Scalability of k-Tridiagonal Matrix Singular Value Decomposition

Singular value decomposition has recently seen a great theoretical improvement for k-tridiagonal matrices, obtaining a considerable speed up over all previous implementations, but at the cost of not ordering the singular values. We provide here a refinement of this method, proving that reordering singular values does not affect performance. We complement our refinement with a scalability study on a real physical cluster setup, offering surprising results. Thus, this method provides a major step up over standard industry implementations.

On Generating Tridiagonal Matrices of Generalized (s,t)-Pell, (s,t)-Pell Lucas and (s,t)-Modified Pell Sequences

In this study, we define some tridigional matrices depending on two real parameters. By using the determinant of these matrices, the elements of (s,t)-Pell, (s,t)-Pell Lucas and (s,t)-modified Pell sequences with even or odd indices are generated. Then we construct the inverse matrices of these tridigional matrices. We also investigate eigenvalues of these matrices.

Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

Numerical algorithm for fractional order population dynamics model with delay

For a fractional-diffusion equation with nonlinearity in the differentiation operator and with the effect of functional delay, an implicit numerical method is constructed based on the approximation of the fractional derivative and the use of interpolation and extrapolation of discrete history. The source of this problem is a generalized model from population theory. Using a fractional discrete analogue of Gronwall's lemma, the convergence of the method is proved under certain conditions. The resulting system of nonlinear equations using Newton's method is reduced to a sequence of linear systems with tridiagonal matrices. Numerical results are given for a test example with distributed delay and a model example from the theory of population with concentrated constant delay.