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Software 331

Software

For most of the algorithms described in this book there exists rather so­phisticated software, which is public domain. Of central importance is the netlib, a library of mathematical software, data, documents, etc. Its address IS

http://www.netlib.org/

Linear algebra (LAPACK):

http://www.netlib.org/lapack

Especially linear eigenvalue problems (EISPACK):

http://www.netlib.org/eispack

Please study the therein given hints carefully (e.g., README, etc.) to make sure that you download all necessary material. Sometimes a bit of additional browsing in the neighborhood is needed.

The commercial program package MATLAB also offers a variety of methods associated with topics of this book.

In addition, the book presents a series of algorithms as informal algo­rithms which can be easily programmed from this description-such as the fast summation of spherical harmonics.

Numerous further programs (not only by the authors) can be downloaded from the electronic library Elib by ZIB, either via the ftp-oriented address

http://elib.zib.de/pub/elib/codelib/

or via the web-oriented address

http://www.zib.de/SciSoft/CodeLib/

All of the there available programs are free as long as they are exclusively used for research or teaching purposes.

Index

A-orthogonal, 250 Abel's theorem, 126 Aigner, M., 306 Aitken's L~.2-method, 114 Aitken-Neville algorithm, 184 algorithm

invariance, 12 reliability, 2 speed, 2

almost singular, 45, 73 Arnoldi method, 254 Arrhenius law, 79 asymptotic expansion, 292 automatic differentiation, 92

B-spline basis property, 224 recurrence relation, 221

Bezier curve, 208 points, 208

Babuska, 1., 315 backward substitution, 4 Bernoulli numbers, 288 Bessel

functions, 159, 177 maze, 159

bi-cg-method, 255 bifurcation point, 100 Bjorck, A., 66, 78 Bock, H. G., 96 Bornemann, F. A., 261 Brent, R. P., 85 Brusselator, 112 Bulirsch sequence, 297 Bulirsch, R., 293 Businger, P., 72

cancellation, 27 cascadic principle, 267 Casorati determinant, 157, 177 cg-method, 252

preconditioned, 257 termination criterion, 252, 260

Chebyshev abscissas, 195 approximation problem, 60 iteration, 247 nodes, 184, 196 polynomials, 193, 248

min-max property, 193, 246, 253 Cholesky decomposition

rational, 15 Christoffel-Darboux formula, 285

334 Index

complexity of problems, 2

condition intersection point, 24

condition number absolute, 26 componentwise, 32 of addition, 27 of multiplication, 32 of scalar product, 33 relative, 26 Skeel's, 33

conjugate gradients, 252 continuation method, 92

classical, 102 order, 104 tangent, 103, 108

convergence linear, 85 model, 309 monitor, 307 quadratic, 85 super linear, 85

Cooley, W., 202 cost

QR-factorization, 69, 72 Cholesky decomposition, 16 Gaussian elimination, 7 QR method

for singular values, 137 QR-algorithm, 132

Cramer's rule, 1 Cullum, J., 266 cylinder functions, 159

de Boor algorithm, 235 de Boor, C., 204, 309 de Casteljau algorithm, 213 detailed balance, 143 Deuflhard, P., 73, 90, 261, 308

eigenvalue derivative, 120 Perron, 140

elementary operation, 23 Ericsson, T., 266 error

absolute, 25 analysis

backward, 36 forward, 35

equidistribution, 316 linearised theory, 26 relative, 25

extrapolation algorithm, 295 local, 316 methods, 291, 295

sub diagonal error criterion, 304 tableau, 292

Farin, G., 204 FFT,203 fixed-point

Banach theorem, 84 equation, 82 iteration, 82, 239 method

symmetrizable, 242 Fletcher, R., 255 floating point number, 22 forward substitution, 4 Fourier

series, 152, 200 transform, 197

fast, 201 Francis, J. G. F., 127 Frobenius, F. G., 140

Gaches, J., 50 Gauss

Jordan decomposition, 3 Newton method, 109 Seidel method:, 240

Gauss, C. F., 4, 57 Gautschi, W., 164 generalized inverse, 76 Gentleman, W. M., 70 Givens

fast, 70 rational, 70 rotations, 68

Givens, W., 68 Goertzel algorithm, 171 Goertzel, G., 171 Golub, G. H., 47, 72, 119 graph, 140

irreducible, 140

strongly connected, 140 greedy algorithm, 306 Green's function

discrete, 157, 177 Griewank, A., 92

Hackbusch, W., 244 Hagemann, L. A., 244 Hall, C. A., 230 Hammarling, S., 70 Hermite interpolation

cubic, 186 Hestenes, M. R., 252 Higham, N. J., 46 homotopy, 111

method,l11 Horner algorithm, 169

generalized, 170 Householder

reflections, 70 Householder, A. S., 70

incidence matrix, 141 information theory

Shannon, 308 initial value problem, 270 interpolation

Hermite, 185 nodes, 179

iterative refinement for linear equations, 13 for linear least-squares problems,

66,78

Jacobi method, 240

Kato, T., 122 Krylov spaces, 250 Kublanovskaja, V. N., 127

Lagrange polynomials, 181 representation, 182

Lagrange, J. L., 4 Lanczos method

spectral, 266 Lanczos, C., 262 Landau symbol, 26 LAPACK,13

Index 335

Lebesgue constant, 183 Leibniz formula, 191 Leinen, P., 261 Levenberg-Marquardt method, 98,

117, 149

Manteuffel, T. A., 249 Markov

chain, 137 nearly uncoupled, 147, 150 reversible, 144 uncoupled, 145

process, 137 Markov, A. A., 137 Marsden identity, 223 matrix

bidiagonal, 134 determinant, 1, 12, 30 Hessenberg, 132, 254 incidence, 141 irreducible, 140 norms, 53 numerical range, 17 permutation, 9 primitive, 142 Spd-, 14 stochastic, 137 triangular, 3 Vandermonde, 181

maximum likelihood method, 59 measurement tolerance, 59 Meixner, J., 162 Meyer, C. D., 119 Meyer, W. W., 230 Miller algorithm, 167 Miller, J. C. P., 166, 168 monotonicity test

natural, 90, 106 standard, 90

multigrid methods, 244, 313, 320

Nashed, M. Z., 76 needle impulse, 298, 309, 319 Neumann

functions, 159, 177 series, 29

Neville scheme, 185 Newton

correction, 88

336 Index

simplified correction, 91 Newton method

affine invariance, 88, 90 complex, 115 for square root, 86

nodes Gauss-Christoffel, 282 of a quadrature formula, 273

nonlinear least-squares problem almost compatible, 96 compatible, 93

norm £1_,271 energy, 249 Frobenius, 53 matrix, 53 spectral, 53 vector, 53

normal distribution, 59 numerical rank, 73 numerically singular, 45

Oettli, W., 50, 55 Ohm's law, 58

pcg-method, 257, 267 Penrose axioms, 76 Perron cluster, 147

analysis, 143 Perron, 0., 138 pivot

element, 7 row, 5

pivoting column, 8 conditional, 238 partial, 8 total, 9

polynomials Bernstein, 205 Chebyshev, 154, 176, 193, 248, 285 Hermite, 186, 285 Laguerre, 285 Legendre, 164, 176, 285 orthogonal, 153, 279, 285 trigonometric, 197

power method direct, 124 inverse, 125

Prager, W., 50, 55 preconditioning

diagonal, 260 incomplete Cholesky, 260

pseudo-inverse, 75, 93, 109 QR-factorization, 76 singular value decomposition, 133

QR decomposition column permutation, 72

QR-algorithm shift strategy, 131

quadratic equation, 28, 81 quadrature

condition of problem, 271 error, 283

estimator, 301 formula, 273

Gauss-Christoffel, 282 Newton-Cotes, 275

Gauss-Chebyshev, 285 Gauss-Christoffel, 285 Gauss-Hermite, 285 Gauss-Laguerre, 285 Gauss-Legendre, 285 numerical, 270 parameter-dependent, 312

rank decision, 73 determination, 96

Rayleigh quotient, 262 generalized, 265

refinement global, 317 local, 317

Reinsch, C., 41, 132, 171 residual, 49 Rheinboldt, W. C., 315 Richardson method, 240

relaxed, 243 Rigal, J. L., 50 Ritz-Galerkin approximation, 249 Romberg quadrature

adaptive algorithm, 306 Romberg sequence, 296 Ruhe, A., 266 Rutishauser, H., 127

Sautter, W., 47, 73 scaling, 12

column, 12 row, 12

Schiiffke, W., 162 Schreiber, R. S., 49 Schur normal form, 132 Schur, I., 20 Shannon, C. E., 308 shift strategy, 130 Skeel, R. D., 14, 33, 51, 56 Sonneveld, P., 255 sparse solvers, 238, 266 sparsing, 92 special functions, 151 spectral equivalence, 259 spherical harmonics

algorithm, 163, 166 fast summation, 171

splines complete, 232 minimization property, 229 natural, 232

stability indicator, 37, 42 statistical model

inadequate, 96 steepest descent method, 255 step size, 103

basic, 295, 299 internal, 295

step-size control, 300

Stewart, G. W., 147 Stiefel, E., 252 stochastic process, 137 Stoer, J., 255 Sturm sequence, 178 subcondition, 73 substitution

backward, 4 forward, 4

Taylor interpolation, 186 three-term recurrence relation

adjoint, 170 condition, 161 dominant solution, 162 homogeneous, 156 inhomogeneous, 156, 158

Index 337

minimal solution, 162 symmetric, 156 trigonometric, 40, 162, 170

Traub, J., 2 Trefethen, L. N., 49 Tukey, J. W., 202 turning point, 100

van Loan, C., 47, 119 van Veldhuizen, R., 321 von Mises, R., 124

weight function, 279 weights

Gauss-Christoffel, 282, 285 Newton-Cotes, 275 of a quadrature formula, 273

Wielandt, H., 125, 143 Wilkinson

pathological example, 48, 56 Wilkinson, J. H., 36, 46, 47, 123, 129,

131, 132 Willoughby, R., 266 Wittum, G., 244 work per unit step, 306 Wozniakowski, H., 2 Wronski determinant, 158

Xu, J., 261

Young, D. M., 244 Yserentant, H., 261

Texts in Applied Mathematics

(continued from page ii)

3l. Bremaud: Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. 32. Durran: Numerical Methods for Wave Equations in Geophysical Fluid

33.

34. 3 ~ ::J.

Dynamics. Thomas: Numerical Partial Differential Equations: Conservation Laws and

Elliptic Equations. Chicone: Ordinary Differential Equations with Applications.

Kevorkian: Partial Differential Equations: Analytical Solution Techniques, 2nd ed.

36. Dllllerlld/Paganini: A Course in Robust Control Theory: A Convex Approach. 37. Quarteroni/Sacco/Saleri: Numerical Mathematics. 38. Gallier: Geometric Methods and Applications: For Computer Science and

Engineering. 39. Atkinson/Han: Theoretical Numerical Analysis: A Functional Analysis

Framework. 40. Braller/Castill(}-Chimez: Mathematical Models in Population Biology and

Epidemiology. 41. Davies: Integral Transforms and Their Applications, 3rd ed. 42. Deuflhard/Bornemann: Scientific Computing with Ordinary Differential

Equations. 43. Deuflhard/Hohmann: Numerical Analysis in Modern Scientific Computing: An

Introduction, 2nd ed.