A translation of Tim Davis's Concise LDLᵀ Factorization, part of SuiteSparse with several improvements.

Please cite this repository if you use LDLFactorizations.jl in your work: see `CITATION.bib`

.

This package is appropriate for matrices A that possess a factorization of the
form LDLᵀ without pivoting, where L is unit lower triangular and D is *diagonal* (indefinite in general), including definite and quasi-definite matrices.

LDLFactorizations.jl should not be expected to be as fast, as robust or as accurate as factorization packages such as HSL.jl, MUMPS.jl or Pardiso.jl. Those are multifrontal and/or implement various forms of parallelism, and employ sophisticated pivot strategies.

The main advantages of LDLFactorizations.jl are that

- it is very short and has a small footprint;
- it is in pure Julia, and so

2.a. it does not require external compiled dependencies;

2.b. it will work with multiple input data types.

Whereas MUMPS.jl, HSL.jl and Pardiso.jl only work with single and double precision reals and complex data types, LDLFactorizations.jl accepts any numerical data type.

```
julia> ]
pkg> add LDLFactorizations
```

The only exported functions are `ldl()`

, `\`

and `ldiv!`

.
Calling `ldl()`

with a dense array converts it to a sparse matrix.
A permutation ordering can be supplied: `ldl(A, p)`

where `p`

is an `Int`

array representing a permutation of the integers between 1 and the order
of `A`

.
If no permutation is supplied, one is automatically computed using AMD.jl.
Only the upper triangle of `A`

is accessed.

`ldl`

returns a factorization in the form of a `LDLFactorization`

object.
The `\`

and `ldiv!`

methods are implemented for objects of type `LDLFactorization`

so that
solving a linear system is as easy as

```
LDLT = ldl(A) # LDLᵀ factorization of A
x = LDLT \ b # solves Ax = b
ldiv!(LDLT, b) # computes LDLT \ b in-place and overwriting b to store the result
y = similar(b)
ldiv!(y, LDLT, b) # computes LDLT \ b in-place and store the result in y
```

The factorization can of course be reused to solve for multiple right-hand sides.

Factors can be accessed as `LDLT.L`

and `LDLT.D`

, and the permutation vector as `LDLT.P`

.
Because the L factor is unit lower triangular, its diagonal is not stored.
Thus the factors satisfy: PAPᵀ = (L + I) D (L + I)ᵀ.

Timothy A. Davis. 2005. Algorithm 849: A concise sparse Cholesky factorization package. ACM Trans. Math. Softw. 31, 4 (December 2005), 587-591. DOI 10.1145/1114268.1114277.

Like the original LDL, this package is distributed under the LGPL.

07/23/2017

25 days ago

88 commits