Numerical Mathematics Alfio Quarteroni Riccardo Sacco Fausto Saleri Springer Texts in Applied Mathematicsm 37 Editors J.E Marsden L Sirovich M Golubitsky W Jäger Advisors G Iooss P Holmes D Barkley M Dellnitz P Newton Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo Alfio QuarteroniMMRiccardo Sacco Fausto Saleri Numerical Mathematics With 134 Illustrations 123 Alfio Quarteroni Department of Mathematics Ecole Polytechnique MFe ´rale de Lausanne ´de CH-1015 Lausanne Switzerland alfio.quarteroni@epfl.ch Riccardo Sacco Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 20133 Milan Italy ricsac@mate.polimi.it Fausto Saleri Dipartimento di Matematica, M“F Enriques” Università degli Studi di MMilano Via Saldini 50 20133 Milan Italy fausto.saleri@unimi.it Series Editors J.E Marsden Control and Dynamical Systems, 107–81 California Institute of Technology Pasadena, CA 91125 USA L Sirovich Division of Applied Mathematics Brown University Providence, RI 02912 USA M Golubitsky Department of Mathematics University of Houston Houston, TX 77204-3476 USA W Jă ger a Department of Applied Mathematics Universit a t Heidelberg ă Im Neuenheimer Feld 294 69120 Heidelberg Germany Mathematics Subject Classification (1991): 15-01, 34-01, 35-01, 65-01 Library of Congress Cataloging-in-Publication Data Quarteroni, Alfio Numerical mathematics/Alfio Quarteroni, Riccardo Sacco, Fausto Saleri p.Mcm — (Texts in applied mathematics; 37) Includes bibliographical references and index ISBN 0-387-98959-5 (alk paper) Numerical analysis.MI Sacco, Riccardo.MII Saleri, Fausto.MIII Title.MIV Series I Title.MMII Series QA297.Q83M2000 519.4—dc21 99-059414 © 2000 Springer-Verlag New York, Inc All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or herafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone ISBN 0-387-98959-5nSpringer-VerlagnNew YorknBerlinnHeidelbergMSPIN 10747955 Preface Numerical mathematics is the branch of mathematics that proposes, develops, analyzes and applies methods from scientific computing to several fields including analysis, linear algebra, geometry, approximation theory, functional equations, optimization and differential equations Other disciplines such as physics, the natural and biological sciences, engineering, and economics and the financial sciences frequently give rise to problems that need scientific computing for their solutions As such, numerical mathematics is the crossroad of several disciplines of great relevance in modern applied sciences, and can become a crucial tool for their qualitative and quantitative analysis This role is also emphasized by the continual development of computers and algorithms, which make it possible nowadays, using scientific computing, to tackle problems of such a large size that real-life phenomena can be simulated providing accurate responses at affordable computational cost The corresponding spread of numerical software represents an enrichment for the scientific community However, the user has to make the correct choice of the method (or the algorithm) which best suits the problem at hand As a matter of fact, no black-box methods or algorithms exist that can effectively and accurately solve all kinds of problems One of the purposes of this book is to provide the mathematical foundations of numerical methods, to analyze their basic theoretical properties (stability, accuracy, computational complexity), and demonstrate their performances on examples and counterexamples which outline their pros viii Preface and cons This is done using the MATLAB software environment This choice satisfies the two fundamental needs of user-friendliness and widespread diffusion, making it available on virtually every computer Every chapter is supplied with examples, exercises and applications of the discussed theory to the solution of real-life problems The reader is thus in the ideal condition for acquiring the theoretical knowledge that is required to make the right choice among the numerical methodologies and make use of the related computer programs This book is primarily addressed to undergraduate students, with particular focus on the degree courses in Engineering, Mathematics, Physics and Computer Science The attention which is paid to the applications and the related development of software makes it valuable also for graduate students, researchers and users of scientific computing in the most widespread professional fields The content of the volume is organized into four parts and 13 chapters Part I comprises two chapters in which we review basic linear algebra and introduce the general concepts of consistency, stability and convergence of a numerical method as well as the basic elements of computer arithmetic Part II is on numerical linear algebra, and is devoted to the solution of linear systems (Chapters and 4) and eigenvalues and eigenvectors computation (Chapter 5) We continue with Part III where we face several issues about functions and their approximation Specifically, we are interested in the solution of nonlinear equations (Chapter 6), solution of nonlinear systems and optimization problems (Chapter 7), polynomial approximation (Chapter 8) and numerical integration (Chapter 9) Part IV, which is the more demanding as a mathematical background, is concerned with approximation, integration and transforms based on orthogonal polynomials (Chapter 10), solution of initial value problems (Chapter 11), boundary value problems (Chapter 12) and initial-boundary value problems for parabolic and hyperbolic equations (Chapter 13) Part I provides the indispensable background Each of the remaining Parts has a size and a content that make it well suited for a semester course A guideline index to the use of the numerous MATLAB Programs developed in the book is reported at the end of the volume These programs are also available at the web site address: http://www1.mate.polimi.it/˜ calnum/programs.html For the reader’s ease, any code is accompanied by a brief description of its input/output parameters We express our thanks to the staff at Springer-Verlag New York for their expert guidance and assistance with editorial aspects, as well as to Dr MATLAB is a registered trademark of The MathWorks, Inc Preface ix Martin Peters from Springer-Verlag Heidelberg and Dr Francesca Bonadei from Springer-Italia for their advice and friendly collaboration all along this project We gratefully thank Professors L Gastaldi and A Valli for their useful comments on Chapters 12 and 13 We also wish to express our gratitude to our families for their forbearance and understanding, and dedicate this book to them Lausanne, Switzerland Milan, Italy Milan, Italy January 2000 Alfio Quarteroni Riccardo Sacco Fausto Saleri Contents Series Preface v Preface vii PART I: Getting Started Foundations of Matrix Analysis 1.1 Vector Spaces 1.2 Matrices 1.3 Operations with Matrices 1.3.1 Inverse of a Matrix 1.3.2 Matrices and Linear Mappings 1.3.3 Operations with Block-Partitioned Matrices 1.4 Trace and Determinant of a Matrix 1.5 Rank and Kernel of a Matrix 1.6 Special Matrices 1.6.1 Block Diagonal Matrices 1.6.2 Trapezoidal and Triangular Matrices 1.6.3 Banded Matrices 1.7 Eigenvalues and Eigenvectors 1.8 Similarity Transformations 1.9 The Singular Value Decomposition (SVD) 1.10 Scalar Product and Norms in Vector Spaces 1.11 Matrix Norms 1 7 10 10 11 11 12 14 16 17 21 References 639 [Sch81] Schumaker L (1981) Splines Functions: Basic Theory Wiley, New York [Sel84] Selberherr S (1984) Analysis and Simulation of Semiconductor Devices Springer-Verlag, Wien and New York [SG69] Scharfetter D and Gummel H (1969) Large-signal analysis of a silicon Read diode oscillator IEEE Trans on Electr Dev 16: 64–77 [Ske79] Skeel R (1979) Scaling for Numerical Stability in Gaussian Elimination J Assoc Comput Mach 26: 494–526 [Ske80] Skeel R (1980) Iterative Refinement Implies Numerical Stability for Gaussian Elimination Math Comp 35: 817–832 [SL89] Su B and Liu D (1989) Computational Geometry: Curve and Surface Modeling Academic Press, New York [Sla63] Slater J (1963) Introduction to Chemical Physics McGrawHill Book Co [Smi85] Smith G (1985) Numerical Solution of Partial Differential Equations: Finite Difference Methods Oxford University Press, Oxford [Son89] Sonneveld P (1989) Cgs, a fast lanczos-type solver for nonsymmetric linear systems SIAM Journal on Scientific and Statistical Computing 10(1): 36–52 [SR97] Shampine L F and Reichelt M W (1997) The MATLAB ODE Suite SIAM J Sci Comput 18: 1–22 [SS90] Stewart G and Sun J (1990) Matrix Perturbation Theory Academic Press, New York [SS98] Schwab C and Schătzau D (1998) Mixed hp-FEM on o Anisotropic Meshes Mat Models Methods Appl Sci 8(5): 787–820 [Ste71] Stetter H (1971) Stability of discretization on infinite intervals In Morris J (ed) Conf on Applications of Numerical Analysis, pages 207–222 Springer-Verlag, Berlin [Ste73] Stewart G (1973) Introduction to Matrix Computations Academic Press, New York [Str69] Strassen V (1969) Gaussian Elimination is Not Optimal Numer Math 13: 727–764 640 References [Str80] Strang G (1980) Linear Algebra and Its Applications Academic Press, New York [Str89] Strikwerda J (1989) Finite Difference Schemes and Partial Differential Equations Wadsworth and Brooks/Cole, Pacific Grove [Sze67] Szegă G (1967) Orthogonal Polynomials AMS, Providence, o R.I [Tit37] Titchmarsh E (1937) Introduction to the Theory of Fourier Integrals Oxford [Var62] Varga R (1962) Matrix Iterative Analysis Prentice-Hall, Englewood Cliffs, New York [vdV92] van der Vorst H (1992) Bi-cgstab: a fast and smoothly converging variant of bi-cg for the solution of non-symmetric linear systems SIAM Jour on Sci and Stat Comp 12: 631644 [Ver96] Verfărth R (1996) A Review of a Posteriori Error Estimation u and Adaptive Mesh Refinement Techniques Wiley, Teubner, Germany [Wac66] Wachspress E (1966) Iterative Solutions of Elliptic Systems Prentice-Hall, Englewood Cliffs, New York [Wal75] Walsh G (1975) Methods of Optimization Wiley [Wal91] Walker J (1991) Fast Fourier Transforms CRC Press, Boca Raton [Wen66] Wendroff B (1966) Theoretical Numerical Analysis Academic Press, New York [Wid67] Widlund O (1967) A Note on Unconditionally Stable Linear Multistep Methods BIT 7: 65–70 [Wil62] Wilkinson J (1962) Note on the Quadratic Convergence of the Cyclic Jacobi Process Numer Math 6: 296–300 [Wil63] Wilkinson J (1963) Rounding Errors in Algebraic Processes Prentice-Hall, Englewood Cliffs, New York [Wil65] Wilkinson J (1965) The Algebraic Eigenvalue Problem Clarendon Press, Oxford [Wil68] Wilkinson J (1968) A priori Error Analysis of Algebraic Processes In Intern Congress Math., volume 19, pages 629–639 Izdat Mir, Moscow References 641 [Wol69] Wolfe P (1969) Convergence Conditions for Ascent Methods SIAM Review 11: 226–235 [Wol71] Wolfe P (1971) Convergence Conditions for Ascent Methods II: Some Corrections SIAM Review 13: 185–188 [Wol78] Wolfe M (1978) Numerical Methods for Unconstrained Optimization Van Nostrand Reinhold Company, New York [You71] Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York [Zie77] Zienkiewicz O C (1977) The Finite Element Method (Third Edition) McGraw Hill, London Index of MATLAB Programs forward row forward col backward col lu kji lu jki lu ijk chol2 mod grams LUpivtot lu band forw band back band mod thomas cond est JOR SOR basicILU ilup gradient conjgrad arnoldi alg arnoldi met GMRES Lanczos Lanczosnosym Forward substitution: row-oriented version 66 Forward substitution: column-oriented version 66 Backward substitution: column-oriented version 66 LU factorization of matrix A kji version 77 LU factorization of matrix A jki version 77 LU factorization of the matrix A: ijk version 79 Cholesky factorization 81 Modified Gram-Schmidt method 84 LU factorization with complete pivoting 88 LU factorization for a banded matrix 92 Forward substitution for a banded matrix L 92 Backward substitution for a banded matrix U 92 Thomas algorithm, modified version 93 Algorithm for the approximation of K1 (A) 109 JOR method 135 SOR method 136 Incomplete LU factorization 141 ILU(p) factorization 143 Gradient method with dynamic parameter 149 Preconditioned conjugate gradient method 157 The Arnoldi algorithm 161 The Arnoldi method for linear systems 164 The GMRES method for linear systems 166 The Lanczos method for linear systems 167 The Lanczos method for unsymmetric systems 170 644 Index of MATLAB Programs powerm invpower basicqr houshess hessqr givensqr vhouse givcos garow gacol qrshift qr2shift psinorm symschur cycjacobi sturm givsturm chcksign bound eiglancz bisect chord secant regfalsi newton fixpoint horner newthorn mulldefl aitken adptnewt newtonxsys broyden fixposys hookejeeves explore backtrackr lagrpen lagrmult interpol dividif hermpol par spline bernstein bezier Power method Inverse power method Basic QR iteration Hessenberg-Householder method Hessenberg-QR method QR factorization with Givens rotations Construction of the Householder vector Computation of Givens cosine and sine Product G(i, k, θ)T M Product MG(i, k, θ) QR iteration with single shift QR iteration with double shift Evaluation of Ψ(A) Evaluation of c and s Cyclic Jacobi method for symmetric matrices Sturm sequence evaluation Givens method using the Sturm sequence Sign changes in the Sturm sequence Calculation of the interval J = [α, β] Extremal eigenvalues of a symmetric matrix Bisection method The chord method The secant method The Regula Falsi method Newton’s method Fixed-point method Synthetic division algorithm Newton-Horner method with refinement Muller’s method with refinement Aitken’s extrapolation Adaptive Newton’s method Newton’s method for nonlinear systems Broyden’s method for nonlinear systems Fixed-point method for nonlinear systems The method of Hooke and Jeeves (HJ) Exploration step in the HJ method Backtraking for line search Penalty method Method of Lagrange multipliers Lagrange polynomial using Newton’s formula Newton divided differences Osculating polynomial Parametric splines Bernstein polynomials B´zier curves e 197 198 203 208 210 211 213 214 214 214 217 220 229 229 229 232 232 232 232 234 250 254 255 255 255 260 263 266 269 274 276 285 290 293 296 297 303 316 319 334 336 342 359 361 361 Index of MATLAB Programs midpntc trapezc simpsonc newtcot trapmodc romberg simpadpt redmidpt redtrap midptr2d traptr2d coeflege coeflagu coefherm zplege zplagu zpherm dft idft fftrec compdiff multistep predcor ellfem femmatr H1error artvisc sgvisc bern thetameth pardg1cg1 ipeidg0 ipeidg1 Midpoint composite formula Composite trapezoidal formula Composite Cavalieri-Simpson formula Closed Newton-Cotes formulae Composite corrected trapezoidal formula Romberg integration Adaptive Cavalieri-Simpson formula Midpoint reduction formula Trapezoidal reduction formula Midpoint rule on a triangle Trapezoidal rule on a triangle Coefficients of Legendre polynomials Coefficients of Laguerre polynomials Coefficients of Hermite polynomials Coefficients of Gauss-Legendre formulae Coefficients of Gauss-Laguerre formulae Coefficients of Gauss-Hermite formulae Discrete Fourier transform Inverse discrete Fourier transform FFT algorithm in the recursive version Compact difference schemes Linear multistep methods Predictor-corrector scheme Linear FE for two-point BVPs Construction of the stiffness matrix Computation of the H1 -norm of the error Artificial viscosity Optimal artificial viscosity Evaluation of the Bernoulli function θ-method for the heat equation dG(1)cG(1) method for the heat equation dG(0) implicit Euler dG(1) implicit Euler 645 375 376 377 383 387 391 397 404 404 406 406 430 430 430 430 430 430 439 439 441 446 490 507 557 557 558 570 570 571 592 596 621 622 Index A-conjugate directions, 151 A-stability, 481 absolute value notation, 62 adaptive error control, 43 adaptivity, 43 Newton’s method, 275 Runge-Kutta methods, 512 algorithm Arnoldi, 160, 164 Cuthill-McKee, 98 Dekker-Brent, 256 Remes, 435 synthetic division, 262 Thomas, 91 amplification coefficient, 609 error, 612 analysis a priori for an iterative method, 132 a posteriori, 42 a priori, 42 backward, 42 forward, 41 B-splines, 353 parametric, 361 backward substitution, 65 bandwidth, 452 Bernoulli function, 565 numbers, 389 bi-orthogonal bases, 168 binary digits, 46 boundary condition Dirichlet, 541 Neumann, 541, 582 Robin, 579 breakdown, 160, 165 B´zier curve, 360 e B´zier polygon, 359 e CFL condition, 606 number, 606 characteristic curves, 598 variables, 600 characteristic polygon, 359 chopping, 51 648 Index cofactor, condition number, 34 asymptotic, 38 interpolation, 332 of a matrix, 36, 58 of a nonlinear equation, 246 of an eigenvalue, 189 of an eigenvector, 190 Skeel, 111 spectral, 59 consistency, 37, 124, 474, 493, 510 convex function, 295, 321 strongly, 312 convex hull, 98 critical point, 295 Dahlquist first barrier, 499 second barrier, 500 decomposition real Schur, 201, 210, 211 generalized, 225 Schur, 14 singular value, 16 computation of the, 222 spectral, 15 deflation, 207, 216, 263 degree of exactness, 380 of a vector, 160 of exactness, 372, 380, 405, 420 of freedom, 552 determinant of a matrix, discrete truncation of Fourier series, 417 Chebyshev transform, 426 Fourier transform, 438 Laplace transform, 458 Legendre transform, 428 maximum principle, 567, 611 scalar product, 425 dispersion, 448, 612 dissipation, 612 distribution, 547 derivative of a, 547 divided difference, 267, 334 domain of dependence, 600 numerical, 606 eigenfunctions, 589 eigenvalue, 12 algebraic multiplicity of an, 13 geometric multiplicity of an, 13 eigenvector, 12 elliptic operator, 602 equation characteristic, 12 difference, 482, 483, 499 heat, 581, 592 error absolute, 40 cancellation, 39 global truncation, 474 interpolation, 329 local truncation, 474, 605 quadrature, 372 rounding, 45 estimate a posteriori, 64, 195, 196, 381, 392, 395 a priori, 60, 381, 392, 395 exponential fitting, 565 factor asymptotic convergence, 125 convergence, 125, 245, 259 growth, 104 factorization block LU, 94 Cholesky, 80 compact forms, 78 Crout, 78 Doolittle, 78 incomplete, 140 LDMT , 79 Index LU, 68 QR, 82, 209 fill-in, 98, 141 level, 141 finite differences, 118, 177, 237, 533 backward, 443 centered, 443, 444 compact, 444 forward, 442 finite elements, 118, 347 discontinuous, 594, 619 fixed-point iterations, 257 flop, 53 FOM, 163, 164 form divided difference, 334 Lagrange, 329 formula Armijo’s, 304 Goldstein’s, 304 Sherman-Morrison, 95 forward substitution, 65 Fourier coefficients, 436 discrete, 437 function gamma, 528 Green’s, 532 Haar, 460 stability, 516 weight, 415 Galerkin finite element method, 364, 550 stabilized, 568 generalized method, 559 method, 544 pseudo-spectral approximation, 591 Gauss elimination method, 68 multipliers in the, 69 GAXPY, 77 generalized inverse, 17 649 Gershgorin circles, 184 Gibbs phenomenon, 439 gradient, 294 graph, 97 oriented, 97, 186 Gronwall lemma, 471, 476 hyperbolic operator, 602 hypernorms, 63 ILU, 140 inequality Cauchy-Schwarz, 340, 568 Hălder, 19 o Kantorovich, 305 Poincar, 536, 569 e triangular, 569 Young’s, 544 integration adaptive, 391 automatic, 391 multidimensional, 402 non adaptive, 391 interpolation Hermite, 341 in two dimensions, 343 osculatory, 342 piecewise, 338 Taylor, 369 interpolation nodes, 328 piecewise, 345 IOM, 164 Jordan block, 15 canonical form, 15 kernel of a matrix, 10 Krylov method, 160 subspace, 159 Lagrange interpolation, 328 multiplier, 312, 317 650 Index Lagrangian function, 312 augmented, 317 penalized, 315 Laplace operator, 572 least-squares, 417 discrete, 431 Lebesgue constant, 331, 332 linear map, linear regression, 433 linearly independent vectors, LU factorization, 72 M-matrix, 29, 145 machine epsilon, 49 machine precision, 51 mass-lumping, 588 matrix, block, companion, 242 convergent, 26 defective, 13 diagonalizable, 15 diagonally dominant, 29, 145 Gaussian transformation, 73 Givens, 206 Hessenberg, 12, 203, 211, 212 Hilbert, 70 Householder, 204 interpolation, 330 irreducible, 185 iteration, 124 mass, 587 norm, 21 normal, orthogonal, permutation, preconditioning, 126 reducible, 185 rotation, similar, 14 stiffness, 548 transformation, 204 trapezoidal, 11 triangular, 11 unitary, Vandermonde, 368 matrix balancing, 110 maximum principle, 533, 534 discrete, 29 method θ−, 584 Regula Falsi, 252 conjugate gradient, 153 Aitken, 272 alternating-direction, 158 backward Euler, 473 backward Euler/centred, 604 BiCG, 171 BiCGSTab, 171 bisection, 248 Broyden’s, 289 CGS, 171 chord, 252, 260 conjugate gradient, 168 with restart, 156 CR, 168 Crank-Nicolson, 473, 593 cyclic Jacobi, 228 damped Newton, 321 damped Newton’s, 308 finite element, 573 fixed-point, 290 Fletcher-Reeves, 306 forward Euler, 473 forward Euler/centred, 603 forward Euler/uncentred, 603 frontal, 102 Gauss Seidel symmetric, 133 Gauss-Jordan, 121 Gauss-Seidel, 128 nonlinear, 324 Givens, 230 GMRES, 166 with restart, 166 gradient, 300 Gram-Schmidt, 83 Heun, 473 Horner, 262 Index Householder, 207 inverse power, 195 Jacobi, 127 JOR, 127 Lanczos, 167, 233 Lax-Friedrichs, 603, 608 Lax-Wendroff, 603, 608 Leap-Frog, 604, 611 Merson, 530 modified Euler, 529 modified Newton’s, 284 Monte Carlo, 407 Muller, 267 Newmark, 604, 611 Newton’s, 253, 261, 283 Newton-Horner, 263, 264 Nystron, 529 ORTHOMIN, 168 Polak-Ribi´re, 307 e Powell-Broyden symmetric, 311 power, 192 QMR, 171 QR, 200 with double shift, 218 with single shift, 215, 216 quasi-Newton, 288 reduction formula, 403 Richardson, 136 Richardson extrapolation, 387 Romberg integration, 389, 409 Rutishauser, 202 secant, 252, 257, 288 secant-like, 309 Simplex, 299 SSOR, 134 steepest descent, 305 Steffensen, 280 successive over-relaxation, 128 upwind, 603, 607 minimax property, 418 minimizer global, 294, 311 local, 294, 311 model computational, 43 module of continuity, 386 nodes Gauss, 426 Gauss-Lobatto, 424, 426 norm absolute, 32 compatible, 21, 22 consistent, 21 energy, 29 equivalent, 20 essentially strict, 432 Frobenius, 22 Hălder, 19 o matrix, 21 maximum, 19, 330 spectral, 23 normal equations, 112 numbers de-normalized, 48 fixed-point, 46 floating-point, 47 numerical flux, 602 numerical method, 37 adaptive, 43 consistent, 37 convergent, 39 efficiency, 44 ill conditioned, 38 reliability, 44 stable, 38 well posed, 38 orbit, 523 overflow, 51 P´clet number, 561 e local, 563 Pad´ approximation, 370 e parabolic operator, 602 pattern of a matrix, 97, 575 penalty parameter, 315 651 652 Index phase angle, 612 pivoting, 85 complete, 86 partial, 86 Poisson equation, 572 polyalgorithm, 277 polynomial Bernstein, 359 best approximation, 330, 433 characteristic, 12, 329 Fourier, 435 Hermite, 429 interpolating, 328 Lagrange piecewise, 346 Laguerre, 428 nodal, 329 orthogonal, 415 preconditioner, 126 block, 139 diagonal, 140 ILU, 142 least-squares, 145 MILU, 144 point, 139 polynomial, 145 principal root of unity, 437 problem Cauchy, 469 generalized eigenvalue, 146, 224, 238, 589 ill posed, 34, 35 ill-conditioned, 34 stiff, 520 well conditioned, 34 well posed, 33 programming linear, 282 nonlinear, 282, 313 pseudo-inverse, 17, 114 pseudo-spectral derivative, 449 differentiation matrix, 449 quadrature formula, 371 Cavalieri-Simpson, 377, 385, 400, 401, 409 composite Cavalieri-Simpson, 377 composite midpoint, 374 composite Newton-Cotes, 383 composite trapezoidal, 376 corrected trapezoidal, 386 Gauss, 421 on triangles, 406 Gauss-Kronrod, 393 Gauss-Lobatto, 422, 425 Gauss-Radau on triangles, 406 Hermite, 372, 386 Lagrange, 372 midpoint, 373, 385 on triangles, 405 Newton-Cotes, 378 on triangles, 404 pseudo-random, 408 trapezoidal, 375, 385, 438 on triangles, 405 quotient Rayleigh, 12 generalized, 146 QZ iteration, 225 rank of a matrix, rate asymptotic convergence, 125 convergence, 259 reduction formula midpoint, 403 trapezoidal, 404 reference triangle, 345 regularization, 34 representation floating-point, 47 positional, 45 residual, 247 resolvent, 35 restart, 164 round digit, 53 rounding, 51 Index roundoff unit, 51 rule Cramer’s, 58 Descartes, 262 Laplace, Runge’s counterexample, 331, 344, 353 SAXPY, 77 saxpy, 77 scalar product, 18 scaling, 110 by rows, 110 Schur complement, 102 decomposition, 14 semi-discretization, 584, 586 series Chebyshev, 418 Fourier, 416, 582 Legendre, 419 set bi-orthogonal, 189 similarity transformation, 14 singular integrals, 398 singular values, 16 space normed, 19 phase, 523 Sobolev, 543 vector, spectral radius, 13 spectrum of a matrix, 12 spline cardinal, 351 interpolatory cubic, 349 natural, 349 not-a-knot, 350 one-dimensional, 348 parametric, 358 periodic, 348 splitting, 126 stability absolute, 479, 480, 499, 502 region of, 480 asymptotic, 471 factors, 42 Liapunov, 471 of interpolation, 332 relative, 502 zero, 477, 495, 502 standard deviation, 298 statistic mean value, 407 stencil, 445 stopping tests, 171, 269 strong formulation, 547 Sturm sequences, 230 subspace generated, invariant, 13 vector, substructures, 100 Sylvester criterion, 29 system hyperbolic, 599 strictly, 600 overdetermined, 112 underdetermined, 115 theorem Abel, 262 Bauer-Fike, 187 Cauchy, 262 Cayley-Hamilton, 13 Courant-Fisher, 146, 233 de la Vall´e-Poussin, 434 e equioscillation, 433 Gershgorin, 184 Ostrowski, 259 polynomial division, 263 Schur, 14 trace of a matrix, transform fast Fourier, 426 Fourier, 450 Laplace, 455 Zeta, 457 triangulation, 344, 573 underflow, 51 653 654 Index upwind finite difference, 565 weak formulation, 545 solution, 545, 599 wobbling precision, 49 ... Germany Mathematics Subject Classification (1991): 1 5-0 1, 3 4-0 1, 3 5-0 1, 6 5-0 1 Library of Congress Cataloging-in-Publication Data Quarteroni, Alfio Numerical mathematics/ Alfio Quarteroni, Riccardo Sacco,. .. Sacco, Fausto Saleri p.Mcm — (Texts in applied mathematics; 37) Includes bibliographical references and index ISBN 0-3 8 7-9 895 9-5 (alk paper) Numerical analysis.MI Sacco, Riccardo.MII Saleri, Fausto.MIII... accordingly be used freely by anyone ISBN 0-3 8 7-9 895 9-5 nSpringer-VerlagnNew YorknBerlinnHeidelbergMSPIN 10747955 Preface Numerical mathematics is the branch of mathematics that proposes, develops,