https://github.com/jagot/matrixpolynomials.jl
https://github.com/jagot/matrixpolynomials.jl
Last synced: 4 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 (over 5 years ago)
- Default Branch: master
- Last Pushed: 2024-02-23T09:08:52.000Z (over 1 year ago)
- Last Synced: 2025-02-22T14:38:49.622Z (5 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
[](https://jagot.github.io/MatrixPolynomials.jl/stable)
[](https://jagot.github.io/MatrixPolynomials.jl/dev)
[](https://github.com/jagot/MatrixPolynomials.jl/actions)
[](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.