https://github.com/klausc/ldufacts.jl

P' L D L' P factorization of symmetric real or hermitian indefinite matrices
https://github.com/klausc/ldufacts.jl

factorize hermitian indefinite linear-algebra symmetric

Last synced: 8 days ago
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P' L D L' P factorization of symmetric real or hermitian indefinite matrices

• Host: GitHub
• URL: https://github.com/klausc/ldufacts.jl
• Owner: KlausC
• License: mit
• Created: 2022-02-10T11:53:29.000Z (almost 3 years ago)
• Default Branch: main
• Last Pushed: 2022-02-21T16:02:23.000Z (almost 3 years ago)
• Last Synced: 2024-10-19T17:46:50.409Z (3 months ago)
• Topics: factorize, hermitian, indefinite, linear-algebra, symmetric
• Language: Julia
• Homepage:
• Size: 40 KB
• Stars: 0
• Watchers: 2
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

        
# LDUFacts

[![Build Status][gha-img]][gha-url]     [![Coverage Status][codecov-img]][codecov-url]

## Introduction

This package is dedicated to the L-D-Lt factorization of symmetric real and Hermitian matrices.

In contrast to Cholesky factorization, which is appropriate for most use cases, it has the following features:

- handle also the indefinite case (Cholesky requires positive (semi)-definite)

- square-root free algorithm

- exact operation with real and complex rational types

- exact operation with user-defined field types

- option of full pivot search

## Algorithm

Factors `P, L, D` of matrix `A` are found in a way, that `P' * A * P = L * D * L'`.

Here `D` is diagonal and real, `L` is lower unit triangular, and `P` is a 'simple' square matrix.

With diagonal pivot search, `P` is a permutation matrix, with full pivot search up to `n-1` nonzero additional matrix elements may occur.

Without pivot search, `P` is the unit matrix.

When not a full pivot search is done, it is possible, that the algorithm fails before the factors are found.

For positive definite matrices, the no-pivot version is guaranteed to succeed. For positive semi-definite matrices,

a diagonal pivot search always succeeds, for infefinite matrices, a full pivot search is required for that.

There are examples of regular matrices, (e.g. `[0. 1; 1 0]`), which do not have any factorization where `P` is a permutation. For those cases

the full pivot search is required.

## Usage

    ]add LDUFacts

    using LDUFacts

    

    A = Matrix(Symmetric(rand(5, 5)))

    la = ldu(A, FullPivot())

    

    issuccess(la)

    rank(la)

    la.L

    la.d

    la.P' * A * la.P - la.L * la.D * la.L' # should be close to zero

[gha-img]: https://github.com/KlausC/LDUFacts.jl/actions/workflows/CI.yml/badge.svg

[gha-url]: https://github.com/KlausC/LDUFacts.jl/actions/workflows/CI.yml

[codecov-img]: https://codecov.io/gh/KlausC/LDUFacts.jl/branch/main/graph/badge.svg

[codecov-url]: https://codecov.io/gh/KlausC/LDUFacts.jl