66 research outputs found
Differential qd algorithm with shifts for rank-structured matrices
Although QR iterations dominate in eigenvalue computations, there are several
important cases when alternative LR-type algorithms may be preferable. In
particular, in the symmetric tridiagonal case where differential qd algorithm
with shifts (dqds) proposed by Fernando and Parlett enjoys often faster
convergence while preserving high relative accuracy (that is not guaranteed in
QR algorithm). In eigenvalue computations for rank-structured matrices QR
algorithm is also a popular choice since, in the symmetric case, the rank
structure is preserved. In the unsymmetric case, however, QR algorithm destroys
the rank structure and, hence, LR-type algorithms come to play once again. In
the current paper we discover several variants of qd algorithms for
quasiseparable matrices. Remarkably, one of them, when applied to Hessenberg
matrices becomes a direct generalization of dqds algorithm for tridiagonal
matrices. Therefore, it can be applied to such important matrices as companion
and confederate, and provides an alternative algorithm for finding roots of a
polynomial represented in the basis of orthogonal polynomials. Results of
preliminary numerical experiments are presented
A CMV--based eigensolver for companion matrices
In this paper we present a novel matrix method for polynomial rootfinding. By
exploiting the properties of the QR eigenvalue algorithm applied to a suitable
CMV-like form of a companion matrix we design a fast and computationally simple
structured QR iteration.Comment: 14 pages, 4 figure
Row Compression and Nested Product Decomposition of a Hierarchical Representation of a Quasiseparable Matrix
This research introduces a row compression and nested product decomposition of an nxn hierarchical representation of a rank structured matrix A, which extends the compression and nested product decomposition of a quasiseparable matrix. The hierarchical parameter extraction algorithm of a quasiseparable matrix is efficient, requiring only O(nlog(n))operations, and is proven backward stable. The row compression is comprised of a sequence of small Householder transformations that are formed from the low-rank, lower triangular, off-diagonal blocks of the hierarchical representation. The row compression forms a factorization of matrix A, where A = QC, Q is the product of the Householder transformations, and C preserves the low-rank structure in both the lower and upper triangular parts of matrix A. The nested product decomposition is accomplished by applying a sequence of orthogonal transformations to the low-rank, upper triangular, off-diagonal blocks of the compressed matrix C. Both the compression and decomposition algorithms are stable, and require O(nlog(n)) operations. At this point, the matrix-vector product and solver algorithms are the only ones fully proven to be backward stable for quasiseparable matrices. By combining the fast matrix-vector product and system solver, linear systems involving the hierarchical representation to nested product decomposition are directly solved with linear complexity and unconditional stability. Applications in image deblurring and compression, that capitalize on the concepts from the row compression and nested product decomposition algorithms, will be shown
Quasiseparable Hessenberg reduction of real diagonal plus low rank matrices and applications
We present a novel algorithm to perform the Hessenberg reduction of an
matrix of the form where is diagonal with
real entries and and are matrices with . The
algorithm has a cost of arithmetic operations and is based on the
quasiseparable matrix technology. Applications are shown to solving polynomial
eigenvalue problems and some numerical experiments are reported in order to
analyze the stability of the approac
Computing the k-th Eigenvalue of Symmetric -Matrices
The numerical solution of eigenvalue problems is essential in various
application areas of scientific and engineering domains. In many problem
classes, the practical interest is only a small subset of eigenvalues so it is
unnecessary to compute all of the eigenvalues. Notable examples are the
electronic structure problems where the -th smallest eigenvalue is closely
related to the electronic properties of materials. In this paper, we consider
the -th eigenvalue problems of symmetric dense matrices with low-rank
off-diagonal blocks. We present a linear time generalized LDL decomposition of
matrices and combine it with the bisection eigenvalue algorithm
to compute the -th eigenvalue with controllable accuracy. In addition, if
more than one eigenvalue is required, some of the previous computations can be
reused to compute the other eigenvalues in parallel. Numerical experiments show
that our method is more efficient than the state-of-the-art dense eigenvalue
solver in LAPACK/ScaLAPACK and ELPA. Furthermore, tests on electronic state
calculations of carbon nanomaterials demonstrate that our method outperforms
the existing HSS-based bisection eigenvalue algorithm on 3D problems.Comment: 14 pages, 11 figure
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