more further reading
This commit is contained in:
parent
3ae5807516
commit
6f45471a73
@ -183,4 +183,4 @@ The *nice* case of diagonalization is when you have **orthonormal eigenvectors**
|
||||
* Continued summary of large-scale linear algebra from lecture 12, mentioning randomized algorithms (which we will cover in more detail later), such as "sketched" least-squares and randomized SVD, and also specialized algorithms for particular cases.
|
||||
* Krylov methods: defined [Krylov subspaces](https://en.wikipedia.org/wiki/Krylov_subspace) reachable by iterative algorithms, defined a Krylov algorithm (loosely) an iterative algorithm that finds the "best" solution in the whole Krylov space (possibly approximately) on the n-th step. Gave [power iteration](https://en.wikipedia.org/wiki/Power_iteration) for largest |λ| as an example of something *not* a Krylov method. Explained why the basis (b Ab A²b ⋯) is a poor (ill-conditioned) choice, and instead explained the [Arnoldi iteration](https://en.wikipedia.org/wiki/Arnoldi_iteration) to find an orthonormal basis Qₙ by (essentially) Gram–Schmidt, leading to the [GMRES algorithm](https://en.wikipedia.org/wiki/Generalized_minimal_residual_method) for Ax=b.
|
||||
|
||||
**Further reading:** A review of randomized linear algebra can be found in the Strang textbook sec. II.4, and also in [Halko, Martinsson, and Tropp (2011)](https://epubs.siam.org/doi/10.1137/090771806). A recent paper on a variety of new randomized algorithms, e.g. for "sketched" least-square problems or to accelerate iterative algorithms like GMRES, is [Nakatsukasa and Tropp (2022)](https://arxiv.org/pdf/2111.00113.pdf). A nice review of the randomized SVD can be found in a blog post by [Gregory Gundersen (2019)](https://gregorygundersen.com/blog/2019/01/17/randomized-svd/).
|
||||
**Further reading:** Arnoldi iterations and GMRES are covered in the Strang textbook section II.1, and briefly in [OCW lecture 12](https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/resources/lecture-12-computing-eigenvalues-and-singular-values/); much more detail is found other sources (Trefethen, etc.) noted in the further reading for Lecture 12. A review of randomized linear algebra can be found in the Strang textbook sec. II.4, and also in [Halko, Martinsson, and Tropp (2011)](https://epubs.siam.org/doi/10.1137/090771806). A recent paper on a variety of new randomized algorithms, e.g. for "sketched" least-square problems or to accelerate iterative algorithms like GMRES, is [Nakatsukasa and Tropp (2022)](https://arxiv.org/pdf/2111.00113.pdf). A nice review of the randomized SVD can be found in a blog post by [Gregory Gundersen (2019)](https://gregorygundersen.com/blog/2019/01/17/randomized-svd/).
|
||||
|
Loading…
x
Reference in New Issue
Block a user