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Index of MATLAB Programs




forward row Forward substitution: row-oriented version . 66
forward col Forward substitution: column-oriented version 66
backward col Backward substitution: column-oriented version 66
lu kji LU factorization of matrix A. kji version . . 77
lu jki LU factorization of matrix A. jki version . . 77
lu ijk LU factorization of the matrix A: ijk version 79
chol2 Cholesky factorization . . . . . . . . . . . . . 81
mod grams Modi¬ed Gram-Schmidt method . . . . . . . 84
LUpivtot LU factorization with complete pivoting . . . 88
lu band LU factorization for a banded matrix . . . . 92
forw band Forward substitution for a banded matrix L 92
back band Backward substitution for a banded matrix U 92
mod thomas Thomas algorithm, modi¬ed version . . . . . 93
cond est Algorithm for the approximation of K1 (A) . 109
JOR JOR method . . . . . . . . . . . . . . . . . . 135
SOR SOR method . . . . . . . . . . . . . . . . . . 136
basicILU Incomplete LU factorization . . . . . . . . . 141
ilup ILU(p) factorization . . . . . . . . . . . . . . 143
gradient Gradient method with dynamic parameter . 149
conjgrad Preconditioned conjugate gradient method . 157
arnoldi alg The Arnoldi algorithm . . . . . . . . . . . . 161
arnoldi met The Arnoldi method for linear systems . . . 164
GMRES The GMRES method for linear systems . . . 166
Lanczos The Lanczos method for linear systems . . . 167
Lanczosnosym The Lanczos method for unsymmetric systems 170
644 Index of MATLAB Programs

powerm Power method . . . . . . . . . . . . . . . . . 197
invpower Inverse power method . . . . . . . . . . . . . 198
basicqr Basic QR iteration . . . . . . . . . . . . . . . 203
houshess Hessenberg-Householder method . . . . . . . 208
hessqr Hessenberg-QR method . . . . . . . . . . . . 210
givensqr QR factorization with Givens rotations . . . 211
vhouse Construction of the Householder vector . . . 213
givcos Computation of Givens cosine and sine . . . 214
Product G(i, k, θ)T M . . . . . . . . . . . . .
garow 214
gacol Product MG(i, k, θ) . . . . . . . . . . . . . . 214
qrshift QR iteration with single shift . . . . . . . . . 217
qr2shift QR iteration with double shift . . . . . . . . 220
psinorm Evaluation of Ψ(A) . . . . . . . . . . . . . . 229
symschur Evaluation of c and s . . . . . . . . . . . . . 229
cycjacobi Cyclic Jacobi method for symmetric matrices 229
sturm Sturm sequence evaluation . . . . . . . . . . 232
givsturm Givens method using the Sturm sequence . . 232
chcksign Sign changes in the Sturm sequence . . . . . 232
Calculation of the interval J = [±, β] . . . .
bound 232
eiglancz Extremal eigenvalues of a symmetric matrix 234
bisect Bisection method . . . . . . . . . . . . . . . 250
chord The chord method . . . . . . . . . . . . . . . 254
secant The secant method . . . . . . . . . . . . . . 255
regfalsi The Regula Falsi method . . . . . . . . . . . 255
newton Newton™s method . . . . . . . . . . . . . . . 255
¬xpoint Fixed-point method . . . . . . . . . . . . . . 260
horner Synthetic division algorithm . . . . . . . . . 263
newthorn Newton-Horner method with re¬nement . . . 266
mullde¬‚ Muller™s method with re¬nement . . . . . . . 269
aitken Aitken™s extrapolation . . . . . . . . . . . . . 274
adptnewt Adaptive Newton™s method . . . . . . . . . . 276
newtonxsys Newton™s method for nonlinear systems . . . 285
broyden Broyden™s method for nonlinear systems . . . 290
¬xposys Fixed-point method for nonlinear systems . . 293
hookejeeves The method of Hooke and Jeeves (HJ) . . . 296
explore Exploration step in the HJ method . . . . . 297
backtrackr Backtraking for line search . . . . . . . . . . 303
lagrpen Penalty method . . . . . . . . . . . . . . . . 316
lagrmult Method of Lagrange multipliers . . . . . . . 319
interpol Lagrange polynomial using Newton™s formula 334
dividif Newton divided di¬erences . . . . . . . . . . 336
hermpol Osculating polynomial . . . . . . . . . . . . . 342
par spline Parametric splines . . . . . . . . . . . . . . . 359
bernstein Bernstein polynomials . . . . . . . . . . . . . 361
bezier B´zier curves . . . . . . . . . . . . . . . . . .
e 361
Index of MATLAB Programs 645

midpntc Midpoint composite formula . . . . . . . . . 375
trapezc Composite trapezoidal formula . . . . . . . . 376
simpsonc Composite Cavalieri-Simpson formula . . . . 377
newtcot Closed Newton-Cotes formulae . . . . . . . . 383
trapmodc Composite corrected trapezoidal formula . . 387
romberg Romberg integration . . . . . . . . . . . . . . 391
simpadpt Adaptive Cavalieri-Simpson formula . . . . . 397
redmidpt Midpoint reduction formula . . . . . . . . . . 404
redtrap Trapezoidal reduction formula . . . . . . . . 404
midptr2d Midpoint rule on a triangle . . . . . . . . . . 406
traptr2d Trapezoidal rule on a triangle . . . . . . . . 406
coe¬‚ege Coe¬cients of Legendre polynomials . . . . . 430
coe¬‚agu Coe¬cients of Laguerre polynomials . . . . . 430
coefherm Coe¬cients of Hermite polynomials . . . . . 430
zplege Coe¬cients of Gauss-Legendre formulae . . . 430
zplagu Coe¬cients of Gauss-Laguerre formulae . . . 430
zpherm Coe¬cients of Gauss-Hermite formulae . . . 430
dft Discrete Fourier transform . . . . . . . . . . 439
idft Inverse discrete Fourier transform . . . . . . 439
¬trec FFT algorithm in the recursive version . . . 441
compdi¬ Compact di¬erence schemes . . . . . . . . . 446
multistep Linear multistep methods . . . . . . . . . . . 490
predcor Predictor-corrector scheme . . . . . . . . . . 507
ellfem Linear FE for two-point BVPs . . . . . . . . 557
femmatr Construction of the sti¬ness matrix . . . . . 557
Computation of the H1 -norm of the error
H1error . 558
artvisc Arti¬cial viscosity . . . . . . . . . . . . . . . 570
sgvisc Optimal arti¬cial viscosity . . . . . . . . . . 570
bern Evaluation of the Bernoulli function . . . . . 571
thetameth θ-method for the heat equation . . . . . . . . 592
pardg1cg1 dG(1)cG(1) method for the heat equation . . 596
ipeidg0 dG(0) implicit Euler . . . . . . . . . . . . . . 621
ipeidg1 dG(1) implicit Euler . . . . . . . . . . . . . . 622
Index




A-conjugate directions, 151 B-splines, 353
parametric, 361
A-stability, 481
backward substitution, 65
absolute value notation, 62
bandwidth, 452
adaptive error control, 43
Bernoulli
adaptivity, 43
function, 565
Newton™s method, 275
numbers, 389
Runge-Kutta methods, 512
bi-orthogonal bases, 168
algorithm
binary digits, 46
Arnoldi, 160, 164
boundary condition
Cuthill-McKee, 98
Dirichlet, 541
Dekker-Brent, 256
Neumann, 541, 582
Remes, 435
Robin, 579
synthetic division, 262
breakdown, 160, 165
Thomas, 91
B´zier curve, 360
e
ampli¬cation B´zier polygon, 359
e
coe¬cient, 609
error, 612 CFL
analysis condition, 606
a priori number, 606
for an iterative method, 132 characteristic
a posteriori, 42 curves, 598
a priori, 42 variables, 600
backward, 42 characteristic polygon, 359
forward, 41 chopping, 51
648 Index

cofactor, 9 distribution, 547
condition number, 34 derivative of a, 547
asymptotic, 38 divided di¬erence, 267, 334
interpolation, 332 domain of dependence, 600
of a matrix, 36, 58 numerical, 606
of a nonlinear equation, 246
eigenfunctions, 589
of an eigenvalue, 189
eigenvalue, 12
of an eigenvector, 190
algebraic multiplicity of an,
Skeel, 111
13
spectral, 59
geometric multiplicity of an,
consistency, 37, 124, 474, 493, 510
13
convex function, 295, 321
eigenvector, 12
strongly, 312
elliptic
convex hull, 98
operator, 602
critical point, 295
equation
Dahlquist characteristic, 12
¬rst barrier, 499 di¬erence, 482, 483, 499
second barrier, 500 heat, 581, 592
decomposition error
real Schur, 201, 210, 211 absolute, 40
generalized, 225 cancellation, 39
Schur, 14 global truncation, 474
singular value, 16 interpolation, 329
computation of the, 222 local truncation, 474, 605
spectral, 15 quadrature, 372
de¬‚ation, 207, 216, 263 rounding, 45
degree estimate
of exactness, 380 a posteriori, 64, 195, 196, 381,
of a vector, 160 392, 395
of exactness, 372, 380, 405, a priori, 60, 381, 392, 395
420 exponential ¬tting, 565
of freedom, 552
factor
determinant of a matrix, 8
asymptotic convergence, 125
discrete
convergence, 125, 245, 259
truncation of Fourier series,
growth, 104
417
factorization
Chebyshev transform, 426
block LU, 94
Fourier transform, 438
Cholesky, 80
Laplace transform, 458
compact forms, 78
Legendre transform, 428
Crout, 78
maximum principle, 567, 611
Doolittle, 78
scalar product, 425
incomplete, 140
dispersion, 448, 612
LDMT , 79
dissipation, 612
Index 649

LU, 68 Gershgorin circles, 184
Gibbs phenomenon, 439
QR, 82, 209
gradient, 294
¬ll-in, 98, 141
graph, 97
level, 141
oriented, 97, 186
¬nite di¬erences, 118, 177, 237,
Gronwall lemma, 471, 476
533
backward, 443
hyperbolic
centered, 443, 444
operator, 602
compact, 444
hypernorms, 63
forward, 442
¬nite elements, 118, 347
ILU, 140
discontinuous, 594, 619
inequality
¬xed-point iterations, 257
Cauchy-Schwarz, 340, 568
¬‚op, 53
H¨lder, 19
o
FOM, 163, 164
Kantorovich, 305
form
Poincar´, 536, 569
e
divided di¬erence, 334
triangular, 569
Lagrange, 329
Young™s, 544
formula
integration
Armijo™s, 304
adaptive, 391
Goldstein™s, 304
automatic, 391
Sherman-Morrison, 95
multidimensional, 402
forward substitution, 65
non adaptive, 391
Fourier coe¬cients, 436
interpolation
discrete, 437
Hermite, 341
function
in two dimensions, 343
gamma, 528
osculatory, 342
Green™s, 532
piecewise, 338
Haar, 460
Taylor, 369
stability, 516
interpolation nodes, 328
weight, 415
piecewise, 345
IOM, 164
Galerkin
¬nite element method, 364,
Jordan
550
block, 15
stabilized, 568
canonical form, 15
generalized method, 559
method, 544 kernel of a matrix, 10
pseudo-spectral approximation, Krylov
591 method, 160
Gauss elimination subspace, 159
method, 68
multipliers in the, 69 Lagrange
GAXPY, 77 interpolation, 328
multiplier, 312, 317
generalized inverse, 17
650 Index

Lagrangian function, 312 unitary, 7
augmented, 317 Vandermonde, 368
matrix balancing, 110
penalized, 315
maximum principle, 533, 534
Laplace operator, 572
discrete, 29
least-squares, 417
method
discrete, 431
θ’, 584
Lebesgue
Regula Falsi, 252
constant, 331, 332
conjugate gradient, 153
linear map, 7
Aitken, 272
linear regression, 433
alternating-direction, 158
linearly independent vectors, 3
backward Euler, 473
LU factorization, 72
backward Euler/centred, 604
M-matrix, 29, 145 BiCG, 171
machine epsilon, 49 BiCGSTab, 171
machine precision, 51 bisection, 248
mass-lumping, 588 Broyden™s, 289
matrix, 3 CGS, 171
block, 4 chord, 252, 260
companion, 242 conjugate gradient, 168
convergent, 26 with restart, 156
defective, 13 CR, 168
diagonalizable, 15 Crank-Nicolson, 473, 593
diagonally dominant, 29, 145 cyclic Jacobi, 228
Gaussian transformation, 73 damped Newton, 321
Givens, 206 damped Newton™s, 308
Hessenberg, 12, 203, 211, 212 ¬nite element, 573
Hilbert, 70 ¬xed-point, 290
Householder, 204 Fletcher-Reeves, 306
interpolation, 330 forward Euler, 473
irreducible, 185 forward Euler/centred, 603
iteration, 124 forward Euler/uncentred, 603
mass, 587 frontal, 102
norm, 21 Gauss Seidel
normal, 7 symmetric, 133
orthogonal, 6 Gauss-Jordan, 121
permutation, 5 Gauss-Seidel, 128
preconditioning, 126 nonlinear, 324
reducible, 185 Givens, 230
rotation, 7 GMRES, 166
similar, 14 with restart, 166
sti¬ness, 548 gradient, 300
transformation, 204 Gram-Schmidt, 83
trapezoidal, 11 Heun, 473
Horner, 262
triangular, 11
Index 651

model
Householder, 207
inverse power, 195 computational, 43
Jacobi, 127 module of continuity, 386
JOR, 127
nodes
Lanczos, 167, 233
Gauss, 426
Lax-Friedrichs, 603, 608
Gauss-Lobatto, 424, 426
Lax-Wendro¬, 603, 608
norm
Leap-Frog, 604, 611
absolute, 32
Merson, 530
compatible, 21, 22
modi¬ed Euler, 529
consistent, 21
modi¬ed Newton™s, 284
energy, 29
Monte Carlo, 407
equivalent, 20
Muller, 267
essentially strict, 432
Newmark, 604, 611
Frobenius, 22
Newton™s, 253, 261, 283
Newton-Horner, 263, 264 H¨lder, 19
o
Nystron, 529 matrix, 21
ORTHOMIN, 168 maximum, 19, 330
Polak-Ribi´re, 307
e spectral, 23
Powell-Broyden normal equations, 112
symmetric, 311 numbers
power, 192 de-normalized, 48
QMR, 171 ¬xed-point, 46
QR, 200 ¬‚oating-point, 47
with double shift, 218 numerical ¬‚ux, 602
with single shift, 215, 216 numerical method, 37
quasi-Newton, 288 adaptive, 43
reduction formula, 403 consistent, 37
Richardson, 136 convergent, 39
Richardson extrapolation, 387 e¬ciency, 44
Romberg integration, 389, 409 ill conditioned, 38
Rutishauser, 202 reliability, 44
secant, 252, 257, 288 stable, 38
secant-like, 309 well posed, 38
Simplex, 299
SSOR, 134 orbit, 523
steepest descent, 305 over¬‚ow, 51
Ste¬ensen, 280
P´clet number, 561
e
successive over-relaxation, 128
local, 563
upwind, 603, 607
Pad´ approximation, 370
e
minimax
parabolic
property, 418
operator, 602
minimizer
pattern of a matrix, 97, 575
global, 294, 311
local, 294, 311 penalty parameter, 315
652 Index

phase angle, 612 Cavalieri-Simpson, 377, 385,
400, 401, 409
pivoting, 85
composite Cavalieri-Simpson,
complete, 86
377
partial, 86
composite midpoint, 374
Poisson equation, 572
composite Newton-Cotes, 383
polyalgorithm, 277
composite trapezoidal, 376
polynomial
corrected trapezoidal, 386
Bernstein, 359
Gauss, 421
best approximation, 330, 433
on triangles, 406
characteristic, 12, 329
Gauss-Kronrod, 393
Fourier, 435
Gauss-Lobatto, 422, 425
Hermite, 429
Gauss-Radau
interpolating, 328
on triangles, 406
Lagrange piecewise, 346
Hermite, 372, 386
Laguerre, 428
Lagrange, 372
nodal, 329
midpoint, 373, 385
orthogonal, 415
on triangles, 405
preconditioner, 126
Newton-Cotes, 378
block, 139
on triangles, 404
diagonal, 140
pseudo-random, 408
ILU, 142
trapezoidal, 375, 385, 438
least-squares, 145
on triangles, 405
MILU, 144
quotient
point, 139
Rayleigh, 12
polynomial, 145
generalized, 146
principal root of unity, 437
QZ iteration, 225
problem
Cauchy, 469 rank of a matrix, 9
generalized eigenvalue, 146, 224, rate
238, 589 asymptotic convergence, 125
ill posed, 34, 35 convergence, 259
ill-conditioned, 34 reduction formula
sti¬, 520 midpoint, 403
well conditioned, 34 trapezoidal, 404
well posed, 33 reference triangle, 345
programming regularization, 34
linear, 282 representation
nonlinear, 282, 313 ¬‚oating-point, 47
pseudo-inverse, 17, 114 positional, 45
pseudo-spectral residual, 247
derivative, 449 resolvent, 35
di¬erentiation matrix, 449 restart, 164
round digit, 53
quadrature formula, 371 rounding, 51
Index 653

roundo¬ unit, 51 asymptotic, 471
rule factors, 42
Cramer™s, 58 Liapunov, 471
Descartes, 262 of interpolation, 332
Laplace, 9 relative, 502
Runge™s counterexample, 331, 344, zero, 477, 495, 502
353 standard deviation, 298
statistic mean value, 407
SAXPY, 77 stencil, 445
saxpy, 77 stopping tests, 171, 269
scalar product, 18 strong formulation, 547
scaling, 110 Sturm sequences, 230
by rows, 110 subspace
Schur generated, 2
complement, 102 invariant, 13
decomposition, 14 vector, 2
semi-discretization, 584, 586 substructures, 100
series Sylvester criterion, 29
Chebyshev, 418 system
Fourier, 416, 582 hyperbolic, 599
Legendre, 419 strictly, 600
set overdetermined, 112
bi-orthogonal, 189 underdetermined, 115
similarity transformation, 14
singular integrals, 398 theorem
singular values, 16 Abel, 262
space Bauer-Fike, 187
normed, 19 Cauchy, 262
phase, 523 Cayley-Hamilton, 13
Sobolev, 543 Courant-Fisher, 146, 233
vector, 1 de la Vall´e-Poussin, 434
e
spectral radius, 13 equioscillation, 433
spectrum of a matrix, 12 Gershgorin, 184
spline Ostrowski, 259
cardinal, 351 polynomial division, 263
interpolatory cubic, 349 Schur, 14
natural, 349 trace of a matrix, 8
not-a-knot, 350 transform
one-dimensional, 348 fast Fourier, 426
parametric, 358 Fourier, 450
periodic, 348 Laplace, 455
splitting, 126 Zeta, 457
stability triangulation, 344, 573
absolute, 479, 480, 499, 502
region of, 480 under¬‚ow, 51
654 Index

upwind ¬nite di¬erence, 565

weak
formulation, 545
solution, 545, 599
wobbling precision, 49

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