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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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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.
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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.