Lecture materials
Before even the first class, I recommend that you refresh your linear algebra background. A more accessible review of linear algebra can be found in these notes by Jonathan Goodman.Also get a CIMS computer account and start playing with MATLAB, Linux, etc.
1. (Sept. 4 and 11) Numerical computing
My lecture slides cover basics only. Try also these nice notes by my colleague Jonathan Goodman.Here are the MATLAB codes for computing the harmonic sum in double and single precision.
You can see the proof of convergence of the Babylonian fixed-point iteration for computing square roots on Wikipedia.
2. (Sept. 11 and 18) Solving
square linear systems: GEM and LU factorization
My lecture slides
summarize the important points. Here is the MATLAB code MyLU.m.3. (Sept. 18 and 25) Nonsquare
and sparse linear systems
My lecture slides
summarize the important points but sparse matrices and
iterative methods are a huge field and you should read more
about them in the suggested readings, and ask questions. In
some sense sparse matrices are the most important in practice
today.
4. (Oct. 2) Eigenvalue problems
Review linear algebra regarding eigenvalues. My lecture slides focus on the important points: conditioning, the power method, and the basic QR iteration. Here is a nice review by Maysum Panju of the power and QR methods and their links, and a summary of why QR works.5. (Oct. 9) Singular value problems
My lecture slides focus on the important points: do the homework to try it out.6. (Oct. 16) Solving nonlinear equations
A key algorithm to understand is Newton's method.7. (Oct. 23) Mathematical
Programming (Optimization)
We will do a whirl-wind tour of optimization to introduce you
to the basic concepts.8. (Oct. 30th) Polynomial
Interpolation
We focus on low-order polynomial interpolation and piecewise
polynomial interpolation in 1D, 2D and 3D, and defer spectral
approximation to the next two lectures.
9. (Nov 6th and Nov 13th) Orthogonal
Polynomials
We will talk about Legendre and Chebyshev polynomials,
preparing us to discuss the Fourier basis and Gaussian
quadrature.
10. (Nov 20th) Fourier
and Wavelet Transforms
This will discuss Fourier Transforms, including the FFT
algorithm, and briefly introduces wavelets.
11. (Dec. 4th) Numerical
Integration
We discuss quadrature rules in low dimensions.12. (Dec. 11th) Monte
Carlo Methods
Review some basic probability concepts. For more details, take
a look at these notes
by
Jonathan
Goodman.