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Index

abs, 8adaptive

Euler, 287interpolation, 100quadrature formulae, 127Runge-Kutta, 301

algorithm, 28backward substitutions, 144forward substitutions, 144Gauss, 145Horner, 68LU factorization, 145Strassen, 29synthetic division, 68Thomas, 163, 334Winograd and Coppersmith, 29

aliasing, 97angle, 9anonymous function, 16ans, 31arpackc, 210artificial

diffusion flux, 359viscosity, 340, 359

average, 111axis, 203

backtracking strategy, 233backward difference formula, 302barycentric interpolation, 90basis, 4bicgstab, 179

boundary conditions, 332, 374Dirichlet, 332Neumann, 332, 374

boundary-value problem, 186, 329Butcher array, 300, 301

cancellation, 7Cauchy point, 245CFL

condition, 361, 372number, 361, 362

characteristiccurves, 356variables, 368

chol, 151clear, 32coefficient

amplification, 362dispersion, 362, 363dissipation, 362, 363Fourier, 361

compass, 9complex, 8complexity, 29computational cost, 28

Cramuer rule, 142LU factorization, 148

cond, 161condest, 161condition number, 161, 183, 344

of interpolation, 87conditions

A. Quarteroni et al., Scientific Computing with MATLAB and Octave,Texts in Computational Science and Engineering 2,DOI 10.1007/978-3-642-45367-0, © Springer-Verlag Berlin Heidelberg 2014

435

436 Index

Karush–Kuhn–Tucker, 256Lagrange, 257LICQ, 256optimality, 218, 256Wolfe, 232strong, 232

conj, 9consistency, 279, 281, 286, 347constraint

active, 254equality, 254inequality, 254

contour lines, 226, 255, 405, 407, 409conv, 21convergence, 26, 65, 286

Euler method, 278, 280factorasymptotic, 60

finite differences, 347Gauss-Seidel method, 173global, 227iterative method, 168, 169Jacobi method, 173local, 227of interpolation, 86order, 26Newton method, 49quadratic, 227secant method, 52super-linear, 52, 222, 238

power method, 199Richardson method, 174

cos, 32cputime, 30cross, 15cumtrapz, 121

Dahlquist barrier, 303, 304dblquad, 134deconv, 21deflation, 68, 210

criterion, 69descent direction, 177, 228

conjugate gradient, 229gradient, 229Newton, 229quasi-Newton, 229

det, 12, 149diag, 13

diff, 24differential equation

ordinary, 271partial, 271

digitsignificant, 5

discretization step, 275disp, 33dispersion, 361–363dissipation, 361, 362domain of dependence, 368dot, 15dot operation, 15, 18

eig, 206eigenvalue, 16, 193

extremal, 197problem, 193

eigenvector, 16, 193eigs, 208end, 30eps, 5, 6equation

Burgers, 357convection-diffusion, 336, 340heat, 330, 348hyperbolic, 355Poisson, 329, 332pure advection, 355telegrapher’s, 331transport, 357, 366Van der Pol, 322wave, 330, 367

equationsnormal, 107

errora-priori estimate, 162absolute, 5, 26computational, 26estimator, 27, 50, 62, 127a-posteriori, 299increment, 180

interpolation, 83local truncation, 279, 360of quadrature, 119perturbation, 291relative, 5, 26roundoff, 4, 5, 8, 26, 155, 158, 282truncation, 26, 279, 347, 350

Index 437

etime, 30exit, 31exp, 32expectation, 131exponent, 4extrapolation

Aitken, 64Richardson, 135

eye, 11

F, 5factorization

Cholesky, 151, 183, 201Gauss LU, 146incomplete Cholesky, 183incomplete LU, 187LU, 143, 146, 150, 158, 201QR, 55, 164, 239

Fast Fourier Transform, 93, 95inverse, 96

FFT, 93, 95fft, 95fftshift, 96Fibonacci sequence, 34, 40figure, 203fill-in, 157find, 45finite difference

backward, 116centered, 116forward, 115

fix, 379fixed point, 57

convergence, 61, 65iteration function, 57iterations, 57

floating-pointnumber, 3, 4operation, 28system, 5

fluxnumerical, 358

fminbnd, 223fminsearch, 225fminunc, 239, 247for, 34format, 4formula

Euler, 8

fplot, 17, 99fsolve, 73, 277function, 16, 17, 35

convex, 218cost, 213derivative, 23graph of, 17interpolant, 81iteration, 57, 62, 64Lagrange characteristic, 83Lagrangian, 256augmented, 264

Lipschitz continuous, 218, 241Lipschitz-continuous, 275, 285objective, 213penalty, 259primitive, 23Runge’s, 85, 89shape, 339strongly convex, 255user-defined, 17, 35

function, 35function handle, 16, 18funtool, 24fzero, 20fzero, 19, 72, 73

gallery, 185Gauss plane, 10Gershgorin circles, 203, 204, 211gmres, 179golden ratio, 219gradient, 217grid, 17griddata, 109griddata3, 109griddatan, 109

help, 32, 37hold off, 203hold on, 203

ichol, 183if, 30ifft, 96ilu, 187imag, 9image, 208imread, 208

438 Index

Inf, 6int, 24interp1, 99interp1q, 99interp2, 109interp3, 109interpft, 96interpolant, 81

Hermite, 104Lagrange, 83trigonometric, 93

interpolationadaptive, 100barycentric formula, 90composite, 98, 109convergence, 86error, 83Hermite piecewise, 104Lagrange, 81Gauss nodes, 87, 88

nodes, 80piecewise linear, 98, 99polynomial, 81rational, 81spline, 100stability, 86trigonometric, 81, 93

inv, 12

Kronecker symbol, 82

Lagrangemultipliers, 243, 256

LAPACK, 167Laplace operator, 329law

Fourier, 331Kirchoff, 273Ohm, 273

least-squaresmethod, 104solution, 164, 166

Lebesgue constant, 87, 89lexicographic order, 342linear system, 137

banded, 183direct methods, 149methodsdirect, 143, 182

iterative, 143, 168, 182overdetermined, 163underdetermined, 163

linearly independent system, 14, 201linspace, 18load, 32loglog, 27Lotka-Volterra equations, 272lu, 149

m-file, 34machine epsilon, 5magic, 188mantissa, 4mass-lumping, 355matlabFunction, 85matrix, 10

bandwidth of, 152, 183, 184bidiagonal, 162companion, 73complex definite positive, 151determinant of, 12, 149diagonal, 13diagonally dominant, 150, 171, 205finite difference, 183full, 185Hankel, 185hermitian, 14, 151Hessian, 217Hilbert, 159, 161, 179, 180, 185identity, 11ill conditioned, 161, 183inverse, 12iteration, 169, 174Jacobian, 322Leslie, 195, 210lower triangular, 13mass, 354non-symmetric, 186norm of, 161orthogonal, 164pattern of, 152permutation, 154product, 11pseudoinverse, 165rank of, 164Riemann, 186semi positive definite, 151similar, 206

Index 439

singular value decomposition of,164

sparse, 152, 156, 163, 166, 186, 344spectrum, 196splitting of, 169square, 10square root of, 400strictly diagonal dominant, 173sum, 11symmetric, 14symmetric positive definite, 151,

173transpose of, 14tridiagonal, 162, 173, 334unitary, 165upper triangular, 13Vandermonde, 147, 185well conditioned, 161Wilkinson, 211

mesh, 344contour, 417meshgrid, 109, 417method

θ−, 349A-stable, 289Adams-Bashforth, 302Adams-Moulton, 302adaptive Newton, 49Aitken, 63backward Euler/centered, 359Bairstow, 73barrier, 268BFGS, 238Bi-CGStab, 179, 187, 258bisection, 43, 58Bogacki and Shampine pair, 301Broyden, 55, 73conjugate gradient, 177, 231consistent, 279, 347Crank-Nicolson, 282, 350, 353cyclic composite, 304deflation, 68Dekker-Brent, 72derivative free, 218descent, 218, 228, 235Dormand-Prince pair, 301Euleradaptive forward, 287backward, 276, 353

forward, 275, 287forward adaptive, 297improved, 305

explicit, 276finite difference, 115, 239, 333, 336,

341, 357finite element, 186, 337, 340, 366,

373forward Euler/centered, 358forward Euler/decentered, 358,

372Gauss Elimination, 147Gauss-Newton, 249damped, 250

Gauss-Seidel, 172, 181GMRES, 179, 185, 258golden section, 219gradient, 175, 231Heun, 305, 306, 327implicit, 276inverse power, 201iterative, 56Jacobi, 170, 181Krylov, 179, 188Lanczos, 179, 210Lax-Friedrichs, 358Lax-Wendroff, 358, 372Leap-Frog, 311, 370least-squares, 104Levenberg-Marquardt, 252line search, 218, 228cubic, 234quadratic, 234

modified Newton, 49Monte Carlo, 131, 379multifrontal, 188multigrid, 188multistep, 285, 302Muller, 73Newmark, 311, 312, 369Newton, 47, 52, 62Newton-Horner, 70one-step, 276, 300power, 197power with shift, 201preconditioned conjugate gradient,

178, 183preconditioned gradient, 175predictor-corrector, 305

440 Index

QR, 206quadratic interpolation, 222quasi-Newton, 55, 73relaxation, 173, 190, 398Richardsondynamic, 174stationary, 174

Runge-Kutta, 300, 305adaptive, 301

secant, 52, 54SOR, 190spectral, 184, 374Steffensen, 64successive over-relaxation, 190trust region, 218, 242, 252upwind, 358, 372

minimizerglobal, 217constrained, 254

local, 217constrained, 254

mkpp, 102model

Leontief, 139Lotka and Leslie, 195

multipliers, 146, 155Lagrange, 243

NaN, 7nargin, 37nargout, 37nchoosek, 379Newton

divided differences, 222nodes

Chebyshev-Gauss, 88Chebyshev-Gauss-Lobatto, 87Gauss-Legendre-Lobatto, 126

normof matrix, 161energy, 174euclidean, 15Frobenius, 215

norm, 15normal equations, 164number

complex, 8floating-point, 4normalized, 3

real, 3

ode, 301ode113, 307ode15s, 304ode15s, 322ode23, 301, 309ode23s, 322, 323, 325ode23tb, 301ode45, 301, 309ones, 14operator

boolean, 32, 33divergence, 330Laplace, 329, 342logical, 32, 33relational, 32short-circuit, 33

optimset, 223order

of convergence, 26overflow, 6, 7

Peclet numberglobal, 336local, 336

partial derivative, 53, 329patch, 203path, 34pcg, 179pchip, 104pde, 346pdetool, 109, 186, 373phase plane, 308pivot, 156

elements, 146pivoting, 154

by row, 154complete, 396total, 156

plot, 18, 27Pn, 19point

admissible, 254critical, 217regular, 217stationary, 217

poly, 39, 85polyder, 22, 86

Index 441

polyfit, 22, 83, 106polyint, 22polynomial, 20

characteristic, 193, 285division of, 21, 69Lagrangian interpolation, 81Legendre, 125product of, 21roots of, 21Taylor, 23, 79

polyval, 83ppval, 102preconditioner, 169, 174, 178

incomplete Cholesky factorization,183

incomplete LU, 187pretty, 378problem

boundary value, 329Cauchy, 274convection-diffusion, 336, 340convection-dominated, 336Dirichlet, 332least squaresnonlinear, 248

Neumann, 332Poisson, 183, 184, 341stiff, 319

productinner, 15scalar, 15

quadl, 126quadratic programming, 257quadrature

nodes, 123weights, 123

quadrature formulae, 117adaptive Simpson, 127, 128composite midpoint, 118composite rectangle, 118composite Simpson, 121composite trapezoidal, 120degree of exactness, 119error, 120, 122Gauss, 133Gauss-Legendre, 125Gauss-Legendre-Lobatto, 184interpolatory, 123

midpoint, 118Newton-Cotes, 133rectangle, 118Simpson, 122trapezoidal, 121

quit, 31quiver, 15quiver3, 15

rand, 30rank, 164Rayleigh quotient, 193real, 9realmax, 6realmin, 6region of absolute stability, 289, 303regression line, 106relaxation method, 190residual, 50, 162, 180, 231

preconditioned, 170relative, 176

return, 36root

multiple, 19, 21, 49simple, 19, 48

root condition, 285roots, 21, 73roundoff

error, 4, 5, 8, 26, 155, 158unity, 5

rpmak, 109rsmak, 109rule

Armijo, 232Cramer, 142Laplace, 12

save, 32scale

linear, 27, 28logarithmic, 27semi-logarithmic, 28

semi-discretization, 348, 353semilogy, 28Sequential Quadratic Programming,

268series

discrete Fourier, 94shift, 201

442 Index

simple, 24, 399simplex, 223sin, 32Singular Value Decomposition, 108,

164, 165singular values, 165sparse, 153spdemos, 109spdiags, 153, 163spectral radius, 169spectrometry, 138spline, 100

error, 103natural cubic, 100not-a-knot condition, 102

spline, 102spy, 152, 183, 344sqrt, 32stability

of interpolation, 86absolute, 287, 289, 290asymptotic, 349of Adams methods, 303region of absolute, 289, 327zero-, 284, 286

statistical mean, 131stencil, 343step adaptivity, 297steplength, 275stopping test, 50, 62, 180Sturm sequences, 73, 210sum, 379svd, 165svds, 165syms, 24, 399system

hyperbolic, 368linear, 137nonlinear equations, 52triangular, 144underdetermined, 145

taylor, 24taylortool, 79theorem

Abel, 67Cauchy, 68

Descartes, 67first mean-value, 23Lax-Richtmyer equivalence, 286mean-value, 23of integration, 22Ostrowski, 59zeros of continuous functions, 43

time-step, 275title, 203toolbox, 2, 32trapz, 121tril, 13triu, 13

UMFPACK, 166, 167, 187underflow, 6

vander, 147varargin, 45variance, 111, 389vector

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

viscosity, 340, 359

wavelet, 109wavelets, 109weak

formulation, 338solution, 357

while, 34wilkinson, 211workspace

workspace base, 32

xlabel, 203

ylabel, 203

zeromultiple, 19of a function, 19simple, 19, 48

zeros, 11, 14

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Texts in Computational Science and Engineering

1. H. P. Langtangen, Computational Partial Differential Equations.Numerical Methods and Diffpack Programming, 2nd Edition.

2. A. Quarteroni, F. Saleri, P. Gervasio, Scientific Computing withMATLAB and Octave, 4th Edition.

3. H. P. Langtangen, Python Scripting for Computational Science, 3rdEdition.

4. H. Gardner, G. Manduchi, Design Patterns for e-Science.

5. M. Griebel, S. Knapek, G. Zumbusch, Numerical Simulation inMolecular Dynamics.

6. H. P. Langtangen, A Primer on Scientific Programming with Python,3rd Edition.

7. A. Tveito, H. P. Langtangen, B. F. Nielsen, X. Cai, Elements ofScientific Computing.

8. B. Gustafsson, Fundamentals of Scientific Computing.

9. M. Bader, Space-Filling Curves.

10.M.G. Larson, F. Bengzon, The Finite Element Method: Theory,Implementation, and Practice.

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Monographs in Computational Science and Engineer-ing

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2. H.P. Langtangen, Computational Partial Differential Equations.Numerical Methods and Diffpack Programming.

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6. S. Turek, Efficient Solvers for Incompressible Flow Problems. AnAlgorithmic and Computational Approach.

7. R. von Schwerin,Multi Body System SIMulation. Numerical Meth-ods, Algorithms, and Software.

8. H.-J. Bungartz, F. Durst, C. Zenger (eds.), High Performance Sci-entific and Engineering Computing.

9. T.J. Barth, H. Deconinck (eds.), High-Order Methods for Computa-tional Physics.

10. H.P. Langtangen, A.M. Bruaset, E. Quak (eds.), Advances in Soft-ware Tools for Scientific Computing.

11. B. Cockburn, G.E. Karniadakis, C.-W. Shu (eds.), DiscontinuousGalerkin Methods. Theory, Computation and Applications.

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18. U. van Rienen, M. Gunther, D. Hecht (eds.), Scientific Computingin Electrical Engineering.

19. I. Babuska, P.G. Ciarlet, T. Miyoshi (eds.), Mathematical Modelingand Numerical Simulation in Continuum Mechanics.

20. T.J. Barth, T. Chan, R. Haimes (eds.), Multiscale and Multiresolu-tion Methods. Theory and Applications.

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

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24. T. Schlick, H.H. Gan (eds.), Computational Methods for Macro-molecules: Challenges and Applications.

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

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

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

29. M.A. Schweitzer, A Parallel Multilevel Partition of Unity Methodfor Elliptic Partial Differential Equations.

30. T. Biegler, O. Ghattas, M. Heinkenschloss, B. van Bloemen Waan-ders (eds.), Large-Scale PDE-Constrained Optimization.

31. M. Ainsworth, P. Davies, D. Duncan, P. Martin, B. Rynne (eds.),Topics in Computational Wave Propagation. Direct and InverseProblems.

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

33. H.P. Langtangen, A. Tveito (eds.), Advanced Topics in Computa-tional Partial Differential Equations. Numerical Methods and Diff-pack Programming.

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

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

36. B.N. Khoromskij, G. Wittum, Numerical Solution of Elliptic Dif-ferential Equations by Reduction to the 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 Sim-ulation.

40. R. Kornhuber, R. Hoppe, J. Periaux, O. Pironneau, O. Wildlund,J. Xu (eds.), Domain Decomposition Methods in Science and Engi-neering.

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

42. A. Schmidt, K.G. Siebert, Design of Adaptive Finite Element Soft-ware. The Finite Element Toolbox ALBERTA.

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44. B. Engquist, P. Lotstedt, O. Runborg (eds.), Multiscale Methods inScience and Engineering.

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

46. D. Kressner, Numerical Methods for General and Structured Eigen-value Problems.

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

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

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54. J. Behrens, Adaptive Atmospheric Modeling.

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61. T. Mathew, Domain Decomposition Methods for the Numerical So-lution of Partial Differential Equations.

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

63. M. Bebendorf, Hierarchical Matrices. A Means to Efficiently SolveElliptic Boundary Value Problems.

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82. B. Engquist, O. Runborg, Y.R. Tsai (eds.), Numerical Analysis andMultiscale Computations.

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90. C. Pechstein, Finite and Boundary Element Tearing and Intercon-necting Solvers for Multiscale Problems.

91. R. Bank, M. Holst, O. Widlund, J. Xu (eds.), Domain Decomposi-tion Methods in Science and Engineering XX.

92. H. Bijl, D. Lucor, S. Mishra, C. Schwab (eds.), Uncertainty Quan-tification in Computational Fluid Dynamics.

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94. M. Ehrhardt, T. Koprucki (eds.), Advanced Mathematical Modelsand Numerical Techniques for Multi-Band Effective Mass Approxi-mations.

95. M. Azaıez, H. El Fekih, J.S. Hesthaven (eds.), Spectral and High Or-der Methods for Partial Differential Equations ICOSAHOM 2012.

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