Numerical Analysis 1 - Introduction: MTH
443/643(Each Fall semester). Prerequisites: MTH 331 Linear
Algebra, and CS 205 Scientific Computing (may take concurrently)
Topics include: floating point number systems; rates of convergence - algebraic and geometric; nonlinear equations and nonlinear systems; introduction to numerical linear algebra: condition numbers, LU factorizations, backward stability; basis functions: monomials, orthogonal polynomials, radial basis functions; interpolation; Runge-Kutta methods for the solution of ODEs.
Text: Numerical Analysis
ODE Lab - Applet that illustrates properties of numerical methods for ODE initial value problems.
Numerical Analysis 2 - Linear Algebra: MTH
442/642 (Each Spring semester). Prerequisites: MTH 443/643.
Topics include: Orthogonal matrices, Householder transformations, QR factorization, least squares problems; the singular value decomposition (SVD); symmetric positive definite matrix factorizations; eigenvalues and pseudospectra; iterative methods for linear systems; and Gaussian quadrature.
Text: Numerical Linear Algebra. Supplemental text(s): Numerical Linear Algebra and Applications
Numerical Analysis 3 - PDEs: MTH 667 (Offered in
response to student demand. A graduate level course,
(well-qualified undergraduates may take the course).
Prerequisites: MTH 443/643 (may take concurrently).
Topics include: numerical differentiation in physical space - differentiation matrices; Fourier and Chebyshev approximation; numerical differentiation in transform space; the fast Fourier transform (FFT) and the fast cosine transform (FCT); Clenshaw-Curtis quadrature; the method of lines and stability; diffusion and advection diffusion problems; wave propagation problems; dispersion and dissipation; boundary value problems; discontinuous problems; spectral viscosity.
Text: Spectral Methods in xxxxxx. Supplemental text(s): Chebyshev and Fourier Spectral Methods.
Numerical Advection - Applet that solves the 1d advection equation with periodic boundary conditions using a large number of different numerical methods.
Removal of Gibbs' oscillations - Applet that illustrates post-processing methods to remove Gibbs' oscillations from Fourier and Chebyshev approximations of functions that have discontinuities.