We present an alternative algorithm and implementation for theHessenberg-triangular reduction, essential step in the QZalgorithm solving generalized eigenvalue problems. Thereduction has a cubic computational complexity, hence,high-performance implementations are compulsory keeping thecomputing time under control. Our is of simplemathematical nature relies on connection betweengeneralized class...

Motivated by a 1993 conjecture of Stanley and Stembridge, Shareshian and Wachs conjectured that the characteristic map takes the dot action of the symmetric group on the cohomology of a regular semisimple Hessenberg variety to ωXG(t), where XG(t) is the chromatic quasisymmetric function of the incomparability graph G of the corresponding natural unit interval order, and ω is the usual involutio...

Random orthogonal matrix (ROM) simulation is a very fast procedure for generating multivariate random samples that always have exactly the same mean, covariance and Mardia multivariate skewness and kurtosis. This paper investigates how the properties of parametric, data-specific and deterministic ROM simulations are influenced by the choice of orthogonal matrix. Specifically, we consider how cy...

In [E. Babolian, M.M. Hosseini, Reducing index, and pseudospectral methods for differential–algebraic equations, Appl. Math. Comput. 140 (2003) 77–90] a reducing index method has been proposed for some cases of semi-explicit DAEs (differential algebraic equations). In this paper, this method is generalized to more cases. Also, it is focused on Hessenberg index 2 systems and proposed reduction i...

A new cache-efficient algorithm for reduction from block Hessenberg form to Hessenberg form is presented and evaluated. The algorithm targets parallel computers with shared memory. One level of look-ahead in combination with a dynamic load-balancing scheme significantly reduces the idle time and allows the use of coarse-grained tasks. The coarse tasks lead to high-performance computations on ea...

Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by...

We propose a new algorithm for block-wise solution of the generalized Sylvesterobserver equation XA − FXE = GC, where the matrices A,E, and C are given, the matrices X,F , and G need to be computed, and the matrix E may be singular. The algorithm is based on an orthogonal decomposition of the triplet (A,E,C) to the observer-Hessenberg-triangular form. It is a natural generalization of the widel...

Three versions of an algorithm for balancing three matrices simultaneously are proposed. The balancing is performed by premultiplication and postmultiplication with positive definite diagonal matrices, in order to reduce the magnitude range of all elements in the involved matrices. Some numerically stable algorithms, when applied to matrices with a wide range in the magnitude of elements, can p...

Some new formulas related to the well-known symmetric Lucas polynomials are primary focus of this article. Different approaches used for establishing these formulas. A matrix approach is followed in order obtain some fundamental properties. Particularly, recurrence relations and determinant forms determined by suitable Hessenberg matrices. Conjugate generating functions derived examined. Severa...

We introduce an iterative method named Gpmr (general partitioned minimum residual) for solving block unsymmetric linear systems. is based on a new process that simultaneously reduces two rectangular matrices to upper Hessenberg form and closely related the block-Arnoldi process. tantamount Block-Gmres with right-hand sides in which approximate solutions are summed at each iteration, but its sto...