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

[@bradyan53838]

The interpretation of Arnoldi iteration is extremely useful, thank you

[@kingarth0r]

I just noticed the units have colors of the rainbow also

[@shinzon0]

Great demonstration in python, very nice!

[@AJ-et3vf]

Great video. Thank you

[@martinpuente7526]

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

[@chrisrycroft2010]

The material for videos 5.7 and 5.8 was given as a stand-alone lecture that was separate from the main course content. Because of this, I did not record videos for that material along with the others. I am intending to record videos on this material, but I want to do a good job so it will take some time. I may complete them around Jan/Feb 2024, outside of my usual teaching responsibilities. In the meantime, you may find the associated notes useful: people.math.wisc.edu/~chr/am205/notes/iter_lecture.pdf

[@isaaccastro7853]

Very nice!

[@dragonghost7018]

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?

[@chrisrycroft2010]

I think the only thing that you need to change is that when you compute the scalar products, such as alpha_m = q_m^T v, then they use the Hermitian transpose instead of the regular transpose. Therefore alpha_m = q_m^* v. That also affects the computation of the Euclidean norm. Instead of ||b||_2 = sqrt(b^T b), it becomes ||b||_2 = sqrt(b^* b). The Wikipedia page on the Lanczos algorithm presents the algorithm for the complex case: en.wikipedia.org/wiki/Lanczos_algorithm

[@sohailtabarhossain6096]

Thanks a lot 🙏 🌹

[@sophiaxiao5071]

Thank you

[@guidotoschi7284]

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.