lagged Fibonacci generator 1142, 1148ff. linear congruential generator 267ff., 1142machine language 269Marsaglia shift register 1142, 1148ff.Minimal Standard, Park and Miller’s 269, 1142 nonrandomness of low-order bits 268f. parallel 1009 perfect 272, 1141, 1150f.planes, numbers lie on 268portable 269ff., 1142 primitive polynomials modulo 2 287ff. pseudo-DES 291, 1144, 1156ff.quasi-random sequences 299ff., 881, 888, 1160f. quick and dirty 274quicker and dirtier 275 in Quicksort 324 random access to nth number 293 Index to Volumes 1 and2 random bits 287ff., 1159f. recommendations 276f. rejection method 281ff.serial 1141f. shufﬂing procedure 270, 272 in simulated annealing method 438spectral test 274 state space 1143f. state space exhaustion 1141subtractive method 273, 1143 system-supplied 267f. timings 276f., 1151transformation method 277ff. trick for trigonometric functions 280 Random numbers seeMonte Carlo; Random deviates Random walk 20 random number() intrinsic function 1141, 1143 randomseed() intrinsic function 1141 RANDU, infamous routine 268Range 53f. Rank (matrix) 53 kernel of ﬁnite 785 Rank (sorting) 320, 332, 1176 Rank (statistics) 633ff., 694f., 1277 Kendall’s tau 637ff., 1279Spearman correlation coefﬁcient 634f., 1277ff. sum squared differences of 634, 1277 Ratio variable (statistics) 623 Rational Chebyshev approximation 197ff., 1081f. Rational function 99, 167ff., 194ff., 1080f. approximation for Bessel functions 225 approximation for continued fraction 164, 211, 219f. Chebyshev approximation 197ff., 1081f. evaluation of 170, 1072f.extrapolation in Bulirsch-Stoer method 718ff., 726, 1306f. interpolation and extrapolation using 99, 104ff., 194ff., 718ff., 726 as power spectrum estimate 566 interpolation and extrapolation using 1043f., 1080ff., 1306 minimax 198 Re-entrant procedure 1052 real() intrinsic function, ambiguity of 947Realizable (causal) 552, 554f. reallocate() utility function 955, 990, 992, 1070, 1302 Rearranging seeSorting Reciprocal, multiple precision 910f., 1355f. Record, in data ﬁle 329Recurrence relation 172ff., 971ff. arithmetic progression 971f., 996 associated Legendre polynomials 247Bessel function 172, 224, 227f., 234 binomial coefﬁcients 209 Bulirsch-Stoer 105f.characteristic polynomial of tridiagonal matrix 469 Clenshaw’s recurrence formula 176f.and continued fraction 175continued fraction evaluation 164f. convergence 175 cosine function 172, 500 cyclic reduction 974 dominant solution 174exponential integrals 172 gamma function 206 generation of random bits 287f. geometric progression 972, 996 Golden Mean 21Legendre polynomials 172 minimal vs. dominant solution 174 modiﬁed Bessel function 232 Neville’s 103, 182 orthonormal polynomials 142Perron’s theorems 174f. Pincherle’s theorem 175 for polynomial cumulants 977, 999, 1071f. polynomial interpolation 103, 183 primitive polynomials modulo 2 287f.random number generator 268 rational function interpolation 105f., 1043 recursive doubling 973, 977, 988, 999, 1071f., 1073 second order 974f., 1074 sequence of trig functions 173 sine function 172, 500 spherical harmonics 247stability of 21, 173ff., 177, 224f., 227f., 232, 247, 975 trig functions 572 weight of Gaussian quadrature 144f. Recursion in Fortran 90 958 in multigrid method 865, 1009, 1336 Recursive doubling 973f., 979 cumulants of polynomial 977, 999, 1071f. linear recurrences 973, 988, 1073tridiagonal systems 976 RECURSIVE keyword 958, 1065, 1067 Recursive Monte Carlo integration 306ff., 1161 Recursive procedure 2/xiv, 958, 1065, 1067, 1166 as parallelization tool 958 base case 958for multigrid method 1009, 1336 re-entrant 1052 Recursive stratiﬁed sampling 314ff., 1164ff.Red-black seeOdd-even ordering Reduction functions 948ff. Reduction of variance in Monte Carlo integra- tion 299, 306ff. References (explanation) 4f.References (general bibliography) 916ff., 1359f. Reﬂection formula for gamma function 206Regula falsi (false position) 347ff., 1185f. Regularity condition 775 Regularization compared with optimal ﬁltering 801 constrained linear inversion method 799ff.of inverse problems 796ff. linear 799ff. nonlinear 813 Indexto Volumes1 and 2 objective criterion 802 Phillips-Twomey method 799ff. Tikhonov-Miller 799ff.trade-off curve 799 two-dimensional 803 zeroth order 797see alsoInverse problems Regularizing operator 798 Reid, John 2/xiv, 2/xviRejection method for random number genera- tor 281ff. Relaxation method for algebraically difﬁcult sets 763automated allocation of mesh points 774f., 777 computation of spheroidal harmonics 764ff., 1319ff. for differential equations 746f., 753ff., 1316ff. elliptic partial differential equations 823, 854ff., 1332f. example 764ff., 1319ff. Gauss-Seidel method 855, 864ff., 876, 1338, 1341 internal boundary conditions 775ff. internal singular points 775ff. Jacobi’s method 855f., 864successive over-relaxation (SOR) 857ff., 862, 866, 1332f. see alsoMultigrid method Remes algorithms exchange algorithm 553 for minimax rational function 199 reshape() intrinsic function 950 communication bottleneck 969 order keyword 1050, 1246 Residual 49, 54, 78 in multigrid method 863, 1338 Resolution function, in Backus-Gilbert method 807 Response function 531Restriction operator 864, 1337 RESULT keyword 958, 1073 Reward, $1000 offered 272, 1141, 1150f. Richardson’s deferred approach to the limit 134, 137, 182, 702, 718ff., 726, 788, 869 see alsoBulirsch-Stoer method Richtmyer artiﬁcial viscosity 837 Ridders’ method, for numerical derivatives 182, 1075 Ridders’ method, root ﬁnding 341, 349, 351, 1187 Riemann shock problem 837Right eigenvalues and eigenvectors 451 Rise/fall time 548f. Robust estimation 653, 694ff., 700, 1294 Andrew’s sine 697average deviation 605 double exponential errors 696 Kalman ﬁltering 700Lorentzian errors 696f. mean absolute deviation 605 nonparametric correlation 633ff., 1277Tukey’s biweight 697use of a priori covariances 700 see alsoStatistical tests Romberg integration 124, 134f., 137, 182, 717, 788, 1054f., 1065 Root ﬁnding 143, 340ff., 1009, 1059 advanced implementations of Newton’s rule 386 Bairstow’s method 364, 370, 1193 bisection 343, 346f., 352f., 359, 390, 469, 698, 1184f. bracketing of roots 341, 343ff., 353f., 362, 364, 369, 1183f. Brent’s method 341, 349, 660f., 1188f., 1286 Broyden’s method 373, 382f., 386, 1199 compared with multidimensional minimiza- tion 375 complex analytic functions 364 in complex plane 204 convergence criteria 347, 374deﬂation of polynomials 362ff., 370f., 1192 without derivatives 354double root 341eigenvalue methods 368, 1193 false position 347ff., 1185f. Jenkins-Traub method 369Laguerre’s method 341, 366f., 1191f. Lehmer-Schur algorithm 369 Maehly’s procedure 364, 371matrix method 368, 1193 Muller’s method 364, 372 multiple roots 341Newton’s rule 143f., 180, 341, 355ff., 362, 364, 370, 372ff., 376, 469, 740, 749f., 754, 787, 874, 876, 911f., 1059, 1189, 1194, 1196, 1314ff., 1339, 1341,1355f. pathological cases 343, 356, 362, 372 polynomials 341, 362ff., 449, 1191f.in relaxation method 754, 1316Ridders’ method 341, 349, 351, 1187 root-polishing 356, 363ff., 369ff., 1193 safe Newton’s rule 359, 1190secant method 347ff., 358, 364, 399, 1186f. in shooting method 746, 749f., 1314f.singular Jacobian in Newton’s rule 386stopping criterion for polynomials 366 use of minimum ﬁnding 341 using derivatives 355ff., 1189zero suppression 372 see alsoRoots Root polishing 356, 363ff., 369ff., 1193Roots Chebyshev polynomials 184 complex nth root of unity 999f., 1379 cubic equations 179f.Hermite polynomials, approximate 1062 Jacobi polynomials, approximate 1064 Laguerre polynomials, approximate 1061multiple 341, 364ff., 1192 nonlinear equations 340ff. polynomials 341, 362ff., 449, 1191f.quadratic equations 178 Index to Volumes 1 and2 reﬂection in unit circle 560, 1257 square, multiple precision 912, 1356 see alsoRoot ﬁnding Rosenbrock method 730, 1308 compared with semi-implicit extrapolation 739 stepsize control 731, 1308f. Roundoff error 20, 881, 1362 bracketing a minimum 399 compile time vs. run time 1012conjugate gradient method 824 eigensystems 458, 467, 470, 473, 476, 479, 483 extended trapezoidal rule 132general linear least squares 668, 672 graceful 883, 1343 hardware aspects 882, 1343Householder reduction 466IEEE standard 882f., 1343 interpolation 100 least squares ﬁtting 658, 668Levenberg-Marquardt method 679 linear algebraic equations 23, 27, 29, 47, 56, 84, 1022 linear predictive coding (LPC) 564magniﬁcation of 20, 47, 1022 maximum entropy method (MEM) 567 measuring 881f., 1343multidimensional minimization 418, 422 multiple roots 362 numerical derivatives 180f.recurrence relations 173reduction to Hessenberg form 479 series 164f. straight line ﬁtting 658variance 607 Row degeneracy 22 Row-indexed sparse storage 71f., 1030 transpose 73f. Row operations on matrix 28, 31f. Row totals 624 RSS algorithm 314ff., 1164RST properties (reﬂexive, symmetric, transi- tive) 338 Runge-Kutta method 702, 704ff., 731, 740, 1297ff., 1308 Cash-Karp parameters 710, 1299f. embedded 709f., 731, 1298, 1308 high-order 705quality control 722stepsize control 708ff. Run-length encoding 901 Runge-Kutta method high-order 1297 stepsize control 1298f. Rybicki, G.B. 84ff., 114, 145, 252, 522, 574, 600 S-box for Data Encryption Standard 1148 Sampling importance 306f.Latin square or hypercube 305f. recursive stratiﬁed 314ff., 1164 stratiﬁed 308f.uneven or irregular 569, 648f., 1258Sampling theorem 495, 543 for numerical approximation 600ff. Sande-Tukey FFT algorithm 503 SAVE attribute 953f., 958f., 961, 1052, 1070, 1266, 1293 redundant use of 958f. SAVE statements 3Savitzky-Golay ﬁlters for data smoothing 644ff., 1283f.for numerical derivatives 183, 645 scale() intrinsic function 1107Scallop loss 548Scatter-with-combine functions 984, 1002f., 1032, 1366, 1380f. scatter add() utility function 984, 990, 1002, 1032 scattermax() utility function 984, 990, 1003 Schonfelder, Lawrie 2/xiSchrage’s algorithm 269Schr¨odinger equation 842ff. Schultz’s method for matrix inverse 49, 598Scope 956ff., 1209, 1293, 1296Scoping unit 939SDLC checksum 890Searching with correlated values 111, 1046f.an ordered table 110f., 1045f.selection 333, 1177f. Secant method 341, 347ff., 358, 364, 399, 1186f. Broyden’s method 382f., 1199f.multidimensional (Broyden’s) 373, 382f., 1199 Second Euler-Maclaurin summation formula 135f. Second order differential equations 726, 1307Seed of random number generator 267, 1146f.select case statement 2/xiv, 1010, 1036Selection 320, 333, 1177f. ﬁndmlargest elements 336, 1179f. heap algorithm 336, 1179for median 698, 1294operation count 333by packing 1178parallel algorithms 1178by partition-exchange 333, 1177f.without rearrangement 335, 1178f.timings 336use to ﬁnd median 609 Semi-implicit Euler method 730, 735f. Semi-implicit extrapolation method 730, 735f., 1310f. compared with Rosenbrock method 739stepsize control 737, 1311f. Semi-implicit midpoint rule 735f., 1310f.Semi-invariants of a distribution 608Sentinel, in Quicksort 324, 333Separable kernel 785Separation of variables 246Serial computing convergence of quadrature 1060random numbers 1141sorting 1167 Serial data port 892 Indexto Volumes1 and 2 Series 159ff. accelerating convergence of 159ff. alternating 160f., 1070 asymptotic 161 Bessel function Kν241 Bessel function Yν235 Bessel functions 160, 223 cosine integral 250divergent 161 economization 192f., 195, 1080 Euler’s transformation 160f., 1070exponential integral 216, 218 Fresnel integral 248 hypergeometric 202, 263, 1138 incomplete beta function 219 incomplete gamma function 210, 1090f.Laurent 566 relation to continued fractions 163f. roundoff error in 164f.sine and cosine integrals 250 sine function 160 Taylor 355f., 408, 702, 709, 754, 759 transformation of 160ff., 1070 van Wijngaarden’s algorithm 161, 1070 Shaft encoder 886 Shakespeare 9 Shampine’s Rosenbrock parameters 732, 1308shape() intrinsic function 938, 949 Shell algorithm (Shell’s sort) 321ff., 1168 Sherman-Morrison formula 65ff., 83, 382Shifting of eigenvalues 449, 470f., 480 Shockwave831, 837 Shooting method computation of spheroidal harmonics 772, 1321ff. for differential equations 746, 749ff., 770ff., 1314ff., 1321ff. for difﬁcult cases 753, 1315f. example 770ff., 1321ff. interior ﬁtting point 752, 1315f., 1323ff. Shufﬂing to improve random number generator 270, 272 Side effects prevented by data hiding 957, 1209, 1293, 1296 and PURE subprograms 960 Sidelobe fall-off 548Sidelobe level 548 sign() intrinsic function, modiﬁed in Fortran 95 961 Signal, bandwidth limited 495 Signiﬁcance (numerical) 19 Signiﬁcance (statistical) 609f. one- vs. two-sided 632 peak in Lomb periodogram 570 of 2-d K-S test 640, 1281 two-tailed 613 SIMD machines (Single Instruction Multiple Data) 964, 985f., 1009, 1084f. Similarity transform 452ff., 456, 476, 478, 482 Simplex deﬁned 402 method in linear programming 389, 402, 423ff., 431ff., 1216ff.method of Nelder and Mead 389, 402ff., 444, 697f., 1208f., 1222ff. use in simulated annealing 444, 1222ff. Simpson’s rule 124ff., 128, 133, 136f., 583, 782, 788f., 1053f. Simpson’s three-eighths rule 126, 789f.Simulated annealing seeAnnealing, method of simulated Simulation seeMonte Carlo Sine function evaluated from tan(θ/2)173 recurrence 172series 160 Sine integral 248, 250ff., 1123, 1125f. continued fraction 250series 250 see alsoCosine integral Sine transform seeFast Fourier transform (FFT); Fourier transform Singleton’s algorithm for FFT 525 Singular value decomposition (SVD) 23, 25, 51ff., 1022 approximation of matrices 58f. backsubstitution 56, 1022f. and bases for nullspace and range 53 conﬁdence levels from 693f. covariance matrix 693f.fewer equations than unknowns 57 for inverse problems 797 and least squares 54ff., 199f., 668, 670ff., 1081, 1290f. in minimization 410 more equations than unknowns 57f. parallel algorithms 1026and rational Chebyshev approximation 199f., 1081f. of square matrix 53ff., 1023use for ill-conditioned matrices 56, 58, 449 use for orthonormal basis 58, 94 Singularities of hypergeometric function 203, 263 in integral equations 788ff., 1328in integral equations, worked example 792, 1328ff. in integrands 135ff., 788, 1055, 1328ff.removal in numerical integration 137ff., 788, 1057ff., 1328ff. Singularity, subtraction of the 789SIPSOL 824 Six-step framework, for FFT 983, 1240 size() intrinsic function 938, 942, 945, 948Skew array section 2/xii, 945, 960, 985, 1284 Skewness of distribution 606, 608, 1269 Smoothing of data 114, 644ff., 1283f. of data in integral equations 781 importance in multigrid method 865 sn function 261, 1137f. Snyder, N.L. 1/xvi Sobol’s quasi-random sequence 300ff., 1160f.Sonata 9 Sonnet 9 Sorting 320ff., 1167ff. bubble sort 1168 Index to Volumes 1 and2 bubble sort cautioned against 321 compared to selection 333 covariance matrix 669, 681, 1289eigenvectors 461f., 1227 Heapsort 320, 327f., 336, 1171f., 1179 index table 320, 329f., 1170, 1173ff., 1176 operation count 320ff. by packing 1171parallel algorithms 1168, 1171f., 1176 Quicksort 320, 323ff., 330, 333, 1169f. radix sort 1172rank table 320, 332, 1176 ranking 329, 1176 by reshaping array slices 1168Shell’s method 321ff., 1168 straight insertion 321f., 461f., 1167, 1227 SP, deﬁned 937SPARC or SPARCstation 1/xxii, 2/xix, 4 Sparse linear equations 23, 63ff., 732, 1030 band diagonal 43, 1019ff.biconjugate gradient method 77, 599, 1034 data type for 1030indexed storage 71f., 1030 in inverse problems 804 minimum residual method 78named patterns 64, 822 partial differential equations 822ff. relaxation method for boundary value prob- lems 754, 1316 row-indexed storage 71f., 1030 wavelet transform 584, 598see alsoMatrix Spearman rank-order coefﬁcient 634f., 694f., 1277 Special functions seeFunction Spectral analysis seeFourier transform; Peri- odogram Spectral density 541 and data windowing 545ff. ﬁgures of merit for data windows 548f.normalization conventions 542f. one-sided PSD 492 periodogram 543ff., 566, 1258ff.power spectral density (PSD) 492f. power spectral density per unit time 493 power spectrum estimation by FFT 542ff., 1254ff. power spectrum estimation by MEM 565ff., 1258 two-sided PSD 493variance reduction in spectral estimation 545 Spectral lines, how to smooth 644Spectral methods for partial differential equa- tions 825 Spectral radius 856ff., 862Spectral test for random number generator 274 Spectrum seeFourier transform Spherical Bessel functions 234 routine for 245, 1121 Spherical harmonics 246ff. orthogonality 246routine for 247f., 1122 stable recurrence for 247 table of 246see alsoAssociated Legendre polynomials Spheroidal harmonics 764ff., 770ff., 1319ff. boundary conditions 765 normalization 765 routine for 768ff., 1319ff., 1323ff. Spline 100 cubic 107ff., 1044f. gives tridiagonal system 109natural 109, 1044f. operation count 109 two-dimensional (bicubic) 120f., 1050f. spread() intrinsic function 945, 950, 969, 1000, 1094, 1290f. and dimensional expansion 966ff. Spread matrix 808 Spread spectrum 290 Square root, complex 172Square root, multiple precision 912, 1356f. Square window 546, 1254ff. SSP (small-scale parallel) machines 965ff., 972, 974, 984, 1011, 1016ff., 1021, 1059f., 1226ff., 1250 Stability 20f. of Clenshaw’s recurrence 177 Courant condition 829, 832ff., 836, 846 diffusion equation 840of Gauss-Jordan elimination 27, 29 of implicit differencing 729, 840 mesh-drift in PDEs 834f.nonlinear 831, 837 partial differential equations 820, 827f. of polynomial deﬂation 363in quadrature solution of Volterra equation 787f. of recurrence relations 173ff., 177, 224f., 227f., 232, 247 and stiff differential equations 728f. von Neumann analysis for PDEs 827f., 830, 833f., 840 see alsoAccuracy Stabilized Kolmogorov-Smirnov test 621Stabilizing functional 798 Staggered leapfrog method 833f. Standard (probable) errors 1288, 1290Standard deviation of a distribution 605, 1269 of Fisher’s z 632 of linear correlation coefﬁcient 630 of sum squared difference of ranks 635, 1277 Standard (probable) errors 610, 656, 661, 667, 671, 684 Stars, as text separator 1009 Statement function, superseded by internal sub- program 1057, 1256 Statement labels 9Statistical error 653 Statistical tests 603ff., 1269ff. Anderson-Darling 621 average deviation 605, 1269 bootstrap method 686f.chi-square 614f., 623ff., 1272, 1275f. Indexto Volumes1 and 2 contingency coefﬁcient C 625, 1275 contingency tables 622ff., 638, 1275f. correlation 603f. Cramer’s V 625, 1275 difference of distributions 614ff., 1272difference of means 609ff., 1269f. difference of variances 611, 613, 1271 entropy measures of association 626ff., 1275f. F-test 611, 613, 1271Fisher’s z-transformation 631f., 1276 general paradigm 603 Kendall’s tau 634, 637ff., 1279 Kolmogorov-Smirnov 614, 617ff., 640, 694, 1273f., 1281 Kuiper’s statistic 621 kurtosis 606, 608, 1269 L-estimates 694 linear correlation coefﬁcient 630ff., 1276 M-estimates 694ff.mean 603ff., 608ff., 1269f. measures of association 604, 622ff., 1275 measures of central tendency 604ff., 1269 median 605, 694 mode 605 moments 604ff., 608, 1269 nonparametric correlation 633ff., 1277Pearson’s r 630ff., 1276 for periodic signal 570 phi statistic 625 R-estimates 694 rank correlation 633ff., 1277robust 605, 634, 694ff. semi-invariants 608 for shift vs. for spread 620f. signiﬁcance 609f., 1269ff. signiﬁcance, one- vs. two-sided 613, 632skewness 606, 608, 1269 Spearman rank-order coefﬁcient 634f., 694f., 1277 standard deviation 605, 1269 strength vs. signiﬁcance 609f., 622Student’s t 610, 631, 1269 Student’s t, for correlation 631 Student’s t, paired samples 612, 1271 Student’s t, Spearman rank-order coefﬁcient 634, 1277 Student’s t, unequal variances 611, 1270 sum squared difference of ranks 635, 1277 Tukey’s trimean 694 two-dimensional 640, 1281ff. variance 603ff., 607f., 612f., 1269ff. Wilcoxon 694 see alsoError; Robust estimation Steak, without sizzle 809 Steed’s method Bessel functions 234, 239 continued fractions 164f. Steepest descent method 414 in inverse problems 804 Step doubling 130, 708f., 1052 tripling 136, 1055 Stieltjes, procedure of 151Stiff equations 703, 727ff., 1308ff. Kaps-Rentrop method 730, 1308 methods compared 739 predictor-corrector method 730 r.h.s. independent of x729f. Rosenbrock method 730, 1308 scaling of variables 730 semi-implicit extrapolation method 730, 1310f. semi-implicit midpoint rule 735f., 1310f. Stiff functions 100, 399Stirling’s approximation 206, 812 Stoermer’s rule 726, 1307 Stopping criterion, in multigrid method 875f.Stopping criterion, in polynomial root ﬁnding 366 Storage band diagonal matrix 44, 1019 sparse matrices 71f., 1030 Storage association 2/xiv Straight injection 867 Straight insertion 321f., 461f., 1167, 1227Straight line ﬁtting 655ff., 667f., 1285ff. errors in both coordinates 660ff., 1286ff. robust estimation 698, 1294ff. Strassen’s fast matrix algorithms 96f. Stratiﬁed sampling, Monte Carlo 308f., 314 Stride (of an array) 944 communication bottleneck 969 Strongly implicit procedure (SIPSOL) 824Structure constructor 2/xii Structured programming 5ff. Student’s probability distribution 221f.Student’s t-test for correlation 631 for difference of means 610, 1269for difference of means (paired samples) 612, 1271 for difference of means (unequal variances) 611, 1270 for difference of ranks 635, 1277 Spearman rank-order coefﬁcient 634, 1277 Sturmian sequence 469 Sub-random sequences seeQuasi-random se- quence Subprogram 938 for data hiding 957, 1209, 1293, 1296internal 954, 957, 1057, 1067, 1226, 1256 in module 940 undeﬁned variables on exit 952f., 961, 1070, 1266, 1293, 1302 Subscript triplet (for array) 944 Subtraction, multiple precision 907, 1353Subtractive method for random number genera- tor 273, 1143 Subvector scaling 972, 974, 996, 1000Successive over-relaxation (SOR) 857ff., 862, 1332f. bad in multigrid method 866Chebyshev acceleration 859f., 1332f. choice of overrelaxation parameter 858 with logical mask 1333f. parallelization 1333 sum() intrinsic function 945, 948, 966Sum squared difference of ranks 634, 1277 Index to Volumes 1 and2 SumsseeSeries Sun 1/xxii, 2/xix, 886 SPARCstation 1/xxii, 2/xix, 4 Supernova 1987A 640 SVDseeSingular value decomposition (SVD) swap() utility function 987, 990f., 1015, 1210Symbol, of operator 866f.Synthetic division 84, 167, 362, 370 parallel algorithms 977ff., 999, 1048, 1071f., 1079, 1192 repeated 978f. Systematic errors 653 Tableau (interpolation) 103, 183 Tangent function, continued fraction 163 Target, for pointer 938f., 945, 952f. Taylor series 180, 355f., 408, 702, 709, 742, 754, 759 Test programs 3Thermodynamics, analogy for simulated an- nealing 437 Thinking Machines, Inc. 964Threshold multiply of sparse matrices 74, 1031 Tides 560f.Tikhonov-Miller regularization 799ff. Time domain 490 Time splitting 847f., 861tiny() intrinsic function 952Toeplitz matrix 82, 85ff., 195, 1038 LU decomposition 87 new, fast algorithms 88f.nonsymmetric 86ff., 1038 Tongue twisters 333Torus 297f., 304 Trade-off curve 795, 809 Trademarks 1/xxii, 2/xixf.Transformation Gauss 256 Landen 256 method for random number generator 277ff. Transformational functions 948ff.Transforms, number theoretic 503f. Transport error 831ff. transpose() intrinsic function 950, 960, 969, 981, 1050, 1246 Transpose of sparse matrix 73f.Trapezoidal rule 125, 127, 130ff., 134f., 579, 583, 782, 786, 1052, 1326f. Traveling salesman problem 438ff., 1219ff.Tridiagonal matrix 42, 63, 150, 453f., 488, 839f., 1018f. in alternating-direction implicit method (ADI) 861f. from cubic spline 109cyclic 67, 1030 in cyclic reduction 853 eigenvalues 469ff., 1228with fringes 822from operator splitting 861f. parallel algorithm 975, 1018, 1229f. recursive splitting 1229f.reduction of symmetric matrix to 462ff., 470, 1227f.serial algorithm 1018f. see alsoMatrix Trigonometric functions, linear sequences 173 functions, recurrence relation 172, 572 functions, tan(θ/2)as minimal 173 interpolation 99solution of cubic equation 179f. Truncation error 20f., 399, 709, 881, 1362 in multigrid method 875in numerical derivatives 180 Tukey’s biweight 697 Tukey’s trimean 694 Turbo Pascal (Borland) 8Twin errors 895Two-dimensional seeMultidimensional Two-dimensional K–S test 640, 1281ff. Two-pass algorithm for variance 607, 1269Two-point boundary value problems 702, 745ff., 1314ff. automated allocation of mesh points 774f., 777 boundary conditions 745ff., 749, 751f., 771, 1314ff. difﬁcult cases 753, 1315f.eigenvalue problem for differential equa- tions 748, 764ff., 770ff., 1319ff. free boundary problem 748, 776 grid (mesh) points 746f., 754, 774f., 777 internal boundary conditions 775ff.internal singular points 775ff.linear requires no iteration 751 multiple shooting 753 problems reducible to standard form 748regularity condition 775relaxation method 746f., 753ff., 1316ff. relaxation method, example of 764ff., 1319 shooting to a ﬁtting point 751ff., 1315f., 1323ff. shooting method 746, 749ff., 770ff., 1314ff., 1321ff. shooting method, example of 770ff., 1321ff. singular endpoints 751, 764, 771, 1315f., 1319ff. see alsoElliptic partial differential equa- tions Two-sided exponential error distribution 696 Two-sided power spectral density 493Two-step Lax-Wendroff method 835ff.Two-volume edition, plan of 1/xiii Two’s complement arithmetic 1144 Type declarations, explicit vs. implicit 2 Ubound() intrinsic function 949 ULTRIX 1/xxiii, 2/xixUncertainty coefﬁcient 628Uncertainty principle 600 Undeﬁned status, of arrays and pointers 952f., 961, 1070, 1266, 1293, 1302 Underﬂow, in IEEE arithmetic 883, 1343 Underrelaxation 857Uniform deviates seeRandom deviates, uni- form Indexto Volumes1 and 2 Unitary (function) 843f. Unitary (matrix) seeMatrix unitmatrix() utility function 985, 990, 1006, 1216, 1226, 1325 UNIX 1/xxiii, 2/viii, 2/xix, 4, 17, 276, 293, 886 Upper Hessenberg matrix seeHessenberg ma- trix U.S. Postal Service barcode 894unpack() intrinsic function 950, 964 communication bottleneck 969 Upper subscript 944 upper triangle() utility function 990, 1006, 1226, 1305 Upwind differencing 832f., 837USE statement 936, 939f., 954, 957, 1067, 1384 USES keyword in program listings 2Utility functions 987ff., 1364ff. add vector to matrix diagonal 1004, 1234, 1366, 1381 alphabetical listing 988ff. argument checking 994f., 1370f.arithmetic progression 996, 1072, 1127, 1365, 1371f. array reallocation 992, 1070f., 1365, 1368f. assertion of numerical equality 995, 1022, 1365, 1370f. compared to intrinsics 990ff.complex nth root of unity 999f., 1379 copying arrays 991, 1034, 1327f., 1365f. create unit matrix 1006, 1382 cumulative product of an array 997f., 1072, 1086, 1375 cumulative sum of an array 997, 1280f., 1365, 1375 data types 1361 elemental functions 1364error handling 994f., 1036, 1370f.generic functions 1364geometric progression 996f., 1365, 1372ff.get diagonal of matrix 1005, 1226f., 1366, 1381f. length of a vector 1008, 1383linear recurrence 996location in an array 992ff., 1015, 1017ff.location of ﬁrst logical “true” 993, 1041, 1369 location of maximum array value 993, 1015, 1017, 1365, 1369 location of minimum array value 993, 1369f. logical assertion 994, 1086, 1090, 1092, 1365, 1370 lower triangular mask 1007, 1200, 1382masked polynomial evaluation 1378 masked swap of elements in two arrays 1368 moving data 990ff., 1015multiply vector into matrix diagonal 1004f., 1366, 1381 nrutil.f90 (module ﬁle) 1364ff. outer difference of vectors 1001, 1366, 1380 outer logical and of vectors 1002outer operations on vectors 1000ff., 1379f. outer product of vectors 1000f., 1076, 1365f., 1379 outer quotient of vectors 1001, 1379 outer sum of vectors 1001, 1379f. overloading 1364partial cumulants of a polynomial 999, 1071, 1192f., 1365, 1378f. polynomial evaluation 996, 998f., 1258, 1365, 1376ff. scatter-with-add 1002f., 1032f., 1366, 1380f. scatter-with-combine 1002f., 1032f., 1380f.scatter-with-max 1003f., 1366, 1381set diagonal elements of matrix 1005, 1200, 1366, 1382 skew operation on matrices 1004ff., 1381ff.swap elements of two arrays 991, 1015, 1365ff. upper triangular mask 1006, 1226, 1305, 1382 V-cycle 865, 1336 vabs() utility function 990, 1008, 1290 Validation of Numerical Recipes procedures 3f. Valley, long or narrow 403, 407, 410 Van Cittert’s method 804Van Wijngaarden-Dekker-Brent method see Brent’s method Vandermonde matrix 82ff., 114, 1037, 1047 Variable length code 896, 1346ff.Variable metric method 390, 418ff., 1215 compared to conjugate gradient method 418 Variable step-size integration 123, 135, 703, 707ff., 720, 726, 731, 737, 742ff., 1298ff., 1303, 1308f., 1311ff. Variance(s) correlation 605 of distribution 603ff., 608, 611, 613, 1269 pooled 610reduction of (in Monte Carlo) 299, 306ff.statistical differences between two 609, 1271 two-pass algorithm for computing 607, 1269 see alsoCovariance Variational methods, partial differential equa- tions 824 VAX 275, 293 Vector(s) length 1008, 1383norms 1036outer difference 1001, 1366, 1380 outer operations 1000ff., 1379f. outer product 1000f., 1076, 1365f., 1379 Vector reduction 972, 977, 998 Vector subscripts 2/xiif., 984, 1002, 1032, 1034 communication bottleneck 969, 981, 1250 VEGAS algorithm for Monte Carlo 309ff., 1161 Verhoeff’s algorithm for checksums 894f., 1345 Index to Volumes 1 and2 Vi`ete’s formulas for cubic roots 179 Vienna Fortran 2/xv Virus, computer 889Viscosity artiﬁcial 831, 837 numerical 830f., 837 Visibility 956ff., 1209, 1293, 1296VMS 1/xxii, 2/xix Volterra equations 780f., 1326 adaptive stepsize control 788analogy with ODEs 786 block-by-block method 788 ﬁrst kind 781, 786nonlinear 781, 787second kind 781, 786ff., 1326f. unstable quadrature 787f. von Neuman, John 963, 965von Neumann-Richtmyer artiﬁcial viscosity 837 von Neumann stability analysis for PDEs 827f., 830, 833f., 840 Vowellish (coding example) 896f., 902 W-cycle 865, 1336 Warranty, disclaimer of 1/xx, 2/xviiWave equation 246, 818, 825f.Wavelet transform 584ff., 1264ff. appearance of wavelets 590ff. approximation condition of order p585 coefﬁcient values 586, 589, 1265 contrasted with Fourier transform 584, 594 Daubechies wavelet ﬁlter coefﬁcients 584ff., 588, 590f., 594, 598, 1264ff. detail information 585 discretewavelet transform (DWT) 586f., 1264 DWT (discrete wavelet transform) 586f., 1264ff. eliminating wrap-around 587fast solution of linear equations 597ff. ﬁlters 592f. and Fourier domain 592f.image processing 596f.for integral equations 782 inverse 587 Lemarie’s wavelet 593 of linear operator 597ff. mother-function coefﬁcient 587 mother functions 584multidimensional 595, 1267f.nonsmoothness of wavelets 591 pyramidal algorithm 586, 1264 quadrature mirror ﬁlter 585smooth information 585 truncation 594f. wavelet ﬁlter coefﬁcient 584, 587wavelets 584, 590ff. Wavelets seeWavelet transform Weber function 204 Weighted Kolmogorov-Smirnov test 621Weighted least-squares ﬁtting seeLeast squares ﬁttingWeighting, full vs. half in multigrid 867 Weights for Gaussian quadrature 140ff., 788f., 1059ff., 1328f. nonclassical weight function 151ff., 788f., 1064f., 1328f. Welch window 547, 1254ff.WG5 (ISO/IEC JTC1/SC22/WG5 Committee) 2/xiff. where construct 943, 1291 contrasted with merge 1023for iteration of a vector 1060 nested 2/xv, 943, 960, 1100 not MIMD 985 While iteration 13 Wiener ﬁltering 535, 539ff., 558, 644 compared to regularization 801 Wiener-Khinchin theorem 492, 558, 566f.Wilcoxon test 694 Window function Bartlett 547, 1254ff.ﬂat-topped 549 Hamming 547 Hann 547Parzen 547square 544, 546, 1254ff. Welch 547, 1254ff. Windowing for spectral estimation 1255f.Windows 95 2/xix Windows NT 2/xix Winograd Fourier transform algorithms 503Woodbury formula 68ff., 83Wordlength 18 Workspace, reallocation in Fortran 90 1070f. World Wide Web, Numerical Recipes site 1/xx, 2/xvii Wraparound in integer arithmetic 1146, 1148 order for storing spectrum 501problem in convolution 533 Wronskian, of Bessel functions 234, 239 X.25 protocol 890 X3J3 Committee 2/viii, 2/xff., 2/xv, 947, 959, 964, 968, 990 XMODEM checksum 889X-ray diffraction pattern, processing of 805 Yale Sparse Matrix Package 64, 71 Z-transform 554, 559, 565 Z-transformation, Fisher’s 631f., 1276Zaman, A. 1149 Zealots 814 Zebra relaxation 866Zero contours 372Zero-length array 944 Zeroth-order regularization 796ff. Zip code, barcode for 894Ziv-Lempel compression 896zroots unity() utility function 974, 990, 999