Numerical Analysis. A Comprehensive Introduction. H. R. Schwarz University of Zürich Switzerland. with a contribution by

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1 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

2 ", Contents Preface Preface to the English Edition v vii 1 Systems of Linear Equations Gaussian algorithm The fundamental process Pivotal strategies Supplements Accuracy, error estimates Norms Error estimates, condition Systems with special properties Symmetrie, positive definite Systems Banded Systems Tridiagonal Systems Exchange step and inversion of matrices Linear funetions, exchange Matrix inversion Exercises 46 2 Linear Programming Introductory examples, graphical Solution..." The simplex algorithm Supplements to the simplex algorithm Degeneracy Nonunique Solution Unbounded pbjeetive funetion General linear programs Treatment of free variables Coordinate shift method 69 ix

3 x Contents The two-phase method Discrete Chebyshev approximation Exercises 83 3 Interpolation Existence and uniqueness of polynomial interpolation Lagrange interpolation Technique of computation Applications Error estimates Newton interpolation Aitken-Neville interpolation Aitken's and Neville's algorithms Extrapolation and the Romberg scheme Inverse interpolation Rational interpolation Formulation of the problem, difficulties Special interpolation problem, Thiele's continued fraction Spline interpolation Characterization of the spline function Computation of the cubic spline function General cubic spline functions Periodic cubic spline interpolation Smooth two-dimensional curves Exercises Approximation of Functions Fourier series Efficient evaluation of Fourier coefficients Runge's algorithm The fast Fourier transform Orthogonal polynomials The Chebyshev polynomials Chebyshev interpolation Legendre polynomials Exercises Nonlinear Equations The Banach fixed point theorem Behaviour and order of convergence Equations in one unknown Bisection, regula falsi, secant method Newton's method Interpolation methods 213

4 Contents 5.4 Equations in several unknowns Fixed point iteration and convergence Newton's method Zeros of polynomials Exercises 235 xi 6 Eigenvalue Problems The characteristic polynomial, difficulties Jacobi methods Elementary rotations The classical Jacobi method Cyclic Jacobi method Transformation methods Transformation into Hessenberg form Transformation into tridiagonal form Fast Givens transformation Hyman's method QR algorithm Fundamentals of the QR transformation Practical implementation, real eigenvalues QR double step, complex eigenvalues QR algorithm for tridiagonal matrices Computation of eigenvectors Exercises Method of Least Squares Linear problems, normal equations Methods of orthogonal transformation Givens transformation Special computational techniques Householder transformation Singular value decomposition Nonlinear problems The Gauss-Newton method Minimization methods Exercises Numerical Quadrature The trapezoidal method Problem and notation Definition of the trapezoidal method and improvements The Euler-MacLaurin formula The Romberg procedure Adaptive quadrature. 339

5 xii Contents 8.2 Transformation methods Periodic integrands Integrals over R Transformation methods Interpolation quadrature formulae Newton-Cotes quadrature formulae Spline quadrature formulae Gaussian quadrature formulae Exercises Ordinary Differential Equations Single step methods Euler and the Taylor series method Discretization errors, order of convergence Improved polygonal method, trapezoidal method, Heun's method Runge-Kutta methods Implicit Runge-Kutta methods Differential equations of higher order and Systems Multistep methods ' Adams-Bashforth methods Adams-Moulton methods General linear multistep methods Stability Inherent instability Absolute stability Stiff differential equations Exercises Partial Differential Equations r Elliptic boundary value problems, finite differences Formulation of the problem Discretization of the problem Grid points near the boundary, general boundary conditions Discretization errors Supplements Parabolic initial boundary value problems One-dimensional problems, explicit method One-dimensional problems, implicit method Diffusion equation with variable coefficients Two-dimensional problems Finite element method Fundamentals Principle of the finite element method Elementwise treatment 492

6 Contents Compilation and Solution of the linear equations Examples Exercises 503 xiii References 507 Index 513

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