# 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
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

### 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
 multidimensional case
 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