ralphas / genericmrrr.jl

Symmetric Tridiagonal eigensolver in Julia

GenericMRRR.jl

A Symmetric Tridiagonal Eigen-solver in Julia: MRRR and dqds methods

This package provides methods for computing some or all of the eigenvalues and (optionally) eigenvectors of a real symmetric tridiagonal matrix. It is designed to work with element types supported by arithmetic similar to IEEE standards (especially NaN propagation), such as BigFloat, Quadmath.Float128, and DoubleFloat.

Usage

The package exports geigen! and geigvals! functions with the same API as the SymTridiagonal versions of eigen! and eigvals! in the LinearAlgebra standard library. For example,

n=128
D = rand(n)
D1 = rand(n-1)
A = SymTridiagonal(D, D1)
# only compute the first 10 eigenvalues and corresponding eigenvectors
E = geigen!(copy(A), 1:10)
# only compute eigenvalues in an interval
E = geigvals!(copy(A), -1.0, 1.0)

It also implements geigen! and geigvals! methods for dense Hermitian matrices, when the MRRR algorithm is selected.

A = Hermitian(rand(n,n))
E = geigen!(copy(A), MRRR())

Algorithms

The methods implement the Relatively Robust Representation scheme of Dhillon, Parlett, and associates, along with the dqds scheme for computing eigenvalues to high precision where feasible. The implementations closely follow those in LAPACK.

Notes

There are a few matrices for which the Float64 implementations in LAPACK fail; our implementation typically also fails for them - but often succeeds after promoting the same matrices to a wider type. In most such cases a RepresentationFailure exception is thrown.

Although these algorithms are usually more efficient than the Francis-QR scheme (which is implemented in GenericLinearAlgebra), they are far more complicated so the initial compilation takes several seconds.

A small pseudo-random perturbation is used in the algorithm. Unlike the LAPACK implementation, we use the default Julia RNG(s) without altering the seed, so there will be variations between results on a given matrix. These should almost always be roundoff-level, except for eigenvectors of effectively indistinguishable eigenvalues.

Credits

Almost all of this package was translated from LAPACK subroutines written by Dhillon, Parlett, Marques, et al. Translation errors are (of course) not their responsibility. Efforts have been made to be consistent with LAPACK behavior, but the agreement is not perfect.

Special thanks to Osni Marques for providing a collection of matrices for testing and debugging.

References

I.S.Dhillon & B.N.Parlett, "Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices," Linear Algebra Appl. 387, 1-28 (2004).

I.S.Dhillon, B.N.Parlett & C. Vömel, "The design and implementation of the MRRR algorithm," ACM TOMS 32, 533-560 (2006).

B.N.Parlett & O.A.Marques, "An implementation of the dqds algorithm," Linear Algebra Appl. 309, 217-259 (2000).