Index

1

1-norm 1102, 1103, 1105

A

Adams-Moulton’s method 662 Akima interpolant 432 algebraic-logarithmic singularities

607

array permutation 1139

ASCII collating sequence 1157 ASCII values 1155, 1156

B

B-spline coefficients 450, 545, 553 B-spline representation 471, 473,

476, 479, 500, 504 B-splines 413

band Hermitian storage mode 179, 181, 187, 189, 191, 193, 1210 band storage mode 124, 127, 130,

132, 134, 136, 138, 162, 164, 167, 170, 172, 175, 1060, 1062, 1063, 1064, 1065,, 1073, 1093, 1095, 1099, 1102, 1103, 1208

band symmetric storage mode 140, 143, 147, 148, 149, 151, 153, 351, 353, 356, 358, 361, 363, 366, 1074, 1209

band triangular storage mode 1211 Basic Linear Algebra Subprograms

1034

basis functions 541 bilinear form 1086

BLAS 1034, 1035, 1047, 1048, 1049 Level 1 1034, 1035

Level 2 1047, 1048 Level 3 1047, 1048, 1049

boundary conditions 678 Broyden’s update 854, 860

C

Cauchy principal value 586, 610 central differences 1007 character arguments 1156 character sequence 1159 character string 1160

character workspace 1202 Chebyshev approximation 418, 583 Cholesky decomposition 307 Cholesky factorization 61, 64, 65,

69, 101, 145, 148, 153, 193, 228, 232, 240, 244, 304

circulant linear system 251 circulant matrices 8

classical weight functions 621, 632 codiagonal band Hermitian storage

mode 1213

codiagonal band hermitian storage mode 182

codiagonal band symmetric storage mode 143, 1212

coefficient matrix 20, 37, 65, 81, 97, 112, 132, 149, 170, 191, 207, 217, 232, 244

coefficients 786, 791 column pivoting 299 complex coefficients 839 complex function 827, 830

complex periodic sequence 772, 774 complex sparse Hermitian positive

definite system 236, 244 complex sparse system 207, 217 complex triangular matrix 55, 54 complex triangular system 50 complex tridiagonal system 155 complex vectors 814, 823 condition number 46, 52 conjugate gradient algorithm 902,

905

conjugate gradient method 253, 259 continuous Fourier transform 763 continuous function 581 convolution 810, 814, 1109 coordinate transformation 292 correlation 818, 823

cosine 601

cosine Fourier coefficients 793 cosine Fourier transform 791

CPU time 1162 crossvalidation 578

cubic spline 440, 441, 443, 445 cubic spline approximation 575, 578 cubic spline interpolant 420, 423,

425, 429, 432, 434, 438 cubic splines 415

cyclic reduction algorithm 120, 156

D

data points 532

date 1162, 1163, 1164, 1165 degree of accuracy 1193 deprecated routines 1217 determinants 7, 24, 42, 48, 54, 69,

87, 101, 117, 136, 153, 175, 193 differential algebraic equations 643 differential equations 641, 678 direct search complex algorithm 948 direct search polytope algorithm 909 discrete Fourier cosine

transformation 782

discrete Fourier sine transformation 779

discrete Fourier transform 763 dot product 1039, 1040 double precision iii, 1111

DOUBLE PRECISION options 1178 DOUBLE PRECISION types v

E

eigensystem

complex 336, 398, 400, 403 Hermitian 382

real 330, 350, 391, 393, 396 symmetric 366, 409

eigenvalues 325, 327, 331, 333, 337, 339, 341, 343, 345, 347, 351, 353, 356, 358, 361, 363, 367, 369, 372, 374, 376, 379, 383, 385, 387, 388, 391, 393, 398, 400, 405, 407

eigenvectors 327, 333, 339, 343, 347, 353, 358, 363, 369, 374, 379, 385, 388, 393, 400, 407

endpoint singularities 589 error detection 572

error handling vi, 1195

errors 1193, 1194, 1195, 1196 alert 1194

detection 1193 fatal 1195

informational 1194 multiple 1193 note 1194

severity 1193 terminal 1194, 1195 warning 1195

Euclidean (2-norm) distance 1104 even sequence 782

extended precision arithmetic 1111

F

factored secant update 854, 860 Fast Fourier Transforms 762 Fejer quadrature rule 633

finite difference gradient 996 finite-difference approximation 848,

854

finite-difference gradient 881, 902, 923

finite-difference Hessian 891 finite-difference Jacobian 912 first derivative 636

first derivative evaluations 875 first order differential 696 forward differences 1009, 1011,

1013, 1015

Fourier coefficients 765, 768, 772, 774, 797, 803

Fourier integral 604 Fourier transform 800, 806 Frobenius norm 1100

full storage mode 1065

G

Gauss quadrature 587

Gauss quadrature rule 621, 625 Gauss-Kronrod rules 591 Gauss-Lobatto quadrature rule 621,

625

Gauss-Radau quadrature rule 621, 625

Gaussian elimination 195, 207, 220, 236

Gear’s BDF method 662 Givens plane rotation 1043

Givens transformations 1044, 1045, 1046

globally adaptive scheme 591 gradient 1007, 1009, 1013, 1018

H

Hadamard product 1041, 1084 Helmholtz’s equation 734 Helmholtz's equation 739 Hermite interpolant 429 Hermite polynomials 717

Hermitian positive definite system 87, 89, 92, 97, 101, 179, 182, 191, 193

Hermitian system 103, 105, 112, 114 Hessian 896, 936, 942, 1011, 1013,

1021

Householder transformations 275, 286

hyper-rectangle 619

I

infinite interval 598 infinity norm 1098

infinity norm distance 1106 informational errors 1194 initial-value problem 645, 652, 662 integer options 1173

INTEGER types v integrals 445

integration 589, 591, 594, 598, 601, 607, 610, 613, 619

interpolation 419 cubic spline 420, 423 quadratic 417 scattered data 417

iterated integral 615

iterative refinement 7, 10, 22, 27, 39, 56, 67, 72, 83, 87, 99, 103, 114, 116, 122, 134, 138, 151, 159, 172, 176, 191, 279

J

Jacobian 836, 848, 851, 854, 860, 918, 952, 958, 1015, 1024

Jenkins-Traub three-stage algorithm 838, 839

L

Laguerre’s method 836 Laplace transform 827, 830 LDU factorization 156

least squares 417, 532, 535, 551 least-squares approximation 543,

547

least-squares problem 292 least-squares solution 275 Level 1 BLAS 1034, 1035 Level 2 BLAS 1047, 1048

Level 3 BLAS 1047, 1048, 1049 Levenberg-Marquardt algorithm 868,

912, 918, 952, 958

linear algebraic equations 195, 220 linear constraints 282

linear equality/inequality constraints 984, 990

linear equations

solving 10, 12, 20, 22, 27, 30, 37, 39, 45, 50, 56, 59, 65, 67, 75, 83, 87, 89, 97, 99, 103, 112, 114, 118, 122, 132, 138, 149, 151, 155, 159, 162, 170, 172, 176, 179, 189, 191, 195, 204, 207, 217, 232, 236, 244, 253

linear least-squares problem 275, 282,

linear programming problem 973, 976

LU factorization 15, 18, 20, 24, 32, 35, 37, 42, 127, 130, 136, 167, 170, 175, 199, 204, 212, 217,

M

machine-dependent constants 1201 mathematical constants 1185 matrices 1058, 1059, 1060, 1062,

1063, 1064, 1065, 1067, 1068, 1069, 1070, 1071, 1073, 1074, 1075, 1077, 1081, 1083, 1085, 1092, 1093, 1095, 1099, 1102, 1103, 1116, 1118, 1121, 1123, 1125, 1127, 1130

complex 167, 170, 176, 311, 331, 333, 1067, 1071

band 5, 1062, 1093, 1102, 1103 general 5, 32, 42, 43, 1058,

1063, 1067 general sparse 212

Hermitian 5, 94, 96, 103, 108, 112, 117, 184, 187, 189, 193, 367, 369, 372, 374, 376, 379, 1071, 1074

rectangular 1068, 1075, 1081, 1083, 1084, 1092, 1125, 1127

sparse 5 triangular 52, 54 tridiagonal 5, 156

upper Hessenberg 383, 388 copying 1059, 1060, 1067, 1068,

1073, 1074 general 1206 Hermitian 1207

multiplying 1078, 1079, 1081, 1083, 1090, 1092, 1093

permutation 1139

printing 1116, 1118, 1121, 1123, 1125, 1127, 1130

real 127, 130, 132, 136, 325, 327, 1063, 1069

band 5, 1060, 1090, 1102 general 5, 15, 18, 24, 26, 1058,

1062, 1067 general sparse 199

rectangular 1075, 1075, 1081, 1083, 1089, 1100, 1116, 1118

sparse 5

symmetric 5, 61, 64, 69, 71, 77, 80, 85, 145, 147, 150, 156, 304, 337, 339, 341, 343, 345, 347, 351, 353, 356, 358, 361, 363, 1070, 1073

triangular 46, 48, 49 tridiagonal 5, 120

upper Hessenberg 383, 385 rectangular 1075, 1206 sparse

Hermitian 240 symmetric 224 symmetrical 228 symmetric 301, 1206

transposing 1075, 1077, 1079 triangular 1207

matrix inversion 7 types 5

matrix permutation 1139 matrix storage modes 1206

matrix-matrix multiply 1053, 1055, matrix-matrix solve 1055 matrix-vector multiply 1049, 1050,

1051

matrix/vector operations 1056 method of lines 717 minimization 868, 869, 870, 872,

875, 878, 881, 886, 891, 896, 902, 905, 909, 923, 930, 936, 942, 948, 952, 984, 990

minimum degree ordering 224 minimum point 872, 875, 878

modified Powell hybrid algorithm 848, 851

monic polynomials 628, 630 Muller’s method 836, 841 multiple right sides 6 multivariate functions 868 multivariate quadrature 587

N

naming conventions v Newton algorithm 868

Newton method 891, 896, 936, 942 noisy data 575, 5780

nonadaptive rule 613

nonlinear equations 848, 851, 854, 860

nonlinear least-squares problem 868, 912, 918, 952, 958, 964

nonlinear programming problem 997, 1003

not-a-knot condition 420, 423 numerical differentiation 588

O

odd sequence 779

odd wave numbers 786, 788, 791, 793

ordinary differential equations 641, 642, 645, 652, 662

orthogonal matrix 289 overflow vi

P

page length 1137 page width 1137

parameters 770, 777, 781, 784, 790, 795

partial differential equations 642, 643, 717

performance index 330, 336, 350, 366, 382, 396, 403, 409

periodic boundary conditions 438 Petzold 696

physical constants 1185

piecewise polynomial 413, 505, 507, 510, 512

plane rotation 1043 plots 1181

Poisson solver 734, 739 Poisson's equation 735, 740 polynomial 1087

polynomial curve 535 prime factors 1183 printing 1137, 1181, 1194 printing results vii

programming conventions vi pseudorandom number generators

1170

pseudorandom numbers 1170, 1171

Q

QR decomposition 8, 286 QR factorization 289, 295

quadratic interpolation 514, 516, 518, 520, 523, 525

quadratic polynomial interpolation 417

quadratic programming algorithm 996, 1003

quadrature formulas 587 quadrature rule 630 quadruple precision 1111

quasi-Newton method 881, 886, 923, 930

quintic polynomial 531

R

random number generators 1167, 1168

rank-2k update 1054, 1055 rank-k update 1054 rank-one matrix 301 rank-one matrix update 1053 rank-two matrix update 1053 rational weighted Chebyshev

approximation 581

real periodic sequence 765, 768 real sparse symmetric positive

definite system 232

real symmetric definite linear system 253, 259

real symmetric positive definite system 56, 59, 65, 67, 149, 151

real symmetric system 72, 75, 81, 83 real triangular system 45

real tridiagonal system 118 REAL types v

real vectors 810, 818 rectangular domain 487

rectangular grid 519, 520, 523, 525 recurrence coefficients 625, 628, 630 reserved names 1216

reverse communication 262 Runge-Kutta-order method 652 Runge-Kutta-Verner fifth-order

method 645 Runge-Kutta-Verner sixth-order

method 645

S

scattered data 531

scattered data interpolation 417 search 1150, 1152, 1153 second derivative 636 semi-infinite interval 598 sequence 788, 793

serial number 1166 simplex algorithm 973, 976 sine 601

sine Fourier coefficients 788 sine Fourier transform 786 single precision iii

SINGLE PRECISION options 1175 singular value decomposition 311 singularity 7

singularity points 594

smooth bivariate interpolant 529 smoothing 572

smoothing spline routines 417 solving linear equations 4 sorting 1142, 1143, 1144, 1145,

1146, 1148, 1149, 1150, 1152, sparse linear programming 976 sparse matrix storage mode 1215 sparse system 195, 207

spline approximation 543, 551 spline interpolant 451, 459 spline knot sequence 454, 457

splines 418, 447, 469, 471, 473, 476 cubic 415

tensor product 416 Sturm-Liouville problem 745, 757 symmetric Markowitz strategy 210

T

tensor product splines 416 tensor-product B-spline coefficients

459, 464, 561, 566 tensor-product B-spline

representation 479, 480, 483, 487, 490, 491, 495, 500

tensor-product spline 479, 480, 484, 488, 491, 492, 496, 501

tensor-product spline approximant 561, 566

tensor-product spline interpolant 464 terminal errors 1193

third derivative 636 time 1161

Toeplitz linear system 248, 249 Toeplitz matrices 8

traceback 1198

triple inner product 1040

U

unconstrained minimization 868 underflow vi

uniform (0, 1) distribution 1170, 1171

uniform mesh 740 univariate functions 868 univariate quadrature 586 user errors 1193

user interface iii user-supplied function 636

user-supplied gradient 905, 930, 1003

V

variable knot B-spline 547 variable order 678

vectors 1037, 1038, 1041, 1050, 1090, 1092, 1093, 1107, 1109

complex 1109 real 1107

version 1166

W

work arrays vi

workspace allocation 1199, 1200

Z

zero of a real function 843

zeros of a polynomial 836, 838, 839 zeros of a univariate complex

function 841

zeros of the polynomial 835