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In linear algebra, the order-r Krylov subspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspace spanned by the images of b under the first r powers of A (starting from A 0 = I {\displaystyle A^{0}=I} ), that is,
The analysis highlights Applications, Art and Measurement as prominent areas in the source structure around Krylov subspace.
Source areas are shown by the number of related topics found in each part of the analysis. Use smaller areas too: they can reveal specialized angles and content gaps.
Smaller areas are not necessarily less important. They contain fewer connections in this analysis and can be useful for finding specialized angles or coverage gaps.
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The extracted context around Krylov subspace shows recurring relationship patterns in the source. For example, Krylov subspace → BiCGSTAB, Conjugate, GMRES, IDR, Induced, Krylov, MINRES, QMR, TFQMR Another extracted example is Krylov subspace → Ab, Arnoldi, Gramians, Krylov, Many, Modern, Starting. Use these groups to spot repeated connection types before inspecting the individual relationships.
Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.
displaystyle krylov linear methods subspace isbn dimension mathcal operatorname systems algebra matrix dim span ab ldots subspaces deg used finding
TTTA extracted 25 structured relationships around Krylov subspace. Examples in this analysis include Arnoldi iteration can be used for finding one → instance of → These tests are equivalent to finding the span of the Gramians associated with the system/output maps so the uncontrollable and unobservable subspaces are simply the orthogonal… and Krylov subspace → has method → Krylov. The table shows each extracted connection, where it came from and its confidence.
| Subject | Predicate | Object | Confidence | Src |
|---|---|---|---|---|
| Arnoldi iteration can be used for finding one | instance of | These tests are equivalent to finding the span of the Gramians associated with the system/output maps so the uncontrollable and unobservable subspaces are simply the orthogonal… | 0.80 | text |
| Krylov subspace | has method | Krylov | 0.60 | section |
| Krylov subspace | has method | Conjugate | 0.60 | section |
| Krylov subspace | has method | IDR | 0.60 | section |
| Krylov subspace | has method | Induced | 0.60 | section |
| Krylov subspace | has method | GMRES | 0.60 | section |
| Krylov subspace | has method | BiCGSTAB | 0.60 | section |
| Krylov subspace | has method | QMR | 0.60 | section |
| Krylov subspace | has method | TFQMR | 0.60 | section |
| Krylov subspace | has method | MINRES | 0.60 | section |
| Krylov subspace | related to Issues | Krylov | 0.60 | section |
| Krylov subspace | related to Issues | Lanczos | 0.60 | section |
The concept neighborhoods around Krylov subspace bring nearby vocabulary together. In this analysis, examples include Subspace, Methods and Linear. Use the clusters to find adjacent concepts and terminology that may deserve separate research.
For Krylov subspace, one of the stronger structural bridges in this analysis connects Krylov subspace with Use. Bridges highlight paths between different parts of the map and can reveal research angles that are easy to miss in a flat list.
TTTA analyzes the structure around Krylov subspace to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as Applications, Art & Measurement, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.
Source: Wikipedia — Krylov subspace · EN edition · Analysis: TopicsToTalkAbout
