Arrowhead and Diagonal-plus-rank-one Eigenvalue Solvers

Notice

This is a major rewrite of the package with following features:

• works with Julia 1.5+
• uses multi-threading
• memory usage is improved

Basics

N.B. The latest master is obtained by pkg> add Arrowhead#master.

Usage:

using Arrowhead
# Set the dimension
n=10
# Generate SymArrow
A=GenSymArrow(n,n)
# Compute the eigenvalue decomposition
E,Info=eigen(A)
# Extract eigenvalues
Λ=E.values
# Extract eigenvectors
U=E.vectors

For further examples see the file runtests.jl.

The package contains functions for forward stable algorithms which compute:

• all eigenvalues and eigenvectors of a real symmetric arrowhead matrices,
• all eigenvalues and eigenvectors of rank-one modifications of diagonal matrices (DPR1), and
• all singular values and singular vectors of half-arrowhead matrices.

The last class of matrices typically appears in SVD updating problems. The algorithms and their analysis are given in the references.

Eigen/singular values are computed in a forward stable manner. Eigen/singular vectors are computed entry-wise to almost full accuracy, so they are automatically mutually orthogonal. The algorithms are based on a shift-and-invert approach. Only a single element of the inverse of the shifted matrix eventually needs to be computed with double the working precision.

The package also contains functions for applications:

• divide-and-conquer function for symmetric tridiagonal eigenvalue problem
• roots of real polynomials with real distinct roots.

Contents

Arrowhead and DPR1 Eigenvalues

The file arrowhead1.jl contains definitions of types SymArrow (arrowhead) and SymDPR1. Full matrices are accessible with Matrix(A).

The file arrowhead3.jl contains:

• functions to generate random symmetric arrowhead and DPR1 matrices, GenSymArrow and GenSymDPR1, respectively,
• three functions inv() which compute various inverses of SymArrow matrices,
• two functions bisect() which compute outer eigenvalues of SymArrow and SymDPR1 matrices,
• the main computational function eigen() which computes the k-th eigenpair of an ordered unreduced SymArrow, and
• the driver function eigen() which computes all eigenvalues and eigenvectors of a SymArrow.

The file arrowhead4.jl contains:

• three functions inv() which compute various inverses of SymDPR1 matrices,
• the main computational function eigen() which computes the k-th eigenpair of an ordered unreduced SymDPR1, and
• the driver function eigen() which computes all eigenvalues and eigenvectors of a SymDPR1.

Half-arrowhead SVD

The file arrowhead5.jl contains definition of type HalfArrow. The type is of the form [Diagonal(A.D) A.z] where either length(A.z)=length(A.D) or length(A.z)=length(A.D)+1, thus giving two possible forms of the SVD rank one update.

The file arrowhead6.jl contains:

• the function doubledot(),
• three functions inv() which compute various inverses of HalfArrow matrices,
• the main computational function svd() which computes the k-th singular value triplet u, σ, v of an ordered unreduced HalfArrow, and
• the driver function svd() which computes all singular values and vectors of a HalfArrow.

Tridiagonal Divide and Conquer

The file arrowhead7.jl contains a simple function tdc() which implements divide-and-conquer method for SymTridiagonal matrices by spliting the matrix in two parts and connecting the parts via eigenvalue decomposition of arrowhead matrix.

Polynomial Roots

The file arrowhead7.jl conatains the function rootsah() which computes the roots of Int32, Int64, Float32 and Float64 polynomials with all distinct real roots. The computation is forward stable. The program uses SymArrow form of companion matrix in barycentric coordinates and the corresponding eigen() function specially designed for this case. The file also contains three functions inv(). Similarly, the file arrowhead8.jl conatains the function rootsah() which computes the roots of BigInt and BigFloat polynomials with all distinct real roots. The file also contains function rootsWDK(), an implementation of the Weierstrass-Durand-Kerner polynomial root finding algorithm.

Authors and References

The functions for arrowhead and half-arrowhead matrices were developed and analysed by Jakovcevic Stor, Barlow and Slapnicar (2013) (see also arXiv:1302.7203). The routines for DPR1 matrices are described and analysed in Jakovcevic Stor, Barlow and Slapnicar (2015) (the paper is freely downloadable until Nov 15, 2015, see also arXiv:1405.7537). The polynomial root finder is described and analyzed in Jakovcevic Stor and Slapnicar (2015).

Double the working precision is implemented by using routines by T. J. Dekker (1971) from the package DoubleDouble by Simon Byrne. This package is not maintained any more and is deprecated in favor of DoubleFloats. However, we are using lightweight and faster implementation in the file DoubleDouble.jl which is now ported to Julia 1.0+.

Thanks

Highly appreciated help and advice came from Jiahao Chen, Andreas Noack, Jake Bolewski and Simon Byrne.