Topics in Numerical Analysis:
Spectral Methods

We will work through the book Spectral Methods in Matlab by L. N. Trefethen (SIAM, 1999). This gives a nice introduction to the use of spectral collocation methods for obtaining highly accurate approximations to derivatives of smooth functions, along with applications to the solution of both ordinary and partial differential equations.

The chapters in this book are:

• Differentiation Matrices
• Unbounded Grids: The Semidiscrete Fourier Transform
• Periodic Grids: The DFT and FFT
• Smoothness and Spectral Accuracy
• Chebyshev Differentiation Matrices
• Boundary Value Problems
• Chebyshev Series and the FFT
• Eigenvalues and Pseudospectra
• Time-Stepping and Stability Regions
• Polar Coordinates
• Integrals and Quadrature Formulas
• More about Boundary Conditions
• Fourth-Order Problems

Prerequisites: Familiarity with numerical analysis at the level of AMath 584-5-6.

See the webpage for more information: