Harvard AM205 video 5.9 - Krylov methods: Arnoldi iteration and Lanczos interation

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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.

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.
  1. Chris Rycroft
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Комментарии
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The interpretation of Arnoldi iteration is extremely useful, thank you

bradyan
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I just noticed the units have colors of the rainbow also

kingarthr
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Great demonstration in python, very nice!

shinzon
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Just on a side note about pronunciation. The name "Krylov" is pronounced "Krilof" with the stress on the last syllable and the "y" being simply an "i" as in "sit". "Lanczos" is pronounced as "Lanchosh" with the stress on the first syllable.

guidotoschi
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Hi! Where can I find 5.7 and 5.8 lectures? Great content

martinpuente
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Thanks for this video！Maybe I need some help . I want use Lanzcos to slove complex Hermitian Matrix eignproblems. Is there any difference between complex Hermitain Matrix and Symmertic Real Matrix in using Lanzcos Method?

dragonghost