A new class of canonical forms is given proposed in which (A,C) is in Hessenberg observer or Schur form and output normal: I − A∗A = C∗C. Here, C is the d × n measurement matrix and A is the advance matrix. The (C,A) stack is expressed as the product of n orthogonal matrices, each of which depends on d parameters. State updates require onlyO(nd) operations and derivatives of the system with res...

In the paper, the authors recall some known determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, find alternative determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, and present several new recurrence relations for the Bernoulli numbers and polynomials.

In the paper, the authors find closed forms for derangement numbers in terms of the Hessenberg determinants, discover a recurrence relation of derangement numbers, present a formula for any higher order derivative of the exponential generating function of derangement numbers, and compute some related Hessenberg and tridiagonal determinants.

The class of eigenvalue problems for upper Hessenberg matrices of banded-plus-spike form includes companion and comrade matrices as special cases. For this class of matrices a factored form is developed in which the matrix is represented as a product of essentially 2 × 2 matrices and a banded upper-triangular matrix. A non-unitary analogue of Francis’s implicitly-shifted QR algorithm that prese...

Let there be given a probability measure μ on the unit circle T of the complex plane and consider the inner product induced by μ. In this paper we consider the problem of orthogonalizing a sequence of monomials {zk}k, for a certain order of the rk ∈ Z, by means of the Gram-Schmidt orthogonalization process. This leads to a basis of orthonormal Laurent polynomials {ψk}k. We show that the matrix ...

We will present a matrix framework for the conjugate gradient methods, which is expressed in terms of whole vector sequences instead of single vectors or initial parts of sequences. Using this framework extremely concise derivations of the conjugate gradient method, the Lanczos algorithm, and methods such as GMRES, QMR, CGS, can be given. This framework is then used to present some equivalent f...

A square matrix is called Hessenberg whenever each entry below the subdiagonal zero and on nonzero. Let M denote a matrix. Then circular upper-right corner of nonzero every other above superdiagonal zero. pair consists two diagonalizable linear maps finite-dimensional vector space, that act an eigenbasis one in fashion. A,A⁎ pair. We investigate six bases for underlying space we find attractive...

We present two variants of Moler and Stewart’s algorithm for reducing a matrix pair to Hessenberg-triangular (HT) form with increased data locality in the access to the matrices. In one of these variants, a careful reorganization and accumulation of Givens rotations enables the use of efficient level 3 BLAS. Experimental results on four different architectures, representative of current high pe...