THE MULTIDIMENSIONAL D-ANALOGUE OF THE LU-ALGORITHM BASED ON MULTIDIMENSIONAL DIFFERENTIAL TRANSFORMS
Keywords:
multiparametric matrix, matrix decomposition, multidimensional differential transforms, differential image, original, differential analogueAbstract
A sufficiently simple numerical-analytical method for factorization of nonautonomous multiparametric matrix based on Pukhov’s multidimensional differential transforms is proposed.
A differential (D-) analogue of factorization of nonautonomous matrices based on LU algorithm and Pukhov’s multidimensional differential transforms is proposed. For the LU decomposition, the Doolittle’s algorithm for autonomous matrices is used. Firstly, the Danalogue of Doolittle’s method for the multiparametric matrices of size n whose elements depend on n parameters is presented. At the same time, we suppose that all the aij(x1,x2,...,xm) elements of the A(x1,x2,...,xm) matrix are sufficiently smooth across all the x1, x2 ,..., xm parameters. Then, for the observed matrix, for determining the elements of the L(x1,x2,...,xm) and U(x1,x2,...,xm) multidimensional lower and upper triangular matrices correspondingly the multidimensional D-analogue of the Doolittle’s method is constructed with application of multidimensional differential transforms. Previously, for the determination of characteristics of multiparametric matrices with application of multidimensional differential transforms were developed, in particular, the D-analogue of Berkovits’s method for the determination of the characteristic polynomial, D-analogue of Faddeyev’s method for the determination of characteristic polynomial and inverse, the method based on simplex transformations for the determination of the inverse matrix, the D-analogue of the QR algorithm for the decomposition of the multiparametric matrix into orthogonal and upper triangular parametric matrices.
In the proposed paper, during the implementation of the D-analogue, firstly, the predetermined number of matrix-images of decomposed matrices are calculated, then by using them, the elements of multiparametric matrices reconstructed as a ratio of polynomials. The matrix depending on 2 parameters is considered and the procedure of the solution by the proposed Danalogue, as well as the images of some elements and the reconstructed originals as a ratio of the polynomials, which are the elements of decomposed matrices, is presented. The obtained matrices satisfy the A(x1, x2) = L(x1, x2) U(x1, x2) equality.
