lnotes.mws Practical Information Weekly Diary Assessment Examination Continuous Assessment Class Test Exercises Computer labs Syllabus Reference Materials Support Errors Errors associated with the method Truncation error Errors in the initial data Errors associated with computer technology Number representation Basics Example Some facts Number mapping Round-off error Machine epsilon Adding two numbers Some definitions Absolute error Relative error Significant Digits Error propagation Convergence Solutions of equations in one variable Bisection Method (see ws1) Secant Method Newton-Raphson Method (see ws1) Fixed Point Iteration (See ws1) Fixed Point Iteration: theory Definition Theorem 1 (Existence of a fixed point) Theorem 2 (Uniqueness of the fixed point) Example Theorem 3 (Fixed Point Theorem) Theorem 4 (Convergence of Newton-Raphson Method) Convergence of Iteration Sequences Definition Fixed Point Iteration Theorem 5 (Linearity of Fixed Point Iteration) Proof Theorem 6 (Quadratic nature of FPI with g'(p)=0 ) Proof Solving f(x)=0 Newton-Raphson Method Multiple roots to f(x)=0 Definition Properties Calculating multiple roots Aitken's Method Derivation Example Forward Difference Notation Theorem 7 (Aitken's Method) Newton-Raphson Method for systems of equations Interpolation Introduction Polynomial interpolation Polynomial approximation Theorem 8 (Weierstrass Approximation Theorem) Taylor polynomials The Lagrange Polynomial Theorem 9 (Existence and uniqueness) Theorem 10 (Error) Proof Comments Iterated Interpolation Theorem 11 Proof Neville's method Divided Differences Definition of Divided Differences Newton's forward divided difference formula Theorem 12 Proof Equidistant data points Piecewise Polynomial Interpolation piecewise linear approximation cubic spline interpolation Definition Construction Calculation Numerical differentiation Introduction First derivative Forward and backward differences Derivation Example Three-point formulae Derivation Example Five-point formulae Derivation Example Comparison Second order derivatives Derivation Example Alternative derivation Example revisited Effects of round-off error Richardson Extrapolation Derivation Example Higher order corrections Example Numerical Integration Introduction Quadrature Formulae Examples of derivation Trapezoidal Rule Simpson's Rule Alternative derivation Theory Example Newton-Cotes formulae Theorem 13 Definition Some formulae Examples of use Composite Numerical Integration Theorem 14 (Composite Simpson's Method) Proof Theorem 15 (Composite Trapezoidal Rule) Theorem 16 (Composite Midpoint Rule) Gaussian Quadrature Preliminaries Theorem 17 (Gaussian Quadrature) Proof Application Examples 2-point formula Three point formula Four point formula Comparison Integration over (a,b) Romberg Integration Numerical Solutions for Ordinary Differential Equations Preliminary Theory Definition (Lipschitz Condition) Definition (convex set) Theorem 18 Theorem 19 (unique solution) Definition (well-posed problem) Theorem 20 Taylor Method(s) Euler's Method Lemma Lemma Theorem 21 Proof Example Theorem 22 (Round-off error) Example Taylor's method of order n Local truncation error Definition Euler's method Taylor's method of order n Runge-Kutta Methods Theorem 23 (Taylor's Theorem in two variables) Midpoint method Derivation Example Graphical interpretation Modified Euler's method Heun's method Fourth order Runge-Kutta Error control Runge-Kutta-Fehlberg Method Multistep Methods Introduction Adams-Basforth Methods Adams-Moulton Methods Predictor-Corrector Methods Example Systems of ODE's General Approach Example Boundary Value Problems General Principle Automation