Math 411: Numerical Methods

Catalog Information

Title

Numerical Methods.

(Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

Offered

W (odd years)

Prerequisite

Math 334

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. [This official course description appears to differ with current standard practice, in that iterative solvers of linear systems are taught in Math 410.]

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

Students should be able to describe, derive, and implement the numerical methods listed below. They should be able to explain the advantages and disadvantages of each method. They should understand error analysis and be able to make practical decisions based on the outcomes of that analysis.

1. Numerical solution of initial-value problems
   • Taylor methods
     • Euler's method
   • Runge-Kutta methods
     • Runge-Kutta-Fehlberg method
   • Multi-step methods
   • Implicit 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

Textbooks

Possible textbooks for this course include (but are not limited to):

• Richard L. Burden and J. Douglas Faires, Numerical Analysis (9th Edition), Brooks Cole, 2010.

Additional topics

If time permits, students could be given an introduction to finite element methods.

Courses for which this course is prerequisite

None.