Math 411: Numerical Methods

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Catalog Information

Title

Numerical Methods.

(Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

Offered

W

Prerequisite

Math 334, 410.

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

Additional topics

Courses for which this course is prerequisite

None.