This research introduces a row compression and nested product decomposition of an n × n hierarchical representation of a rank structured matrix A, which extends the compression and nested product decomposition of a quasiseparable matrix. The hierarchical parameter extraction algorithm of a quasiseparable matrix is efficient, requiring only O(nlog(n)) operations, and is proven backward stable. T...

AbstractGeometric transformations are fundamental aspects of all computer graphics rendering and are found throughout the computer graphics pipeline. In earlier work we presented mathematical treatment of this topic. The different concepts were combined into a general mathematical procedure based on the concept of vector spaces in linear algebra. While we established the mathematical aspects of...

This paper gives an overview of matrix transformations for finding rightmost eigenvalues of Ax = kx and Ax = kBx with A and B real non-symmetric and B possibly singular. The aim is not to present new material, but to introduce the reader to the application of matrix transformations to the solution of large-scale eigenvalue problems. The paper explains and discusses the use of Chebyshev polynomi...

The asymptotic behaviour of the superpotential of general SUSY transformations for a coupled channel Hamiltonian with different thresholds is analyzed. It is shown that asymptotically the superpotential can tend to a diagonal matrix with an arbitrary number of positive and negative entries depending on the choice of the factorization solution. The transformation of the Jost matrix is generalize...

Rotations are essential transformations in many parts of numerical linear algebra. In this paper, it is shown that there exists a family of matrices unitary with respect to an orthosymmetric scalar product J , that can be decomposed into the product of two J-unitary matrices—a block diagonal matrix and an orthosymmetric block rotation. This decomposition can be used for computing various one-si...

Rotations are essential transformations in many parts of numerical linear algebra. In this paper, it is shown that there exists a family of matrices unitary with respect to an orthosymmetric scalar product J , that can be decomposed into the product of two J-unitary matrices—a block diagonal matrix and an orthosymmetric block rotation. This decomposition can be used for computing various one-si...

Presented in this paper is a new sparse linear solver methodology motivated by multigrid principles and based around general local transformations that diagonalize a matrix while maintaining its sparsity. These transformations are approximate, but the error they introduce can be systematically reduced. The cost of each transformation is independent of matrix size but dependent on the desired ac...

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 (i) and (ii) below: (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 respect to whi...