SOLUTION OF ONE-PARAMETER GENERALIZED PROBLEM OF EIGENVALUE-FUNCTIONS AND EIGENVECTOR-FUNCTIONS
DOI:
https://doi.org/10.53297/18293336-2022.2-9Keywords:
one-parameter generalized problem, transformation of the original problem, analytical solution, differential transformations, numerical-analytical solution, means of modern information technologiesAbstract
A one-parameter generalized problem of eigenvalues-functions and eigenvectors- functions is considered, when the transposed matrix of the left side acts as the matrix of the right side. This is an important special case of the one-parameter generalized problem, consisting of three computational steps.
At the first stage, the problem for a pair of matrices of the left and right parts is reduced to the usual spectral problem with some transforming matrix, which is determined using the analytical and numerical-analytical methods. The analytical method uses methods for determining one-parameter generalized inverse matrices. In the numerical-analytical method, in accordance with well-known approaches, sequential and parallel computational procedures are used. In both methods, differential transformations act as the main mathematical apparatus, which make it possible to obtain computational procedures that are easily implemented by means of modern information technologies and have high computational efficiency.
At the second stage, the problem of determining the eigenvalues-functions of the found transforming matrix is solved. Possible analytical and numerical-analytical solutions are considered. In the latter case, well-known implicit and explicit computational schemes based on differential transformations are used.
At the third stage, the problem of determining the eigenvector-functions of the transforming matrix is solved. Possible analytical and numerical-analytical solutions are considered.
A model example is considered. The exact coincidence of analytically and numerically- analytically determined transforming matrices of the reduced equivalent problem is shown. An acceptable coincidence of analytical and numerical-analytical solutions of eigenvalues- functions in the vicinity of the center of approximation, as well as an acceptable coincidence of analytical and numerical-analytical solutions of eigenvectors-functions in the vicinity of the center of approximation is shown.
At all stages of calculations, numerical-analytical solutions are restored based on the use of differential-Taylor transformations.
