General Literature on Numerical Methods

Ciarlet, P.G., Lions, J.L., Eds. (1990, 1991): Handbook of Numerical Analysis. Vol. I: Finite Difference Methods (Part 1), Solution of Equations in ffin (Part 1). Vol. II: Finite Element Methods (Part 1). Amsterdam: North Holland.

Conte, S.D., de Boor, C. (1980): Elementary Numerical Analysis, an Algorithmic Approach, 3d edition. New York: McGraw-Hill.

Dahlquist, G., Bjorck, A. (1974): Numerical Methods. Englewood Cliffs, N.J.: Prentice Hall.

Deufthard, P., Hohmann, A. (1991): Numerische Mathematik. Eine algorithmisch orientierte Einfuhrung. Berlin, New York: de Gruyter.

Forsythe, G.E., Malcolm. M.A., l\Ioler C.B. (1977): Computer Methods for Math­ematical Computations. Englewood Cliffs, N.J.: Prentice Hall.

Froberg, C.E. (1985): Numerical Mathematics. Menlo Park, Calif.: Benjamin/ Cummings.

Gregory, R.T., Young, D.M. (1972, 1973): A Survey of Numerical Mathematics. Vols. 1, 2. Reading, :'vlass.: Addison-Wesley.

IIiimmerlin, G., Hoffmann, K.-H. (1991): Numerical Mathematics. Berlin, Heidel­berg, New York: Springer-Verlag.

Henrici, P. (1964): Elements of Numerical Analysis. New York: John Wiley. Hildebrand, F.B. (1974): Introduction to Numerical Analysis, 2d edition. New

York: McGraw-Hill. Householder, A.S. (1953): Principles of Numerical Analysis. )Jew York: McGraw­

Hill (republication 1974, New York: Dover Publications). Isaacson, E., Keller, H.B. (1966): Analysis of Numerical Methods, New York: John

Wiley. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T. (1990): Numerical

Recipes. The Art of Scient~fic Computing. Cambridge: Cambridge University Press.

Ralston, A., Rabinowitz. P. (1978): A First Course in Numerical Analysis. New York: McGraw-Hill.

Rutishauser, H. (1990): Lectun:s on Numerical Mathematics. Edited by M. Gutknecht with the assistance of P. Henrici et al. and translated hy vv. Gautschi. Boston: Birkhiiuser.

Schaback, R, "Verner, H. (1991): Numerische Mathematik, 4th edition. Berlin, Heidelberg, New York: Springer-Verlag.

Schwarz, H.-R. (1986): Numerical Analysis. A Comprehensive Introduction. With a contribution by J. Waldvogel. Chichester: Wiley.

Schwetlick, H., Kreztschmar, H. (1991): Numerische Verfahren fur Naturwis­senschaftlel' und Ingenieure. Leipzig: Fachbuchverlag.

Stiefel, E. (1963): An Introduction to Numerical Mathenatir:s. New York, London: Academic Press.

References for Chapter 8 731

Todd, .1. (1962): A Survey of Numerical Analysis. New York: McGraw-Hill. --- (1978): Basic numerical mathematics, Vol. 1. Numerical Analysis. Basel:

Birkhiiuser (also New York: Academic Press 1978). --- (1977): Basic Nnmerical Mathematics, Vol. 2. Numerical Algebra. Basel:

Birkhiiuser (also New York: Academic Press 1977).

Index

In general, page numbers in italics refer to definitions.

Abramowitz 177 absolute error 12, 327 absolute norms 447 absolutely continuous 92, 98, 588 absolutely stable 527 Achieser 92 A-conjugate 353, 660 Adams-Bashforth methods 493ff Adams-Moulton methods 494ff, 513 ADI methods 647ff, 713

optimal parameters of 654ff Ahlberg 99 Aitken's Ll.2-method 184, 344ff, 359 Aitken's interpolation algorithm 43,

344 algebraic multiplicity 366, 370ff algorithm 9

numerically more trustworthy 18 numerically stable 19 well behaved 19

amplification of errors 14 analog computers 2 Andersen 190 approximation error approximation method 94 arithmetic,

complex 333 fixed point 3 floating-point 4 floating-point 3ff floating-point 7 interval 28 statistical error 30ff

Armijo line search 306 Arnoldi 658, 668 artificial variables 270 Ashenhurst 4

732

asymptotic expansion 160, 165f, 169 of central difference quotient 166 of global discretization error 480ff,

513ff attenuation factors 94 Axelsson 620, 662, 665, 717

Babuska 563 back substitution 191

round-off errors in 221 backward analysis of errors 19 backward deflation 325 Bader 529 badly conditioned 337 Bairstow's method 333ff, 360 Banachiewicz' method 198 band matrix 199, 593ff Bank 532 Baptist 357 Barker 272 Bartels 266 Barth 406 Barton 539 basic (feasible) solution 258 basic variables 258 basis inverse method 266 basis of a linear program 258 Bauer 19, 161, 165, 316, 443, 447 BDF-methods 531, 535 Bendixson. theorem of 454 benign 19 Bernoulli numbers 156, 158 Bernoulli polynomials 158 Bessel functions 135 BFGS method ,'151. 355f Bi-CG algorithm 6.58, 686ff

Index

Bi-CGSTAB algorithm 620, 658, 686ff, 717ff

bidiagonal matrix 436ff biharmonic equation 701 bilinear form 587ff, 601 biorthogonal vectors 681 biorthogonalization 658, 680 bisection method 179, 331, 332, 339,

406, 543 bit-cycling permutation 139 bit-reversal permutation 85, 139 Bjorck 232 block Gauss-Seidel method 646 block iterative methods 645 block Jacobi method 645 block relaxation methods 646 block single-step method 646 block total-step method 645 Bloomfield 81 Bock 539 boundary conditions 466

nonlinear, separated 539540 boundary-value problems 97, 539, 550,

562 linear 548 model problem for 650ff, 655 singular 563, 564 with free boundary 542, 568

Bowdler 435 Boyd 596 Braess 620, 702, 711 Bramble 620, 702 Brandt 702, 711 Brent 344 Brezinski 167, 344 Briggs 702 Brigham 81 Broyden 313ff, 350f Broyden's rank-one method 313, 315,

351, 561 Broyden, Fletcher, Goldfarb, Shanno

(see BFGS) 351 B-splines 111, 106, 142, 595 Buchauer 538 Bulirsch 73, 90, 106, 163ff, 167f, 183,

524, 532, 557, 597 Bunch 190 Buneman 620, 692, 697, 699,

algorithm of 620, 691ff, 699, 713ff, Butcher 477 Buzbee 692, 699 Buzbee 701 Byrne 531, 535

Bohmer 97

Callies 572f, 577 cancellation 8, 14 Canuto 596 Caracotsios 539 Cauchy's convergence criterion 293 Cayley-Hamilton Lemma 374 central difference quotient 166 Chan 701 characteristic polynomial 179, 317f,

331, 338, 366ff, 406ff, 454, 503 Chebyschev polynomials 178, 663 Chebyshev system 175

733

Choleski decomposition 204f, 207, 234, 272, 352, 441, 585, 591, 642, 649

Choleski factor 205 Chui 133 Ciarlet 59, 594, 606 Clark 525 clique representation 279 clique 275 Coddington 470 Collatz 213, 302, 475, 577 collocation method 595 collocation point 595 column sum criteria 626f complete pivot selection 219f complex arithmetic 333 complex conjugate roots 333 composite of integration rule 149 computers (digital, analog) 2 cond 211 condition, condition

of linear equations 207 of matrix 211 of eigenvalue problem 386ff, 412,

447f, of linear least squares 236

condition numbers 13, 355 conjugate-gradient method 620, 657f,

716ff consistency condition 534 consistent initial values 534 consistent matrix norm 209 consistently ordered 633, 634 constrained minimization 290 continuation methods 562ff continued fraction 66 contractive mapping 296, 348 convergence factor 294

734

convergence alternating 295 global 294 linear 294, 340 local 294 monotone 295 of order p 293 quadratic 294, 295, 299, 320, superlinear 356 cubic 430

convex 299 Cooley 81 Cooley-Thkey method 81, 139f corrector method 494, 509, 512, 518ff Cotes 146 Courant, Weyl, theorem of 453 covariance matrix 234 Cramer's rule 407 Crane 525 Crout's method 197 cubic convergence 430 cubic splines 38, 97ff, 10l, lO7, 592 Cullum 402 Curry 115 curvature 101

Dahlquist 512, 530 damping of errors 14 Daniel 248, 250 Dantzig 256, 259, 270 data fitting 231 Daubechies 132, 134 Davenport 525 Davidon 350, 356 Davidon, Fletcher, Powell (see DFP)

351 Davis 38, 146, 151 de Boor 97, lO6, 121 decimal number representation 2 decomposition

QR 228 triangular 195

decomposition of maps 10 deflation, 325, 344

backward 325 forward 325

degenerate linear program 259 degree of a node 275 Dekker 339 ~2-method of Aitken 184, 344ff, 359 Dennis 315 derogatory matrix 372

descent direction 349 design point 575 Deuflhard 529, 532, 535, 563 DFP method 351, 357

Index

diagonal matrix 216, 372, 395, 462 diagonalizable matrix 372, 381ff diagonally dominant 626

irreducibly 627 Diekhoff 525 difference equations 130, 500ff

linear 501ff stability condition for 502

difference methods 619 for ordinary differential equations

582ff, 596, 598 for partial differential equations

639ff difference operator 345 difference quotient,

central 166, 170 one-sided 166,

differential equations, implicit 531ff ordinary 97, 465f, 596 partial 97, 639ff

differential error analysis 11 differential operator,

positive definite 588 symmetric 588, 600

differential-algebraic equations 531ff digital computers 2 direct methods for linear equations

190 Dirichlet boundary-value problem 600,

639 discrete Fouriertransformation 78 discretization (discretizing) 166ff discretization error 1, 480f

global 477ff, 513 local 474, 497

divided differences 43, 45, 117 ,341f, divided-difference scheme 45 dividing off 325 Dixon 355 domain decomposition method 701 Dongarra 190 DOPRI 5(4)-method 492 Dormand 490 Dorr 701 double-step method 320ff, 326 Drazin-inverse 534 dual function 126 Duff 272, 280

Index

Eberlein 397 Eich 537 eigenproblem, generalized 356 eigensolution 364 eigenvalue problem 364

algebraic 364 for differential equations 541 generalized 440, 534

eigenvalues 179, 317, 331, 364ff, 366 algebraic multiplicity of ff 366,

370ff, exclusion theorems for 441 extremal properties for 453 eigenvalues, geometric multiplicity

of 367, 370ff, ill-conditioned 412, 450 multiplicity of S69ff numerically acceptable 411 sensitivity of 447 well-conditioned 411, 450

eigenvector 179, 365ff eigenvector 365 S66ff

left 868, 447ff numerically acceptable f 411, 413f

Eisenstat 280 elementary

divisors 372, 411ff, 374, 450 inequalitics 256 maps 10 operations 9

elimination, Gaussian 190 Gauss-Jordan 200

elimination for sparse matrices 272 elimination matrix 248 elimination methods 619 end corrections 151 Engl538 Enright 525, 531 equilibrated matrix 192, 217 Erdelyi 160 Erisman 272, 280 error analysis,

backward 19 differential 11 for linear equations 207

error bounds 207 for spline interpolation 109

error damping 14 error propagation 9 error

absolute 12 discretization 474, 477ff, 480

error [cant 1 inherent 18 interpolation 48ff relative 5, 12

735

Euclidean (vector) norm 208, 224, 355, 387

Euclidean algorithm 329 Euler's method 480, 472, 483, 51:3,

526ff implicit 527ff, 5:~0 modified ff 475ff region of absolute stability for ,528ff

Euler-Maclaurin summation formula 156ff, 150, 160

exact line search 306, 353ff exclusion theorems 441 exponent 3

over (under) flow 6 exponential interpolation 38 exponential spline 143 extended integration rules 149 extrapolation methods 145, 157

for initial value problems 521ff, 544 for initial value problems 544 , for differentiation 166 , for integration160ff, 181 , for differential equations 528f, 483,

527, 597 extrapolation 51

factorization, numerical 274 symbolic 274

fast Fourier transforms 80ff feasi ble point 256 Feehery 539 Fehlberg 477,490 Fellcn 525 field of values 452 Fike 443,447 fineness 107 finite element method 600, 603, 606,

619 Fix 606 fixed point 293 fixed point theorem of Banach 297 fixed-point representation 3 Fletcher 350, 658, 686 floating-point arithmetic 3ff, 7, 15ff,

483ff floating-point operations 7

736

floating-point representation 8 normalized -4 un normalized -4

formula of Lagrange 52 Forsythe 623 forward deflation 325 Fourier coefficients 92 Fourier methods 620, 691 Fourier series 92. 138 Fourier synthesis 71'" 1',8 Fourier transform 80ff Fox 525 fradional iteration 410 Francis 365,415,417,433

observation of 440ff Frazer, Duncan, Collar, method of ;n9 free variables 257 Freund 620, 658. 684 Frobenius matrices 192, 194. 875,

376ff 454, Frobenius normal form 375f, 377, 398,

503, fully nondegenerate rational interpola­

tion problem 63 function, differentiable 298

Galan 539 Gantmacher 534 Garbow 365 Gass 256, 270 Gauss' arithmetic-geometric mean al-

gorithm 654 Gauss-Jordan method 200 Gauss-Newton method 243 Gauss-Seidel method 622, 628f, 630f,

637. 641ff, 703, 705, 713ff Gaussian

elimination 106, 121, 190f, 215, 560, 416

integration (quadrature) 146, I'll, 181. 184

Gautschi 95, 178 Gear 467, 521, 531, 535 generalized

divided differences 56 Lagrange polynomials 52 eigenproblem 356, 534 inverse 243ff

Gentleman 81 geometric multiplicity 867, 370ff geometrically decreasing step lengths

169

George 272, 278. 280, 620 Gershgorin's theorem 444, 583 Gill 248, 316, 537 Givens 394f

matrix 249 reflection 437ff. 250 rotation 250, 394, 672 method of 394ff

Index

global discretization error -477ff, 498, 480f, 513

Glowinski 701 GMRES algorithm 620, 658, 667, 717ff

incomplete 677f, 717ff quasi-minimal 678 with restarts 677, 717ff

Goertzel88 Goldfarb 351 Goldstein 7 Golub 178ff, 190, 248, 316, 365, 424,

436, 440, 692, 699, 701 Golub and Reinsch, method of 365,

436 Gordon 467, 521 Gottlieb 596 gradient 303 Gragg 248, 250, 481, 516f Gragg's function 521 Gram-Schmidt orthogonalization 173,

223, 228, 669, 676 with reorthogonalization 230, 236

graph of matrix 626 (weakly) 2-cyclic 635, 642 connected 626, 642

graph of symmetric matrix 274 great-circle distance 35 Green's formula 601 Greville 97, 106 grid function 702 Griepentrog 535 Griewank 313 Grigorieff 467, 477. 526, 531 Grimm 557, .566, 598 Gropp 701 Grossmann 232 Grobner 145 Guest 2.32

Haar condition 175 Haar-function 128 Ham-wavelet 133 Hackbusch 620, 702, 711 Hadley 256, 270477. 481

Index

Hairer 467, 477, 481, 516, 521, 531£, 535,539

Hall 109 Handbook of Mathematical Functions

177 Hanson 232 harmless roundoff errors 19 hat-function 125 Hausdorff 452 Heim 538 Helmholtz equation 701 Henrici 151, 317, 328, 467f Hermite function space 58 Hermite

interpolation 48,51,117,148,180 polynomials 178, 184 function space 594

Hermitian matrix 204, 365, 379ff, 453, Herriot 106 Hessenberg matrix 251, 219, 365, 387,

392, 402ff, 407, 415f, 425ff irreducible 426gg

Hessian matrix 316, 349 Hestenes 567, 620, 657f Heun, method of 475ff Hiltmann 538, 572 Himmelblau 290 Hindmarsh 531, 535 Hirsch 442, 454 Hockney 701 Hofreiter 145 Holladay's identity 99 homogeneity axiom 208 homotopy methods 562ff, 565 Horneber 532 Horner scheme 44, 88, 317 Householder 224, 320, 447, 451, 631

method of 388, 395, 434, 394ff, 436 matrix 224 orthogonalization 223 reduction 392

Householder transformation, for arbitrary matrix 392 for symmetric matrix 391

Hull 525, 531 Hyman, 407

method of 407

ill-conditioned 13 implicit differential equations 531ff implicit methods for differential equa-

tions 494, 527f

implicit shift techniques 433ff improper integrals 184 inaccessible points 62 inclusion theorems 442 inclusion theorems 450

737

incomplete Choleski factorization 665f index of nil potency 533 index-1 assumption 535 inequalities,

elementary 256 linear 256

inexact line search :306 initial (starting) values

rules for the selection of 319 initial value problem 465, 468, 471 integral

definite 145 improper 184 indefinite 145

integration by parts 93, 157ft error 151ff

integration rules of Simpson 152 composite (extended) 149 of Gauss 171, 181 of Milne 148, 163 of Newton Cotes 147, 162, 181, 493,

497 of Simpson 148, 162 of Weddle 148 3/8 148

interpolation 37ff exponential 38 Hermite 51ff, 180 inverse 344 linear 37 polynomial 38ff rational 38, 59ff spline 38, 97ff trigonometric 37, 74ff

interpolation error 48ff interpolation formula

of Lagrange 39, 146, 167 of Newton 43ff, 342

Interpolation on product spaces 135 interpolation operator 706 interval arithmetic 28 inverse differences 64ff, 65 inverse interpolation 344 inverse iteration 420 inverse iteration of'Vielandt 405, 408,

410, 414, 420, 427f, 435,

738

inverse of matrix 190, 200 irreducible matrix 405, 407, 626ff, 631,

635 irreducibly diagonally dominant 627 iteration function 290, 317 iterative methods 639

construction of 290ff, 621ff convergence of 293ff,349ff, 623ff convergence of 349ff for linear equations 619ff for minimization 290f, 302ff, 349 for roots of polynomials 316 for zeros of functions 289ff, 338ff

iterative refinement 619, 623

Jacobian elliptic function 546, 556ff Jacobian matrix 12, 14ff, 242, 293,

298, Jacobi matrix 388 Jacobi method 394ff, 622, 625f, 629,

631, 641, 643, 703, 713ff damped 704

Jenkins 317 Jordan block 371 Jordan normal form 369ff, 442

Kahan, Theorem of 631 Kaniel401 Kantorovich

(see Newton-Kantorovich) 302 Kaps 529f Karlin 121 Kaufman 248, 250 Keller 467, 550, 557 Keyes 701 Kiehl 538 knots of splines 97 Korneichuk 97 Kramer 539 Krogh 521 Kronrod 181 Kronseder 538 Krylov sequence 375, 398 Krylov sequence 375, Krylov space 398, 657 Krylov space methods 619, 657, 680,

716ff, Kroner 538 Kublanovskaja 415 Kulisch 30 Kutta 475

Index

Liiuchli 239 Lagrange's interpolation formula 39,

146, 167, 493 Lagrange polynomials 39, 134 Laguerre polynomials 178, 184 Lambert 531 Lanczos 398, 658, 681, 686 Lanczos algorithm 398ff Laplace-Operator 600 Lawson 232 least upper bound norm 210 least-squares problem 232, 236, 666

nonlinear 241 left eigenvector 368, 447ff left preconditioning 679 Legendre polynomials 177 Leibniz formula 117 Leis 539 Levinson 470 Lindberg 531 line search 349, 352ff, 356ff

Armijo 306 exact 306, 353ff inexact 306

linear approximation method 94 linear convergence 294, 340 linear equations,

direct methods for 190 elimination methods for 190 iterative methods for 619ff positive definite systems of 204

linear interpolation problem 37 linear least squares problem 232, 235 linear multistep methods 508ff linear programming 256 linearization 292 linked list 273 Lions 606 Lipschitz condition 356, 480, 467 Lipschitz continuous 356, 505 Liu 272, 278, 280 local discretization error 474, 497 local minimum 353 Lory 525 Louis 132, 134 LR method 365, 405, 415ff lub 210 Lubich 481, 516, 532 Lucnbcrgcr 290, 350

machine numbers 4 machine precision 5, 89, 499 Maehly 325

Index

Maehly's modification of Newton's method 325, 406

Miirz 535 Mallat 134 mantissa 3 Marden 320, 392, 403, 406, 435, 441 matrix norms 209 matrix pencil 534 matrix

band 199, 593f bidiagonal 436ff block tridiagonal 634 block tridiagonal 647 consistently ordered ff 633, 634 derogatory 372 diagonalizable 372, 381 diagonally dominant 626f diagonal 372, 395, 462 Frobenius 375ff, 454, 503 Givens 249 Hermitian 204, 224, 365, 379ff, 453 Hessenberg 251, 219, 251, 387, 365,

392, 407, 415f, 402ff Householder 388, 436ff irreducible 405, 407, 626ff, 631, 635 Jacobi 388 nonderogatory 372ff, 376 normal 365, 379, 381, 450f normalizable 372, 446 orthogonal 211 permutation 192 positive (semi) definite 204, 353,

381, 649 sparse 272, 619f, 641 symmetric 204, 331 triangular 191, 221 tridiagonal 220, 338, 365, 388, 415f unitary 211, 224, 379ff upper Hessenberg 219, 387 with property A 633ff, 642

maximum norm 208 Meijerink 665 Merten 532 method of Jacobi 395 Metropolis 4 Meurant 701 Meyer 109 midpoint rule 476, 495, 497, 521

modified 521 Milne,

corrector method of 495, 513 Milne-Thompson 67 minimal degree algorithm 277, 275

minimal polynomial 372ff minimization,

739

constrained, linear 256 constrained, nonlinear 290 unconstrained, general 290, 302ff,

349 minimum point 233, 241, 289, 349

local 349, 353 minimum-norm property 100 M-matrices 665 model problem 639, 650ff, 655ff, 699,

712 modification of matrix decomposition

247 modified Newton method 302ff Moler 190, 441, 623 moments of cubic splines 102 Moore 30 Moore-Penrose inverse 243ff Morrison 539 More 315 Muller's method 342ff multi-resolution

algorithm 128 analysis 121 methods 121ff, 124f

multigrid methods 620, 701f, 711 multigrid V-cycle 711 multiple root 342 multistage Runge-Kutta methods 476 multistep methods 492ff, 495ff, 527,

530 consistency of 497, 509ff convergence of 498, 504, 509ff linear 508ff of order p 497, 511 step control for 517ff weakly unstable 517

Murray 248, 316, 537 Murty 256, 270

Na 563 Nachtigal 620, 658, 684 National Bureau of Standards 177 natural spline 101 neighboring basis 259 Neville type algorithm 68ff Neville's interpolation algorithm 40ff,

161, 344 Newton-Cotes integration formulas

147, 162ff, 181, '193, 497 Newton-Kantorovich, Theorem of 302 Newton-Raphson method 291

740

Newton's interpolation formula 44ff, 342ff, 503, 518

Newton's method approximate 548 for minimization 302ff for roots of polynomials 316 for zeros of functions 291ff, 298ff for systems of equations 559 general 546, 577 Maehly's version of 325, 406 modified 302ff, 548, 559, 561f, 571

Ng 280 Nickel 317 Nielson 692, 699 Nilson 99 nonbasic variables 258 nondegenerate linear program 259 nonderogatory matrix 372ff, 376 norm(s) 207

absolute 447 equivalence of 209 consistent 209 Euclidean 208 maximum 208 Sobolev 601, 606 submultiplicative 209 subordinate 210

normal equations 232, 239, 243 normal matrix 381, 365, 379, 450f normalizable matrix 372, 446 normalized floating-point representa-

tion 4 numerical factorization 274 numerically more trustworthy 18 numerically stable 19, 88f, 697 Numerov 586 Nystrom's predictor method 495 N0rsett 477, 521, 539 Nurnberger 97

Oberle 525, 557, 566, 598 objective function 256 Oden 606 Oettli 213, 223 O'Leary 701 Olver 160 one-step methods 473, 471

consistency of 474 convergence of 477ff of order p 474 step control for 485ff, 490ff

operand set 9

Index

opposite terms 82 optimal block relaxation methods 647 optimal parameters of ADI methods

654ff optimal solution of linear program 257 optimum point 257 ordinary differential equations 465ff

boundary-value problem for 466 initial-value problem for 465 of first order 466ff of mth order 466ff systems of 465ff

Oren 350, 355 Oren-Luenberger class 350,356ff Orszag 596 Ortega 290, 293, 302 orthogonal

polynomials 172ff projector 244

orthogonality relation 138 orthogonalization

to solve least-squares 235 Gram-Schmidt 223, 228 Householder 223

orthonormal wavelet 133 Osborne 557 Ostrowski 344

Reich, theorem of 631 overflow of exponent 6 overrelaxation 630, 642 overshooting 321, 325f

Paige 401f, 679 Parlett 365, 397, 402, 424 partial pivot selection 192, 194, 201£,

220,415 partial pivot selection 220 415 partition of unity 114 Peaceman and Rachford,

ADI method of 647ff formula of 724f

Peano kernel 152 Peano's error representation 151ff Periaux 701 permutation matrix 192, 633 perturbation of polynomial coefficients

335 Pesch 525 Peters 325f, 328, 338f, 415, 435, 441 Petzold 532, 535 phase matrix 418 phase polynomial 75

Index

phase, one of simplex method 260, 268f two of simplex method 260

piecewise polynomial function 111 ff Piessens 181 Pinkus 121 pivot clique 277 pivot element 192

complete selection of 192, 220 partial selection of 192,202, 194,

220 selection 192, 215

pivot vertex 276 Poisson equation 691, 701 discretized 691 polygon method 472 polynomial

characteristic 318ff, 331, 338 366ff, 406, 454

minimal 372ff polynomial interpolation ff 37ff polynomials

Bernoulli 158 Chebyshev 178 Hermite 178 Lagrange 39 Laguerre 178, 184 Legendre 178

Poole 424 positive (semi) definite 204, 353, 381,

631,649 Powell 350, 356 Prager 213, 223, 563 preconditioned conjugate gradient

method 664 preconditioner 664 preconditioning techniques 664, 679 predictor methods 494, 509, 512, 518ff Prince 490 principal vector 371 product rule 117 projection operator 705 property A 633 Proskurowski 701 pseudoinverse of matrix 243ff, 385

QMR algorithm 620, 658, 680, 685, 717ff

QR-decomposition 228 QR method 179f, 365, 405, 407, 415,

417ff, 436, 440 convergence of 430

Q R method [cont 1 implicit shift techniques ff 433ff practical realization of 425ff with shifts 428, 430

QZ method 441

741

quadratic convergence 294, 295, 299, 320

quadratic function 353 quadrature (Gaussian, numerical) 171,

146 Quarteroni 606, 620 quasi-minimal residual method 658 quasi-Newton

equation 350 methods 350ff

quasilinearization 577

Rabinowitz 146, 151 Rachford 724f Rademacher 30 random variable 234 rank-one modification 250, 351 rank-two method modification 351 rational

interpolation 38, 59ff normal form 375

rational expressions 60 equivalence of 61 relatively prime 61 normal form 375f

Rayleigh quotient 451 Rayleigh-Ritz method 603 Rayleigh-Ritz-Galerkin method 97,

586 reciprocal differences 64, 67 Reddy 606 reduced costs 261 reduction method 692 region of absolute stability 528 regula falsi 339ff, 358 regular matrix pencil 534 Reid 272, 280, 662 reinitialization 315 Reinsch 90, 95, 106, 190, 232, 266,

365, 392, 435f, 440, 623, 662 relative error 5, 12 relatively prime 61 relaxation methods 629ff, 630, 643,

713 convergence of 631

optimal 647

742

relaxation parameters 630 optimal 637

Rentrop 525, 529f, 532, 535 reorthogonalization 230 residual 211, 233 restricted variables 257 Reutersberg 701 Rheinboldt 290, 293, 302 Riesz-basis 122 right eigenvector 366, 447f right preconditioning 679 Roche 532, 535 Romberg 161, 163, 167, 171 Romberg's integration method 145,

161ff, 167 roots of polynomials 175, 289, 316

complex (conjugate) 333, 342 multiple 323, 342 simple 323

Rose 272, 275, 277 round-off error distribution 30 round-off error,

in back-substitution 221 in Gaussian elimination 21,5 in linear least squares 238 in one-step methods 48:3ff

rounding 5 roundoff errors 1. 4 row sum (matrix) norm 209 row sum criteria 625 f Rozenvasser 539 r-step method 495

linear 495 Runge 475 Runge-Kutta methods 475, 476, 482,

487ff, 515 Runge-Kutta-Fehlberg method 488,

527 Rutishauser 3, 73, 106, 161, 165, 286,

365, 396f, 415 Rutishauser's semilogarithmic nota­

tion 3

Saad 401, 620, 658, 667 Sande 81 Sande-Tukey method 81, 139 Sargent 539 Saunders 248, 316, 679 Sautter 222 scaling 216, 462 scaling function 122 Scarborough 5

Schloeder 539 Schoenberg 115, 121, 150 Schrijver 256 Schroder 701 Schultz 59, 97, 594, 620, 667, Schulz 539, 658 Schulz' method 359 Schumaker 97 Schur 379 Schur normal form 379 Schur's theorem 379 Schur-Norm 210 Schwarz 286, 397, 606 Schwarz inequality 589, 602 Scott 402 search direction 302 Seber 232 secant method 341 ff Secrest 172 Sedgwick 525 semi-implicit midpoint rule 529 semilogarithmic notation 3 sensitivity analysis 538 sensitivity equations 538 sensitivity to input perturbation

of linear equations 211 of linear least squares 236 of polynomial roots 335ff

Seydel 525 Shampine 467, 521, 525 Shanks 477 Shanno 351 shift paramcter 427 shift tcchniques 427ff shooting methods,

Index

internal subdivision in multiple 561 limiting case of multiple 577ff

multiple ff 557ff, 563ff, 577ff, 595ff simple 542ff,548, 552ff, 561ff, 569,

596f starting trajectory 561ff, 568f

sign changes 328 signal processing 97 significant digits (bits) 4 similarity transformation 368, 386 simplex method 256

general step 260 phase one 260, 268f phase two 260

Simpson's rule 147, 148, 150, 152, 476, single-step method 622 singular value decomposition 247, 384,

436ff

Index

singular values 382ff, 379, 436ff slack variable 256 Smith 365, 392, 403, 415, 435 Sobolev norm 601,606 Sonneveld 687 SOR methods 630

optimal 714ff sparse matrix 272, 619f, 641 spectral method 595 spectral radius of matrix 442, 624, 585 Spedicato 355 spline

convergence of 107ff cubic 97ff, 107, 592 interpolation 98 natural 101

spline function of degree k 111 spline interpolation 38, 97ff spline of degree k 106 SSOR matrix 665, 716 stability condition 502, 504ff, 509,

511ff stable, numerically 19, 88f, 697 stationary point 305f steepest descent method 659 Steffensen 147, 155 Steffensen's convergence acceleration

346ff, 359 Stegun 177 Stein, Rosenberg, theorem of 629 Steinebach 532, 535 step length, size 472, 467

sequences for extrapolation methods 163, 183

Sterbenz 1, 30 Stetter 467, 477 Stewart 190, 248, 250, 285, 441, 539 Stiefel 161, 165, 286, 397, 620, 657f stiff differential equations 525ff Stoer 90, 164, 167f, 183, 357, 447, 524,

597, stopping criterion for extrapolation,

method 164, 171 storage technique 272 Strang 606 Stroud 172 Sturm sequence 328, 406 Stormer 586 submultiplicative norm 209 subordinate norm 210 subspace iteration 419 summation formula of Euler and

Maclaurin 145, 156ff, 160

superlinear convergence 356 support

abscissas 37 ordinates 37 points 37

Swarztrauber 701 switching function 575, 537 switching point 536 symbolic factorization 274 symmetric matrix 3:11

743

symmetric rank-one method of Broy-den 351

SY M M LQ method 679 system of linear equations 190 Szego 179

Tang 563 Taylor's theorem 108 Tewarson 272 Thiele's continued fraction 68, 64 Tornheim 360 total-step method 6:22 transition function 575, 536 translation invariant 94 trapezoidal

rule for integration 148ff, 530 sum 92, 149, 157, 160ff, 170, 183,

476 Traub 290, 317, 342 triangle inequality 208 triangular decomposition 195, 645 triangular decomposition 645

by direct computation 190, 197 by Gaussian elimination 190 by orthogonalization 226s 235 positive definite case 205

triangular matrix 191, 221 triangulation 603 tridiagonal matrix 178, 220, 318, 338,

365, 388, 405 tridiagonalization 388 trigonometric

expression 74 interpolation 37 polynomial 595

Troesch 554 Trottenberg 70lf truncation error 1 trustworthy, numerically 18 Tukey 81 two-grid method 708 two-scale-relation 122, 128

744

unconstrained minimization 290, 302ff, 349ff

underflow of exponent 6 underrelaxation 630 unitary matrix 379ff, 211 upper Hessenberg matrix 251, 387

Valli 606, 620 van der Vorst 620, 658, 665, 687, 691 van Loan 190, 365, 424 Varga 59, 594, 620, 629, 631, 633, 647,

651, 653f, 721, 724, 726 , Young, theorems of 636ff

variable order method 520 variance 234 variational methods,

for ordinary differential equations ff 586ff, 596, 599

for partial differential equations 600 V-cycle, of multigrid method 711 vector iteration 452, 621

simple 405ff, 408, 410, 418 Wielandt's inverse 405,410,408,

414ff, 418, 427, 435, virtual abscissae 53 Vitasek 563 v. Neumann 7 von Stryk 538

Wachspress 654 Wagschal606 Walsh 99

Wanner 477, 521, 531, 535, 539 Watts 525 wavelet 133 weight function 171 weights,

in Gaussian quadrature 175 in Newton-Cotes formulas 146

Weinstein, theorem of 451 well behaved 19 well-conditioned 13 Welsch 178ff Whitney 121 Widlund 701

Index

Wilkinson 19, 190, 198, 219f, 232, 238f, 266, 325f, 328, 337ff, 365, 392, 397, 403, 406, 415, 422, 430, 433, 435, 441, 534, 623, 662

Willoughby 272, 402, 525, 531 Wittmeyer 621 Witzgall 447 word length, double, triple 3 Wright 537

Young 620, 633, 647, 653f, 725 , Varga, theorems of 636ff

Zaglia 167, 344 zero of function 289ff, 338ff zero suppression 326, 344 Zlamal 604, 606 Zugck 532, 535

Texts in Applied Mathematics

(continued from page ii)

31. Bremaud: Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. 32. Durran: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. 33. Thomas: Numerical Partial Differential Equations: Conservation Laws and Elliptic

Equations. 34. Chicone: Ordinary Differential Equations with Applications. 35. Kevorkian: Partial Differential Equations: Analytical Solution Techniques, 2nd ed. 36. Dullerud/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. Brauer/Castillo-Chavez: Mathematical Models in Population BIOlogy and

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