In this manuscript we will present a new fast technique for solving the generalized eigenvalue problem T x = λSx, in which both matrices T and S are symmetric tridiagonal matrices and the matrix S is assumed to be positive definite.1 A method for computing the eigenvalues is translating it to a standard eigenvalue problem of the following form: L−1T L−T (LT x) = λ(LT x), where S = LLT is the Ch...

The main goal of this article is to establish several new \({\mathbb {A}}\)-numerical radius equalities for \(n\times n\) circulant, skew imaginary tridiagonal, and anti-tridiagonal operator matrices, where {A}}\) the diagonal matrix whose entries are positive bounded A. Some special cases our results lead earlier works in literature, which shows that more general. Further, some pinching type i...

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → V that satisfy the following two conditions: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal. (ii) There exists a basis for V with respec...

[15] D. O'Leary and G. W. Stewart. Computing the eigenvalues and eigenvectors of symmetric arrowhead matrices. [17] A. Sameh and D. Kuck. A parallel QR algorithm for symmetric tridiagonal matrices. [21] Zhonggang Zeng. The acyclic eigenproblem can be reduced to the arrowhead one. [22] Hongyuan Zha. A two-way chasing scheme for reducing a symmetric arrowhead matrix to tridiagonal form. Scientic ...

Non-symmetric and symmetric twisted block factorizations of block tridiagonal matrices are discussed. In contrast to non-blocked factorizations of this type, localized pivoting strategies can be integrated which improves numerical stability without causing any extra fill-in. Moreover, the application of such factorizations for approximating an eigenvector of a block tridiagonal matrix, given an...

This paper discusses scalability and data layout issues arising in the development of a parallel algorithm for reducing a banded matrix to tridiagonal form. As it turns out, balancing the memory and computational complexity of the reduction of the matrix and the accumulation of the associated orthogonal matrix is the key to scalability and sustained performance.

An efficient method to solve the eigenproblem of N x N symmetric tridiagonal matrices is proposed. Unlike the standard eigensolvers that necessitate O(N3) operations to compute the eigenvectors of such matrices, the proposed method computes both the eigenvalues and eigenvectors with only O(N2) operations. The method is based on serial implementation of the recently introduced Divide and Conquer...

As part of the Fujitsu-ANU Parallel Mathematical Subroutine Library Project we have developed a suite of parallel eigenvalue decomposition routines to handle a variety of input matrices. For a dense real symmetric matrix there are routines using the Jacobi algorithm and routines which rst reduce the symmetric matrix to tridiagonal form. All algorithms have been developed to fully exploit the ve...