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Index

abs, 8adaptive

interpolation, 94quadrature formulae, 121Runge-Kutta, 230stepsize, 227

algorithm, 29backward substitutions, 136forward substitutions, 136Gauss elimination, 137Horner, 66Strassen, 29synthetic division, 66Thomas, 151, 260Winograd and Coppersmith, 29

aliasing, 91angle, 8anonymous function, 17ans, 31approximation, 78

least-squares, 100arpackc, 197artificial

diffusion flux, 285viscosity, 266, 285

asymptotic convergence factor, 58average, 106axis, 191

backward difference formula, 231baseball trajectory, 202, 254basis, 4

bicgstab, 168biomechanics, 76, 101boundary conditions, 258, 300

Dirichlet, 258Neumann, 258, 300

boundary-value problem, 175, 255Butcher array, 229, 230

cancellation, 6capillary networks, 132, 143CFL

condition, 287, 298number, 287, 288

characteristiccurves, 282Lagrangian function, 81variables, 294

chol, 143cholinc, 172, 177clear, 32climatology, 75, 81coefficient

dissipation, 288Fourier, 287

communications, 257compass, 8complex, 8complexity, 29computational cost, 29

Cramuer rule, 134Gauss factorization, 139

cond, 149

354 Index

condest, 149condition number, 149, 172, 270

of interpolation, 85conj, 9consistency, 209, 211, 216, 272conv, 21convergence, 26, 63, 216

Euler method, 208, 210finite differences, 272Gauss-Seidel method, 162iterative method, 157, 158Newton method, 48of interpolation, 84power method, 187Richardson method, 163

convergence order, 26cos, 32cputime, 30cross, 15cumtrapz, 115

Dahlquist barrier, 232, 233dblquad, 125deconv, 21deflation, 66, 67, 197demography, 108, 116, 126descent directions, 166det, 12, 141diag, 13diff, 23differential equation

ordinary, 201partial, 201

discrete Fourier series, 89discretization step, 205disp, 33dispersion, 287–289dissipation, 287, 288divergence operator, 256domain of dependence, 294dot, 15dot operation, 15, 18

economy, 131eig, 193eigenvalue, 16, 181

extremal, 184problem, 181

eigenvector, 16, 181

eigs, 195elastic

membrane, 272springs, 182

electrical circuits, 203, 239, 242electromagnetism, 108, 128end, 30eps, 5, 6equation

Burgers, 283convection-diffusion, 262, 266heat, 256, 274hyperbolic, 281Poisson, 255, 258pure advection, 281telegrapher’s, 257transport, 283, 292Van der Pol, 250wave, 256, 293

errora-posteriori estimate, 227a-priori estimate, 150absolute, 5, 26amplification, 288computational, 26dispersion, 288dissipation, 288, 289estimator, 27, 50, 60, 121

increment, 169interpolation, 81local truncation, 209, 286of quadrature, 113perturbation, 221relative, 5, 26roundoff, 5, 7, 25, 145, 147, 212truncation, 26, 209, 273, 276

etime, 30Euler formula, 8eval, 17exit, 31exp, 32exponent, 4extrapolation

Aitken, 62Richardson, 127

eye, 11

F, 5factorization

Index 355

Cholesky, 143, 172, 188Gauss, 138, 142incomplete Cholesky, 172incomplete LU, 176LU, 135, 147, 188QR, 152

Fast Fourier Transform (FFT), 88,90

feval, 17fft, 90fftshift, 90Fibonacci sequence, 33, 40figure, 191finance, 75, 99, 101find, 45finite difference

backward, 110centered, 110forward, 109

fix, 306fixed point, 54

convergence, 59, 63iteration function, 55iterations, 55

floating-pointnumber, 3, 5operation, 29

for, 33format, 4Foucault pendulum, 254Fourier

discrete series, 89inverse fast transform, 90

fplot, 16, 94fsolve, 71, 207function, 16

derivative, 23graph, 16iteration, 55, 59, 62Lipschitz continuous, 205, 215primitive, 22shape, 265

function, 35funtool, 24fzero, 19, 70, 71

gallery, 174Gauss plane, 9Gershgorin circles, 190, 192, 198

gmres, 168grid, 17griddata, 104griddata3, 104griddatan, 104

help, 32, 37hold off, 191hold on, 191hydraulic network, 129hydraulics, 107, 111hydrogeology, 256

if, 30ifft, 90imag, 9image compression, 183, 195Inf, 6inline, 17int, 23interp1, 94interp1q, 94interp2, 103interp3, 103interpft, 91interpolant, 79

Hermite, 98Lagrange, 81trigonometric, 88

interpolationadaptive, 94composite, 93, 103convergence, 84error, 81Hermite piecewise, 98Lagrange, 79

Gauss nodes, 86nodes, 78piecewise linear, 93polynomial, 79rational, 79spline, 95stability, 84trigonometric, 79, 88

interurban railway network, 183, 186inv, 12investment fund, 73

Kronecker symbol, 80

356 Index

LAPACK, 155Laplace operator, 255, 268law

Fourier, 257Kirchoff, 203Ohm, 203

least-squaresmethod, 99solution, 152, 154

Lebesgue costant, 84, 87lexicographic order, 268linear system, 129

banded, 172methods

direct, 134, 140, 171iterative, 135, 157, 171

overdetermined, 152underdetermined, 152

linearly independent system, 15, 188linspace, 18load, 32logarithmic scale, 27loglog, 27Lotka-Volterra equations, 202lu, 140luinc, 177

m-file, 34magic, 177mantissa, 4mass-lumping, 281matrix, 10

bandwidth, 154, 172, 173bidiagonal, 151companion, 71complex definite positive matrices,

142determinant, 12, 140diagonal, 12diagonally dominant, 142, 159, 192finite difference, 172full, 174Hankel, 174hermitian, 13, 142Hilbert, 147, 150, 168, 169, 174identity, 11ill conditioned, 150, 172inverse, 12iteration, 157, 163

Leslie, 183, 197lower triangular, 13mass, 280non-symmetric, 175norm of, 149orthogonal, 152pattern of, 140permutation, 145product, 11pseudoinverse, 153rank, 152Riemann, 175similar, 193singular value decomposition of,

152sparse, 140, 147, 151, 154, 175, 269spectrum, 184splitting of, 158square, 10strictly diagonal, 161sum, 11symmetric, 13symmetric positive definite, 142,

161transpose, 13tridiagonal, 151, 162, 260unitary, 153upper triangular, 13Vandermonde, 139, 174well conditioned, 150Wilkinson, 198

mesh, 270contour, 335meshgrid, 104, 335method

θ−, 275A-stable, 220Adams-Bashforth, 231Adams-Moulton, 231adaptive forward Euler, 218, 227adaptive Newton, 49adaptive Runge-Kutta, 230Aitken, 60backward Euler, 206, 279backward Euler/centered, 285Bairstow, 71Bi-CGStab, 168, 176bisection, 43, 55Bogacki and Shampine pair, 230

Index 357

Broyden, 71conjugate gradient, 166consistent, 209, 273Crank-Nicolson, 212, 276, 279cyclic composite, 233Dekker-Brent, 70Dormand-Prince pair, 230dynamic Richardson, 162explicit, 206finite difference, 109, 259, 262, 267,

283finite element, 175, 263, 266, 292,

299forward Euler, 205, 217forward Euler/centered, 284forward Euler/decentered, 284,

298Gauss elimination, 137Gauss-Seidel, 161, 170GMRES, 168, 174gradient, 164Heun, 234, 253implicit, 206improved Euler, 234inverse power, 188Jacobi, 159, 170Krylov, 168, 177Lanczos, 168, 197Lax-Friedrichs, 284Lax-Wendroff, 284, 298Leap-Frog, 240, 296least-squares, 99modified Newton, 49Monte Carlo, 305multifrontal, 177multigrid, 177multistep, 215, 231Muller, 71Newmark, 240, 241, 295Newton, 47, 51, 60Newton-Horner, 68one-step, 206, 229power, 185power with shift, 189preconditioned conjugate gradient,

167, 172preconditioned gradient, 164predictor-corrector, 234QR, 193

quasi-Newton, 71relaxation, 161, 179, 324Runge-Kutta, 229, 234SOR, 179spectral, 299stationary Richardson, 162Steffensen, 62upwind, 284, 298

mkpp, 96model

Leontief, 131Lotka and Leslie, 183predator/prey, 59

multiplicity, 49multipliers, 138, 146

NaN, 7nargin, 36nargout, 36nchoosek, 305nodes

Chebyshev-Gauss, 86Chebyshev-Gauss-Lobatto, 86Gauss-Legendre-Lobatto, 120

normof matrix, 149energy, 163euclidean, 15

norm, 15normal equations, 102, 152not-a-knot condition, 96number

complex, 8floating-point, 5real, 3

numerical flux, 284numerical integration, 111

ode, 230ode113, 236ode15s, 233ode23, 230, 238ode23s, 251ode23tb, 230ode45, 230, 238ones, 14optics, 108, 128overflow, 6, 7

358 Index

Peclet numberglobal, 262local, 262

partial derivative, 52, 255patch, 191path, 34pcg, 167pchip, 98pde, 272pdetool, 104, 175, 299phase plane, 237pivot elements, 138pivoting, 144

by row, 145complete, 322

plot, 18, 27Pn, 19poly, 39, 83polyder, 22, 84polyfit, 22, 81, 101polyint, 22polynomial, 20

characteristic, 181, 215division, 21, 67Lagrangian interpolation, 79Legendre, 119product, 21roots, 21Taylor, 23, 77

polyval, 20, 81population dynamics, 58, 182, 197,

202, 237Verhulst model, 59

ppval, 96preconditioner, 158, 162, 167

incomplete Cholesky factorization,172

incomplete LU factorization, 177predator/prey model, 43pretty, 304problem

Cauchy, 204convection-diffusion, 262, 266convection-dominated, 262Dirichlet, 258Neumann, 258Poisson, 172, 173, 267stiff, 248, 249

quad2dc, 126quad2dg, 125quadl, 120quadrature

nodes, 117weights, 117

quadrature formulae, 111adaptive Simpson, 121, 122composite midpoint, 112composite rectangle, 112composite Simpson, 115composite trapezoidal, 114degree of exactness, 113error, 114, 116Gauss, 125Gauss-Legendre, 119Gauss-Legendre-Lobatto, 173interpolatory, 117midpoint, 112Newton-Cotes, 125rectangle, 112Simpson, 116trapezoidal, 115

quit, 31quiver, 15quiver3, 15

rand, 30rank, 152Rayleigh quotient, 181real, 9realmax, 5realmin, 5region of absolute stability, 219, 232regression line, 101relaxation method, 179residual, 50, 150, 169

preconditioned, 158relative, 165

return, 35robotics, 77, 97rods system, 73root

multiple, 18, 21, 49simple, 18, 48

root condition, 215roots, 21, 71roundoff

error, 4, 5, 7, 25, 145, 147

Index 359

unity, 5rpmak, 104rsmak, 104rule

Cramer, 134

Laplace, 12Runge’s function, 83, 87

save, 32

scalar product, 15scale

linear, 27, 28logarithmic, 27

semi-logarithmic, 27semi-discretization, 274, 279semilogy, 28shift, 189significant digits, 5

simple, 24, 325simpsonc, 116sin, 32Singular Value Decomposition, 102,

152, 153singular values, 153sparse, 140spdemos, 104spdiags, 140, 151

spectral methods, 173spectral radius, 157spectrometry, 130, 138spherical pendulum, 242spline, 94

error, 97natural cubic, 95not-a-knot, 96

spline, 96spy, 172, 269

sqrt, 32stability

of interpolation, 84absolute, 217, 219, 220asymptotic, 275

of Adams methods, 232region of absolute, 219, 253zero-, 214, 216

stencil, 269

stopping test, 49, 60, 169Sturm sequences, 71, 197

successive over-relaxation method,179

sum, 305svd, 154svds, 154syms, 24, 325system

hyperbolic, 294linear, 129nonlinear, 51triangular, 135underdetermined, 137

taylor, 23Taylor polynomial, 23, 77taylortool, 77theorem

Abel, 65Cauchy, 66Descartes, 65first mean-value, 23Lax-Ritchmyer equivalence, 216mean-value, 23of integration, 22Ostrowski, 57zeros of continuous functions, 43

thermodynamics, 201, 253, 257, 277,301

three-body problem, 246title, 191toolbox, 2, 20, 32trapz, 115tril, 13triu, 13

UMFPACK, 155, 156, 176underflow, 6

vander, 139varargin, 45variance, 106, 315vector

column, 10component of a, 15conjugate transpose, 15norm, 15product, 15row, 10

360 Index

wavelet, 104wavelets, 104weak

formulation, 264solution, 282

while, 33wilkinson, 198

xlabel, 191

ylabel, 191

zero

multiple, 18

of a function, 18

simple, 18, 48

zeros, 11, 14

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17. B. I. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition.

18. U. van Rienen, M. Günther, D. Hecht (eds.), Scientific Computing in Electrical Engineering.

19. I. Babuška, P. G. Ciarlet, T. Miyoshi (eds.), Mathematical Modeling and Numerical Simulation inContinuum Mechanics.

20. T. J. Barth, T. Chan, R. Haimes (eds.), Multiscale and Multiresolution Methods. Theory andApplications.

21. M. Breuer, F. Durst, C. Zenger (eds.), High Performance Scientific and Engineering Computing.

22. K. Urban, Wavelets in Numerical Simulation. Problem Adapted Construction and Applications.

23. L. F. Pavarino, A. Toselli (eds.), Recent Developments in Domain Decomposition Methods.

24. T. Schlick, H. H. Gan (eds.), Computational Methods for Macromolecules: Challenges andApplications.

25. T. J. Barth, H. Deconinck (eds.), Error Estimation and Adaptive Discretization Methods inComputational Fluid Dynamics.

26. M. Griebel, M. A. Schweitzer (eds.), Meshfree Methods for Partial Differential Equations.

27. S. Müller, Adaptive Multiscale Schemes for Conservation Laws.

28. C. Carstensen, S. Funken, W. Hackbusch, R. H. W. Hoppe, P. Monk (eds.), ComputationalElectromagnetics.

29. M. A. Schweitzer, A Parallel Multilevel Partition of Unity Method for Elliptic Partial DifferentialEquations.

30. T. Biegler, O. Ghattas, M. Heinkenschloss, B. van Bloemen Waanders (eds.), Large-ScalePDE-Constrained Optimization.

31. M. Ainsworth, P. Davies, D. Duncan, P. Martin, B. Rynne (eds.), Topics in Computational WavePropagation. Direct and Inverse Problems.

32. H. Emmerich, B. Nestler, M. Schreckenberg (eds.), Interface and Transport Dynamics. Computa-tional Modelling.

33. H. P. Langtangen, A. Tveito (eds.), Advanced Topics in Computational Partial Differential Equations.Numerical Methods and Diffpack Programming.

34. V. John, Large Eddy Simulation of Turbulent Incompressible Flows. Analytical and NumericalResults for a Class of LES Models.

35. E. Bänsch (ed.), Challenges in Scientific Computing - CISC 2002.

36. B. N. Khoromskij, G. Wittum, Numerical Solution of Elliptic Differential Equations by Reduction tothe Interface.

37. A. Iske, Multiresolution Methods in Scattered Data Modelling.

38. S.-I. Niculescu, K. Gu (eds.), Advances in Time-Delay Systems.

39. S. Attinger, P. Koumoutsakos (eds.), Multiscale Modelling and Simulation.

40. R. Kornhuber, R. Hoppe, J. Périaux, O. Pironneau, O. Wildlund, J. Xu (eds.), Domain Decomposition

41.

42. A. Schmidt, K.G. Siebert, Design of Adaptive Finite Element Software. The Finite Element ToolboxALBERTA.

43. M. Griebel, M.A. Schweitzer (eds.), Meshfree Methods for Partial Differential Equations II.

44. B. Engquist, P. Lötstedt, O. Runborg (eds.), Multiscale Methods in Science and Engineering.

45. P. Benner, V. Mehrmann, D.C. Sorensen (eds.), Dimension Reduction of Large-Scale Systems.

46. D. Kressner, Numerical Methods for General and Structured Eigenvalue Problems.

47. A. Boriçi, A. Frommer, B. Joó, A. Kennedy, B. Pendleton (eds.), QCD and Numerical Analysis III.

48. F. Graziani (ed.), Computational Methods in Transport.

49. B. Leimkuhler, C. Chipot, R. Elber, A. Laaksonen, A. Mark, T. Schlick, C. Schütte, R. Skeel (eds.), New Algorithms for Macromolecular Simulation.

Methods in Science and Engineering.

T. Plewa, T. Linde, V.G. Weirs (eds.), Adaptive Mesh Refinement – Theory and Applications.

50. M. Bücker, G. Corliss, P. Hovland, U. Naumann, B. Norris (eds.), Automatic Differentiation:Applications, Theory, and Implementations.

51. A.M. Bruaset, A. Tveito (eds.), Numerical Solution of Partial Differential Equations on ParallelComputers.

52. K.H. Hoffmann, A. Meyer (eds.), Parallel Algorithms and Cluster Computing.

53. H.-J. Bungartz, M. Schäfer (eds.), Fluid-Structure Interaction.

54. J. Behrens, Adaptive Atmospheric Modeling.

55. O. Widlund, D. Keyes (eds.), Domain Decomposition Methods in Science and Engineering XVI.

56. S. Kassinos, C. Langer, G. Iaccarino, P. Moin (eds.), Complex Effects in Large Eddy Simulations.

57. M. Griebel, M.A Schweitzer (eds.), Meshfree Methods for Partial Differential Equations III.

58. A.N. Gorban, B. Kégl, D.C. Wunsch, A. Zinovyev (eds.), Principal Manifolds for Data Visualizationand Dimension Reduction.

59. H. Ammari (ed.), Modeling and Computations in Electromagnetics: A Volume Dedicated toJean-Claude Nédélec.

60. U. Langer, M. Discacciati, D. Keyes, O. Widlund, W. Zulehner (eds.), Domain DecompositionMethods in Science and Engineering XVII.

61. T. Mathew, Domain Decomposition Methods for the Numerical Solution of Partial DifferentialEquations.

62. F. Graziani (ed.), Computational Methods in Transport: Verification and Validation.

63. M. Bebendorf, Hierarchical Matrices. A Means to Efficiently Solve Elliptic Boundary ValueProblems.

64. C.H. Bischof, H.M. Bücker, P. Hovland, U. Naumann, J. Utke (eds.), Advances in Automatic Differ-entiation.

For further information on these books please have a look at our mathematics catalogue at the followingURL: www.springer.com/series/3527

65. M. Griebel, M.A. Schweitzer (eds.), Meshfree Methods for Partial Differential Equations IV.

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67. I.H. Tuncer, Ü. Gülcat, D.R. Emerson, K. Matsuno (eds.), Parallel Computational Fluid Dynamics.

68. S. Yip, T. Diaz de la Rubia (eds.), Scientific Modeling and Simulations

66. B. Engquist, P. Lötstedt, O. Runborg (eds.), Multiscale Modeling and Simulation in Science.

Layers.69. A. Hegarty, N. Kopteva, E. O’Riordan, M. Stynes (eds.), BAIL 2008 – Boundary and Interior

70. M. Bercovier, M.J. Gander, R. Kornhuber, O. Widlund (eds.), Domain Decomposition Methods in

72. M. Peters (ed.), Computational Fluid Dynamics for Sport Simulation.

71. B. Koren, C. Vuik (eds.), Advanced Computational Methods in Science and Engineering.

Science and Engineering XVIII.