The principle of minimized iterations in the solution of the matrix eigenvalue problem
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In this paper, an interpretation of Dr. Cornelius Lanczos' iteration method, which he has named ''minimized iterations'' is discussed, expounding the method as applied to the solution of the characteristic matrix equations both in homogeneous and nonhomogeneous form.Abstract:
An interpretation of Dr. Cornelius Lanczos' iteration method, which he has named \"minimized iterations\", is discussed in this article, expounding the method as applied to the solution of the characteristic matrix equations both in homogeneous and nonhomogeneous form. This interpretation leads to a variation of the Lanczos procedure which may frequently be advantageous by virtue of reducing the volume of numerical work in practical applications. Both methods employ essentially the same algorithm, requiring the generation of a series of orthogonal functions through which a simple matrix equation of reduced order is established. The reduced matrix equation may be solved directly in terms of certain polynomial functions obtained in conjunction with the generated orthogonal functions, and the convergence of the solution may be observed as the order of the reduced matrix is successively increased with the order of the original matrix as a limit. The method of minimized iterations is recommended as a rapid means for determining a small number of the larger eigenvalues and modal columns of a large matrix and as a desirable alternative for various series expansions of the Fredholm problem. 1. The conventional iterative procedures. It is frequently required that real latent roots, or eigenvalues, and modal columns be determined for a real numerical matrix, u, of order, n, in the characteristic homogeneous equation,*read more
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References
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An iteration method for the solution of the eigenvalue problem of linear differential and integral operators
TL;DR: In this article, a systematic method for finding the latent roots and principal axes of a matrix, without reducing the order of the matrix, has been proposed, which is characterized by a wide field of applicability and great accuracy, since the accumulation of rounding errors is avoided, through the process of minimized iterations.
Journal ArticleDOI
XII.—Studies in Practical Mathematics. I. The Evaluation, with Applications, of a Certain Triple Product Matrix
TL;DR: In this paper, a matrix product H * A -1 K, where A is square and non-singular is not zero, is computed for determinant multiplication, and the matrix H * is obtained from H by transposition, that is, by changing rows into columns.