https://github.com/jagot/matrixpolynomials.jl

https://github.com/jagot/matrixpolynomials.jl

Last synced: about 2 months ago
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• Host: GitHub
• URL: https://github.com/jagot/matrixpolynomials.jl
• Owner: jagot
• License: mit
• Created: 2019-11-01T22:10:13.000Z (about 5 years ago)
• Default Branch: master
• Last Pushed: 2024-02-23T09:08:52.000Z (11 months ago)
• Last Synced: 2024-10-21T07:15:12.253Z (3 months ago)
• Language: Julia
• Size: 939 KB
• Stars: 8
• Watchers: 3
• Forks: 1
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

        
# MatrixPolynomials.jl

[![Stable](https://img.shields.io/badge/docs-stable-blue.svg)](https://jagot.github.io/MatrixPolynomials.jl/stable)

[![Dev](https://img.shields.io/badge/docs-dev-blue.svg)](https://jagot.github.io/MatrixPolynomials.jl/dev)

[![Build Status](https://github.com/jagot/MatrixPolynomials.jl/workflows/CI/badge.svg)](https://github.com/jagot/MatrixPolynomials.jl/actions)

[![Coverage](https://codecov.io/gh/jagot/MatrixPolynomials.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/jagot/MatrixPolynomials.jl)

This package aids in the computation of the action of a matrix

polynomial on a vector, i.e. `p(A)v`, where `A` is a (square) matrix

(or a linear operator) that is supplied to the polynomial `p`. The

matrix polynomial `p(A)` is never formed explicitly, instead only its

action on `v` is evaluated. This is commonly used in time-stepping

algorithms for ordinary differential equations (ODEs) and discretized

partial differential equations (PDEs) where `p` is an approximation of

the exponential function (or the related `φ` functions:

`φ₀(z) = exp(z)`, `φₖ₊₁ = [φₖ(z)-φₖ(0)]/z`, `φₖ(0)=1/k!`) on the

field-of-values of the matrix `A`, which for the methods in this

package needs to be known before-hand.

## Alternatives

Other packages with similar goals, but instead based on matrix

polynomials found via Krylov iterations are

- https://github.com/JuliaDiffEq/ExponentialUtilities.jl

- https://github.com/Jutho/KrylovKit.jl

Krylov iterations do not need to know the field-of-values of the

matrix `A` before-hand, instead, an orthogonal basis is built-up

on-the-fly, by repeated action of `A` on test vectors: `Aⁿ*v`. This

process is however very sensitive to the condition number of `A`,

something that can be alleviated by iterating a shifted and inverted

matrix instead: `(A-σI)⁻¹` (rational Krylov). Not all matrices/linear

operators are easily inverted/factorized, however.

Moreover, the Krylov iterations for general matrices (then called

Arnoldi iterations) require long-term recurrences with mutual

orthogonalization along with inner products, all of which can be

costly to compute. Finally, a subspace approximation of the polynomial

`p` of a upper Hessenberg matrix needs to computed. The

real-symmetric/complex-Hermitian case (Lanczos iterations) reduces to

three-term recurrences and a tridiagonal subspace matrix. In contrast,

the polynomial methods of this packages two-term recurrences only, no

orthogonalization (and hence no inner products), and finally no

evaluation of the polynomial on a subspace matrix. This could

potentially mean that the methods are easier to implement on a GPU.