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.
The file arrowhead1.jl contains definitions of types
SymArrow (arrowhead) and SymDPR1. Full matrices are accessible
with full(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 eig() which
computes the k-th eigenpair of an ordered unreduced SymArrow,
and the driver function eig() 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 eig()
which computes the k-th eigenpair of an ordered unreduced SymDPR1,
and the driver function eig() which computes all eigenvalues and
eigenvectors of a SymDPR1.
The file arrowhead5.jl contains definition of type HalfArrow.
HalfArrow is of the form [diagm(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, sigma, v of an ordered
unreduced HalfArrow, and the driver function svd() which computes all
singular values and vectors of a HalfArrow.
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.
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 eig() 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.
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).
The Matlab version of the routines used in the papers are written Ivan Slapnicar and Nevena Jakovcevic Stor. The first version of Julia routines was written by Ivan Slapnicar during a visit to MIT, and later version were written by Ivan Slapnicar and Nevena Jakovcevic Stor.
Double the working precision is implemented by using routines by T. J. Dekker (1971) from the package DoubleDouble by Simon Byrne.
Highly appreciated help and advice came from Jiahao Chen, Andreas Noack, Jake Bolewski and Simon Byrne.