AN INTRODUCTION TO NUMERICAL ANALYSIS
CONTENTS SOLUTION OF EQUATIONS BY ITERATION INTRODUCTION SIMPLE ITERATION ITERATIVE SOLUTION OF EQUATIONS RELAXATION AND NEWTON’S METHOD THE SECANT METHOD THE BISECTION METHOD GLOBAL BEHAVIOUR NOTES EXERCISES SOLUTION OF SYSTEMS OF LINEAR EQUATIONS GAUSSIAN ELIMINATION LU FACTORISATION PIVOTING SOLUTION OF SYSTEMS OF EQUATIONS COMPUTATIONAL WORK NORMS AND CONDITION NUMBERS HILBERT MATRIX LEAST SQUARES METHOD SPECIAL MATRICES SYMMETRIC POSITIVE DE?NITE MATRICES TRIDIAGONAL AND BAND MATRICES MONOTONE MATRICES SIMULTANEOUS NONLINEAR EQUATIONS SIMULTANEOUS ITERATION GLOBAL CONVERGENCE EIGENVALUES AND EIGENVECTORS OF A SYMMETRIC MATRIX THE CHARACTERISTIC POLYNOMIAL JACOBI’S METHOD THE GERSCHGORIN THEOREMS HOUSEHOLDER’S METHOD EIGENVALUES OF A TRIDIAGONAL MATRIX THE QR ALGORITHM 5 7 1 THE QR FACTORISATION REVISITED 5 7 2 THE DE?NITION OF THE QR ALGORITHM INVERSE ITERATION FOR THE EIGENVECTORS THE RAYLEIGH QUOTIENT PERTURBATION ANALYSIS POLYNOMIAL INTERPOLATION LAGRANGE INTERPOLATION CONVERGENCE HERMITE INTERPOLATION DI?ERENTIATION NUMERICAL INTEGRATION – I NEWTON–COTES FORMULAE ERROR ESTIMATES THE RUNGE PHENOMENON REVISITED COMPOSITE FORMULAE THE EULER–MACLAURIN EXPANSION EXTRAPOLATION METHODS POLYNOMIAL APPROXIMATION IN THE ?-NORM NORMED LINEAR SPACES BEST APPROXIMATION IN THE ?-NORM CHEBYSHEV POLYNOMIALS INTERPOLATION APPROXIMATION IN THE 2-NORM INNER PRODUCT SPACES BEST APPROXIMATION IN THE 2-NORM ORTHOGONAL POLYNOMIALS COMPARISONS NUMERICAL INTEGRATION – II CONSTRUCTION OF GAUSS QUADRATURE RULES DIRECT CONSTRUCTION ERROR ESTIMATION FOR GAUSS QUADRATURE COMPOSITE GAUSS FORMULAE RADAU AND LOBATTO QUADRATURE NOTE PIECEWISE POLYNOMIAL APPROXIMATION LINEAR INTERPOLATING SPLINES BASIS FUNCTIONS FOR THE LINEAR SPLINE CUBIC SPLINES HERMITE CUBIC SPLINES BASIS FUNCTIONS FOR CUBIC SPLINES INITIAL VALUE PROBLEMS FOR ODES ONE-STEP METHODS CONSISTENCY AND CONVERGENCE AN IMPLICIT ONE-STEP METHOD RUNGE–KUTTA METHODS LINEAR MULTISTEP METHODS ZERO-STABILITY CONSISTENCY DAHLQUIST’S THEOREMS 12 10 SYSTEMS OF EQUATIONS 12 11 STI?SYSTEMS 12 12 IMPLICIT RUNGE–KUTTA METHODS 12 13 NOTES BOUNDARY VALUE PROBLEMS FOR ODES A MODEL PROBLEM ERROR ANALYSIS BOUNDARY CONDITIONS INVOLVING A DERIVATIVE THE GENERAL SELF-ADJOINT PROBLEM THE STURM–LIOUVILLE EIGENVALUE PROBLEM THE SHOOTING METHOD THE ?NITE ELEMENT METHOD INTRODUCTION: THE MODEL PROBLEM RAYLEIGH–RITZ AND GALERKIN PRINCIPLES FORMULATION OF THE ?NITE ELEMENT METHOD ERROR ANALYSIS OF THE ?NITE ELEMENT METHOD A POSTERIORI ERROR ANALYSIS BY DUALITY