Unitary matrices have a rich mathematicalstructure which is closely analogous to real symmetric matrices. For real symmetric matrices this structure can be exploited to develop very eecient numerical algorithms and for some of these algorithms unitary analogues are known. Here we present a unitary analogue of the bisection method for symmetric tridiagonal matrices. Recently Delsarte and Genin i...

Inverse iteration is known to be an effective method for computing eigenvectors corresponding simple and well-separated eigenvalues. In the non-symmetric case, solution of shifted Hessenberg systems a central step. Existing inverse solvers approach with either RQ or LU factorizations and, once factored, solve systems. This has limited level-3 BLAS potential since distinct shifts have factorizat...

A Schur-type algorithm is presented for computing recursively the triangular factorization R = LU of a strongly nonsingular n n matrix satisfying a displacement equation RY V R = GH with Hessenberg matrices Y and V and n matrices G, H. If is small compared with n and the matrices Y and V admit fast matrix-vector multiplication, the new algorithm is fast in the sense that it will require less th...

An operator theoretic approach to orthogonal rational functions on the unit circle with poles in its exterior is presented in this paper. This approach is based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two different alternatives are discussed, depending whether we use for the matrix represen...

Various algorithms connected with the computation of the minimal polynomial of a square n n matrix over a eld K are presented here. The complexity of the rst algorithm, where the complete factorization of the characteristic polynomial is needed, is O(p nn 3). It produces the minimal polynomial and all characteristic subspaces of a matrix of size n. Furthermore an iterative algorithm for the min...

Various algorithms connected with the computation of the minimal polynomial of a square n×n matrix over a field k are presented here. The complexity of the first algorithm, where the complete factorization of the characteristic polynomial is needed, is O( √ nn). It produces the minimal polynomial and all characteristic subspaces of a matrix of size n. Furthermore an iterative algorithm for the ...

Abstract. The Pfaff lattice is an integrable system arising from the SR-group factorization in an analogous way to how the Toda lattice arises from the QR-group factorization. In our recent paper [Intern. Math. Res. Notices, (2007) rnm120], we studied the Pfaff lattice hierarchy for the case where the Lax matrix is defined to be a lower Hessenberg matrix. In this paper we deal with the case of ...

We describe a matrix multiply based block unsymmetric inverse iteration solver for upper Hessenberg matrices. Our kernel is robust in that it prevents overrow by scaling. It uses new techniques to ensure performance is not sacriiced when scaling is not necessary. Finally, we give results on a parallel implementation on an Intel Paragon TM supercomputer.

We present a Hessenberg reduction (HR) algorithm for hybrid multicore + GPU systems that gets more than 16× performance improvement over the current LAPACK algorithm running just on current multicores (in double precision arithmetic). This enormous acceleration is due to proper matching of algorithmic requirements to architectural strengths of the hybrid components. The reduction itself is an i...