45 research outputs found
MRRR-based Eigensolvers for Multi-core Processors and Supercomputers
The real symmetric tridiagonal eigenproblem is of outstanding importance in
numerical computations; it arises frequently as part of eigensolvers for
standard and generalized dense Hermitian eigenproblems that are based on a
reduction to tridiagonal form. For its solution, the algorithm of Multiple
Relatively Robust Representations (MRRR or MR3 in short) - introduced in the
late 1990s - is among the fastest methods. To compute k eigenpairs of a real
n-by-n tridiagonal T, MRRR only requires O(kn) arithmetic operations; in
contrast, all the other practical methods require O(k^2 n) or O(n^3) operations
in the worst case. This thesis centers around the performance and accuracy of
MRRR.Comment: PhD thesi
Darboux transformation and perturbation of linear functionals
28 pages, no figures.-- MSC2000 codes: 42C05; 15A23.MR#: MR2055354 (2005b:15027)Zbl#: Zbl 1055.42016Let L be a quasi-definite linear functional defined on the linear space of polynomials with real coefficients. In the literature, three canonical transformations of this functional are studied: , and where denotes the linear functional , and is the Kronecker symbol. Let us consider the sequence of monic polynomials orthogonal with respect to L. This sequence satisfies a three-term recurrence relation whose coefficients are the entries of the so-called monic Jacobi matrix. In this paper we show how to find the monic Jacobi matrix associated with the three canonical perturbations in terms of the monic Jacobi matrix associated with L. The main tools are Darboux transformations. In the case that the LU factorization of the monic Jacobi matrix associated with xL does not exist and Darboux transformation does not work, we show how to obtain the monic Jacobi matrix associated with as a limit case. We also study perturbations of the functional L that are obtained by combining the canonical cases. Finally, we present explicit algebraic relations between the polynomials orthogonal with respect to L and orthogonal with respect to the perturbed functionals.The work of the authors has been partially supported by Dirección General de Investigación (Ministerio de Ciencia y TecnologÃa) of Spain under grant BFM 2003-06335-C03-02 and NATO collaborative grant PST.CLG.979738.Publicad
New Structured Matrix Methods for Real and Complex Polynomial Root-finding
We combine the known methods for univariate polynomial root-finding and for
computations in the Frobenius matrix algebra with our novel techniques to
advance numerical solution of a univariate polynomial equation, and in
particular numerical approximation of the real roots of a polynomial. Our
analysis and experiments show efficiency of the resulting algorithms.Comment: 18 page
The Orthogonal QD-Algorithm
The orthogonal qd-algorithm is presented to compute the singular
value decomposition of a bidiagonal matrix. This algorithm
represents a modification of Rutishauser's qd-algorithm, and it
is capable of determining all the singular values to high relative
precision. A generalization of the Givens transformation is also
introduced, which has applications besides the orthogonal qd-algorithm.
The shift strategy of the orthogonal qd-algorithm is based on
Laguerre's method, which is used to compute a lower bound for the
smallest singular value of the bidiagonal matrix. Special attention
is devoted to the numerically stable evaluation of this shift.
(Also cross-referenced as UMIACS-TR-94-9.1