Arrowhead and Diagonal-plus-rank-one Eigenvalue Solvers
Notice
This is a major rewrite of the packege with following features:
- works with Julia 1.0+
- 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 furthaer examples see the file runtests.jl.
The package contains routines 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 entrywise 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 routines for applications:
- divide-and-conquer routine 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 routines 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.