﻿ EGNIYA - Solutions, Eigensolvers, Eigenvectors, Eigenvalues, Matrix Decompositions, Metric, Random Statistical Distributions and Stochastic Processes

## Solutions

During the development or Research and Development phases on main products, EGNIYA produces some side material in the form of technical reports, papers, experimental outcomes and case studies some of are applicable for our own patent application and some are useful for other engineering and scientific disciplines. In this page, sort of such materials are presented.

## Solutions Pack1 - The Path Towards Shiftless/Non-Iterative Exact Eigensolvers The Off-diagonals as Eigenshifters: The Path Towards Shiftless/Non-Iterative Exact Eigensolvers

Numerical computation of eigenvalues and eigenvectors of matrices is one of the challenges in scientific and engineering applications,and usually obtained by using Francis QR algorithm with shifts or Jacobi iterations. We conduct some research at MHRS (Matrix Computations and High Resolution Spectra) Labs for novel methods in eigenvector/eigenvalue computations aiming faster and more accurate techniques without use of shifts and iterations. One of our recent work suggests that off-diagonals of a matrix act as eigenshifters, which might open the way towards shiftless and non-iterative eigensolvers. The details of the ongoing research are periodically updated in the form of articles, test/trial results and sample code updates in links below.

A brief info about the concept The Off-diagonals as Eigenshifters: The Path Towards Shiftless/Non-Iterative Eigensolvers- Part1 ...

More info in MHRS pages Matrix Computations and High Resolution Spectra Labs ...

## Solutions Pack2 - Eigenvectors and Eigenvalues of Nonsymetric Matrices Numerical computation of eigenvalues and eigenvectors of nonsymmetric matrices can be considered as slightly more difficult compared to real symmetric matrices for which all eigenvalues are real and can be efficiently obtained by generally using Francis QR algorithm with shifts or Jacobi iterations. For large size matrices and especially for closely spaced eigenvalues, computation of eigenpairs for nonsymmetric matrices (in different problems especially in root finding) is not trivial, requiring mostly double shifts due to possible complex eigenvalues and exceptional shifts due to confinement in local minima/maxima. We propose a new approach for extracting eigenvectors and eigenvalues of nonsymmetric matrices by using real symmetric matrices which are formed by summing the original nonsymmetric matrix and its transpose.

More info in pages related to Eigenvectors and Eigenvalues of a Nonsymmetric Matrix Extracted from Real Symmetric Transpose Sum Matrix ...

## Solutions Pack3 - Pole Space Metric for Stochastic Processes and Dynamic Systems Autoregressive (AR) models are used in a variety of applications and an AR model can be represented by the poles corresponding to describing AR coefficients. Similarly, the behavior of linear dynamic single-input-single-output (SISO) systems can be described as a function of the system poles, which are directly estimated from the given data and represent a system as a set of poles without any identities, which is analogous to the nature of association-free multi-target tracking and corresponding application of set distances known as optimal subpattern assignment (OSPA) distance. In this work, we define a metric in pole-space and provide a measure of the “distance” between AR processes, or linear dynamic systems represented by poles. Main concentration is on the mathematics behind the problem and some variants of the metric which can be used depending on the specific application requirements.

The main contribution is a new approach and derivation of a novel metric based on poles which can be used for classification of AR processes, or classification, identification or change detection in linear dynamic systems represented by system poles.

Download EGNIYA Research Article PSM Metric for Stochastic Processes and Dynamic Systems...