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Introduction (And Things You
Lecture Notes For
Numerical Analysis
M315
Prof. Paul A. Gray
Fall, 1998
Introduction
(And Things You Should Know)
Errors Which We Encounter
Sources of Errors
Definitions
Representing Numbers
Floating-Point Representation
Representational Limitations
The Hidden Bit
Finite Representability Set
Rounding and Chopping
Overflow and Underflow
Loss of Significance Errors
Roots of a Single-Variable Equation
The Bisection Method
Bisection Error Analysis
Newton's Method
Convergence of Newton's Method
The Secant Method
Secant Method Convergence
Fixed-Point Iterations
Numerical O.D.E.'s
Background
Notation
Euler's Method
Convergence and Error Analysis of Euler's Method
Alternative Derivations of Euler's Method
Derivation via Taylor's Theorem
Derivation via Numerical Integration
Derivation via Tangent Line Approximation
Higher-Order Methods
Taylor Series Expansion
Runge-Kutte Methods
Mathematica and Solutions to Ordinary Differential Equations
Summary
References
About this document ...
Paul Gray
Wed Oct 28 11:42:13 EST 1998