This paper presents an improved algorithm for computing the Combinatorial Canonical Form (CCF) of a layered mixed matrix A = Q T , which consists of a numerical matrix Q and a generic matrix T . The CCF is the (combinatorially unique) nest block-triangular form obtained by the row operations on the Q-part, followed by permutations of rows and columns of the whole matrix. The main ingredient of ...

We consider the Schur complement operation for symmetric matrices over GF(2), which we identify with graphs through the adjacency matrix representation. It is known that Schur complementation for such a matrix (i.e., for a graph) can be decomposed into a sequence of two types of elementary Schur complement operations: (1) local complementation on a looped vertex followed by deletion of that ver...

This paper is concerned with the reduction of a unitary matrix U to CMV-like shape. A Lanczos–type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the Hermitian part of U , i.e., of the matrix U +UH . By elaborating on the Lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If U is ran...

We present a fast condition estimation algorithm for the eigenvalues of a class of structured matrices. These matrices are low rank modifications to Hermitian, skew-Hermitian, and unitary matrices. Fast structured operations for these matrices are presented, including Schur decomposition, eigenvalue block swapping, matrix equation solving, compact structure reconstruction, etc. Compact semisepa...

In this paper we describe block algorithms for the reduction of a real symmetric matrix to tridiagonal form and for the reduction of a general real matrix to either bidiagonal or Hessenberg form using Householder transformations. The approach is to aggregate the transformations and to apply them in a blocked fashion, thus achieving algorithms that are rich in matrix-matrix operations. These red...

In this paper we describe block algorithms for the reduction of a real symmetric matrix to tridiagonal form and for the reduction of a general real matrix to either bidiagonal or Hessenberg form using Householder transformations. The approach is to aggregate the transformations and to apply them in a blocked fashion, thus achieving algorithms that are rich in matrix-matrix operations. These red...

In this paper we describe block algorithms for the reduction of a real symmetric matrix to tridiagonal form and for the reduction of a general real matrix to either bidiagonal or Hessenberg form using Householder transformations. The approach is to aggregate the transformations and to apply them in a blocked fashion, thus achieving algorithms that are rich in matrix-matrix operations. These red...

Jacobi-Davidson methods can efficiently compute a few eigenpairs of a large sparse matrix. Block variants of JacobiDavidson are known to be more robust than the standard algorithm, but they are usually avoided as the total number of floating point operations increases. We present the implementation of a block Jacobi-Davidson solver and show by detailed performance engineering and numerical expe...