Numerical Analysis Topics Outline

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Numerical Analysis Topics

 
Floating Point System
Linear Equations
Direct methods
Classical Iterative methods
Modern iterative methods
Least Squares
Eigenvalue problems
Interpolation and Approximations
Polynomial Interpolation
Fourier Transform
Wavelets
Nonlinear Equations
Single variable
System of equations
Differentiation and Integration
Newton Cotes
Extrapolation
Gauss Quadrature
Adaptive Strategy
Ordinary Differential Equations
Runge-Kutta methods
Multistep methods
Partial Differential Equations
Optimization

 

 

 

 

Nonlinear Equations

Nonlinear Equations of one variable

Bisection method
Newton's method
Secant method
Fixed point iterations
Roots of Polynomials:
Muller's method
Horner's scheme
 

Systems of nonlinear equations

Newton's method
Quasi-Newton methods
Rank-1 update

 

Floating Point System

Representation of floating point numbers
Floating point arithmetic
IEEE floating point standard

 

Solution of Linear Equations

Theory in solution of linear equations

Vector and matrix norms
Perturbation estimates

Direct solution of linear equations

Gaussian elimination
Pivoting
LU decomposition
Cholesky factorization
Banded and tridiagonal matrices
Iterative refinement
Cyclic reduction
Fast Poisson solver

Classical Iterative techniques

General iteration via matrix splitting
Classical Iterative schemes
Jacobi iteration
Gauss-Seidel iteration
SOR

Modern iterative techniques

Conjugate gradient 
Krylov subspace methods 
Preconditioning
Multigrid

Least Squares

Discrete least squares
Singular value decomposition
Continuous least squares
Orthogonal polynomials
Legendre
Chebyshev

Eigenvalue problems

Power methods
Deflation
Householder reflection
Givens method
QR
Divide and conquer
Inverse iteration

Interpolation and Approximations

Polynomial Interpolation

Lagrange interpolation
basis form
divided difference forms
Higher dimensional case
Hermite interpolation
Cubic splines
B-Splines
Bezier curves
Subdivision algorithms

Fourier transform

Trigonometric polymonials
FFT

Wavelet

Wavelet basics 
Wavelet transforms

 

Differentiation and Integration

Numerical Differentiation

Finite Differencing
Richardson's extrapolation

Numerical Integration

Newton-Cotes: basic and composite rules
multidimensional case
Romberg Integration
Gauss Quadrature
multidimensional case
Adaptive Integration
Gauss-Kronrod

 

Ordinary Differential Equations

Initial value problems

First order systems of ODE
Taylor's methods
Runge-Kutta methods
Variable step strategy
Runge-Kutta-Fehlberg method
Multistep methods
Predictor-Corrector pairs
Stiff equations

Boundary value problems

Finite difference

Derivation
Stability
Lax Theorem

Finite elements

Variational formulation
Convergence
Superconvergence

 

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Copyrighted by S.-Sum Chow

This page was last updated on 10/08/2002.