# Difference between revisions of "Math 411: Numerical Methods"

### Title

Numerical Methods.

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### Description

Iterative solvers for linear systems, eigenvalue, eigenvector approximations, numerical solutions to nonlinear systems, numerical techniques for initial and boundary value problems, elementary solvers for PDEs.

## Desired Learning Outcomes

### Prerequisites

The formal prerequisites reflect the fact that incoming students should have basic knowledge of ordinary differential equations and have had a first course in numerical methods. Indirectly, the prerequisites ensure that students have had multivariable calculus.

### Minimal learning outcomes

1. Numerical solution of initial-value problems
• Taylor methods
• Euler's method
• Runge-Kutta methods
• Runge-Kutta-Fehlberg method
• Multi-step methods
• Extrapolation methods
• Stability
• Stiff differential equations
2. Numerical solution of boundary-value problems
• Shooting methods
• Finite-difference methods
• Rayleigh-Ritz method
3. Numerical solution of nonlinear systems of equations
• Newton's method
• Quasi-Newton methods
• Steepest-descent methods
4. Approximation theory
• Least-squares approximation
• Orthogonal polynomials
• Chebyshev polynomials
• Rational function approximation
• Trigonometric polynomial approximation
• Fast Fourier transforms
5. Numerical computation of eigenvalues and eigenvectors
• Power Method
6. Partial differential equations
• Finite-difference methods
• For elliptic equations
• For parabolic equations
• For hyperbolic equations
• Introduction to finite-element methods