The objective of this paper is to extend and redesign the block matrix reduction applied for the family of two-sided factorizations, introduced by Dongarra et al. [9], to the context of multicore architectures using algorithms-by-tiles. In particular, the Block Hessenberg Reduction is very often used as a pre-processing step in solving dense linear algebra problems, such as the standard eigenva...

Matrices with a skew-symmetric part of low rank arise in many applications, including path following methods and integral equations. This paper explores the properties of the Arnoldi process when applied to such a matrix. We show that an orthogonal Krylov subspace basis can be generated with short recursion formulas and that the Hessenberg matrix generated by the Arnoldi process has a structure...

Global and asymptotic convergence properties for the QR algorithm with Francis double shift are established for certain orthogonal similarity classes of 4 x 4 real matrices. It is shown that in each of the classes every unreduced Hessenberg matrix will decouple and that the rate of decoupling is almost always linear. The effect of the EISPACK exceptional shift strategy is shown to be negligible.

Let L̂ be a positive definite bilinear functional on the unit circle defined on Pn, the space of polynomials of degree at most n. Then its Geronimus transformation L is defined by L̂(p, q) = L ( (z − α)p(z), (z − α)q(z) ) for all p, q ∈ Pn, α ∈ C. Given L̂, there are infinitely many such L which can be described by a complex free parameter. The Hessenberg matrix that appears in the recurrence rela...

One of the popular test matrices for eigenvalue routines is Frank matrix due to its well-conditioned and poorly conditioned eigenvalues. All eigenvalues are real, positive different. Sturm Theorem a very useful tool computing tridiagonal symmetric matrices. In this paper, we apply generalized which special form Hessenberg examine by using property. Moreover, illustrate our results with an examp...

To every infinite lower Hessenberg matrix D is associated a linear operator on l2. In this paper we prove the similarity of the operator D − ∆, where ∆ belongs to a certain class of compact operators, to the operator D−∆′, where ∆′ is of rank one. We first consider the case when ∆ is lower triangular and has finite rank; then we extend this to ∆ of infinite rank assuming that D is bounded. In S...

In applying the QR algorithm to compute the eigenvalues of a unitary Hessenberg matrix, a projected Wilkinson shift of unit modulus is proposed and proved to give global convergence with (at least) a quadratic asymptotic rate for the QR iteration. Experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence.

In this paper, we propose another version of the full orthogonalization method (FOM) for the resolution of linear system Ax b = , based on an extended definition of Sturm sequence in the calculation of the determinant of an upper hessenberg matrix in o(n2). We will also give a new version of Givens method based on using a tensor product and matrix addition. This version can be used in parallel ...

An algorithm for reduction of a regular matrix pair (A; B) to block Hessenberg-triangular form is presented. This condensed form Q T (A; B)Z = (H; T), where H and T are block upper Hessenberg and upper triangular, respectively, and Q and Z orthogonal, may serve as a rst step in the solution of the generalized eigenvalue problem Ax = Bx. It is shown how an elementwise algorithm can be reorganize...