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