Numerical Analysis A Comprehensive Introduction H. R. Schwarz University of Zürich Switzerland with a contribution by J. Waldvogel Swiss Federal Institute of Technology, Zürich JOHN WILEY & SONS Chichester New York Brisbane Toronto Singapore
", Contents Preface Preface to the English Edition v vii 1 Systems of Linear Equations 1 1.1 Gaussian algorithm 1 1.1.1 The fundamental process 1 1.1.2 Pivotal strategies 9 1.1.3 Supplements 16 1.2 Accuracy, error estimates 20 1.2.1 Norms 20 1.2.2 Error estimates, condition 25 1.3 Systems with special properties 29 1.3.1 Symmetrie, positive definite Systems 30 1.3.2 Banded Systems 35 1.3.3 Tridiagonal Systems 37 1.4 Exchange step and inversion of matrices 41 1.4.1 Linear funetions, exchange 42 1.4.2 Matrix inversion 44 1.5 Exercises 46 2 Linear Programming 49 2.1 Introductory examples, graphical Solution..." 49 2.2 The simplex algorithm 54 2.3 Supplements to the simplex algorithm 63 2.3.1 Degeneracy 63 2.3.2 Nonunique Solution 66 2.3.3 Unbounded pbjeetive funetion 67 2.4 General linear programs. 68 2.4.1 Treatment of free variables 68 2.4.2 Coordinate shift method 69 ix
x Contents 2.4.3 The two-phase method 73 2.5 Discrete Chebyshev approximation 77 2.6 Exercises 83 3 Interpolation 85 3.1 Existence and uniqueness of polynomial interpolation 85 3.2 Lagrange interpolation 86 3.2.1 Technique of computation 86 3.2.2 Applications 90 3.3 Error estimates 96 3.4 Newton interpolation 100 3.5 Aitken-Neville interpolation 108 3.5.1 Aitken's and Neville's algorithms 109 3.5.2 Extrapolation and the Romberg scheme 111 3.5.3 Inverse interpolation 114 3.6 Rational interpolation 116 3.6.1 Formulation of the problem, difficulties 116 3.6.2 Special interpolation problem, Thiele's continued fraction.. 118 3.7 Spline interpolation 125 3.7.1 Characterization of the spline function 126 3.7.2 Computation of the cubic spline function 128 3.7.3 General cubic spline functions 133 3.7.4 Periodic cubic spline interpolation 136 3.7.5 Smooth two-dimensional curves 139 3.8 Exercises 141 4 Approximation of Functions 143 4.1 Fourier series 143 4.2 Efficient evaluation of Fourier coefficients 155 4.2.1 Runge's algorithm 155 4.2.2 The fast Fourier transform 159 4.3 Orthogonal polynomials 169 4.3.1 The Chebyshev polynomials 169 4.3.2 Chebyshev interpolation 178 4.3.3 Legendre polynomials 182 4.4 Exercises 188 5 Nonlinear Equations 191 5.1 The Banach fixed point theorem 191 5.2 Behaviour and order of convergence 195 5.3 Equations in one unknown 203 5.3.1 Bisection, regula falsi, secant method 203 5.3.2 Newton's method 209 5.3.3 Interpolation methods 213
Contents 5.4 Equations in several unknowns 216 5.4.1 Fixed point iteration and convergence 216 5.4.2 Newton's method 218 5.5 Zeros of polynomials 224 5.6 Exercises 235 xi 6 Eigenvalue Problems 238 6.1 The characteristic polynomial, difficulties 238 6.2 Jacobi methods 241 6.2.1 Elementary rotations 242 6.2.2 The classical Jacobi method 244 6.2.3 Cyclic Jacobi method 250 6.3 Transformation methods 253 6.3.1 Transformation into Hessenberg form 254 6.3.2 Transformation into tridiagonal form 258 6.3.3 Fast Givens transformation 260 6.3.4 Hyman's method 264 6.4 QR algorithm 269 6.4.1 Fundamentals of the QR transformation 270 6.4.2 Practical implementation, real eigenvalues 275 6.4.3 QR double step, complex eigenvalues 280 6.4.4 QR algorithm for tridiagonal matrices 286 6.4.5 Computation of eigenvectors 290 6.5 Exercises 291 7 Method of Least Squares 294 7.1 Linear problems, normal equations 294 7.2 Methods of orthogonal transformation 299 7.2.1 Givens transformation 300 7.2.2 Special computational techniques 306 7.2.3 Householder transformation 309 7.3 Singular value decomposition. 315 7.4 Nonlinear problems 319 7.4.1 The Gauss-Newton method 320 7.4.2 Minimization methods 324 7.5 Exercises 328 8 Numerical Quadrature 330 8.1 The trapezoidal method 331 8.1.1 Problem and notation 331 8.1.2 Definition of the trapezoidal method and improvements... 331 8.1.3 The Euler-MacLaurin formula 335 8.1.4 The Romberg procedure 337 8.1.5 Adaptive quadrature. 339
xii Contents 8.2 Transformation methods 342 8.2.1 Periodic integrands 343 8.2.2 Integrals over R 345 8.2.3 Transformation methods 347 8.3 Interpolation quadrature formulae 350 8.3.1 Newton-Cotes quadrature formulae 351 8.3.2 Spline quadrature formulae 359 8.4 Gaussian quadrature formulae 361 8.5 Exercises 368 9 Ordinary Differential Equations 371 9.1 Single step methods 371 9.1.1 Euler and the Taylor series method 371 9.1.2 Discretization errors, order of convergence 375 9.1.3 Improved polygonal method, trapezoidal method, Heun's method 379 9.1.4 Runge-Kutta methods 384 9.1.5 Implicit Runge-Kutta methods 392 9.1.6 Differential equations of higher order and Systems 395 9.2 Multistep methods ' 398 9.2.1 Adams-Bashforth methods 398 9.2.2 Adams-Moulton methods 401 9.2.3 General linear multistep methods 404 9.3 Stability 413 9.3.1 Inherent instability 413 9.3.2 Absolute stability 415 9.3.3 Stiff differential equations 422 9.4 Exercises 427 10 Partial Differential Equations r 430 10.1 Elliptic boundary value problems, finite differences 430 10.1.1 Formulation of the problem 430 10.1.2 Discretization of the problem 432 10.1.3 Grid points near the boundary, general boundary conditions. 439 10.1.4 Discretization errors 451 10.1.5 Supplements 464 10.2 Parabolic initial boundary value problems 468 10.2.1 One-dimensional problems, explicit method 468 10.2.2 One-dimensional problems, implicit method 474 10.2.3 Diffusion equation with variable coefficients 479 10.2.4 Two-dimensional problems 482 10.3 Finite element method 487 10.3.1 Fundamentals 487 10.3.2 Principle of the finite element method 490 10.3.3 Elementwise treatment 492
Contents 10.3.4 Compilation and Solution of the linear equations 499 10.3.5 Examples 500 10.4 Exercises 503 xiii References 507 Index 513