Harvard AM205 video 5.9 - Krylov methods: Arnoldi iteration and Lanczos interation
Channel: Chris Rycroft
Date: 2022-12-12

Harvard Applied Math 205 is a graduate-level course on scientific computing and numerical methods. This video introduces Krylov methods, which are a family of methods for computing eigenvalues of matrices, and solving them. Krylov methods do not require direct access and manipulation of the matrix elements; it is only necessary to perform matrix multiplication, making the methods well suited to sparse matrices.

The video first introduces the Arnoldi iteration that can apply to a general matrix, and then specializes to the Lanczos iteration that applies to symmetric matrices, resulting in a large algorithmic speedup. A Python example of the Lanczos iteration is presented and discussed. The Python example is based on an example in the textbook "Numerical Linear Algebra" by Lloyd N. Trefethen and David Bau III.

For more information see the main course website at https://people.math.wisc.edu/~chr/am205

NOTE: this video has been posted prior to videos 5.7 & 5.8. The videos 5.7 & 5.8 form a self-contained section and will be posted at a later date.