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Feedback Algorithm for System of Equations

IP.com Disclosure Number: IPCOM000085352D
Original Publication Date: 1976-Mar-01
Included in the Prior Art Database: 2005-Mar-02
Document File: 2 page(s) / 18K

Publishing Venue

IBM

Related People

Verkhovsky, BS: AUTHOR

Abstract

This algorithm has immediate application for a system of equations with a stochastic matrix, and is particularly efficient for the case where the method of simple iterations converges very slowly. The algorithm also has application to the problems which have dealt with Markovian processes, as well as the problems with a Minkovski-Leontieff matrix. While other procedures have been used for such problems, the present one is believed to be more efficient. (Image Omitted)

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Feedback Algorithm for System of Equations

This algorithm has immediate application for a system of equations with a stochastic matrix, and is particularly efficient for the case where the method of simple iterations converges very slowly. The algorithm also has application to the problems which have dealt with Markovian processes, as well as the problems with a Minkovski-Leontieff matrix. While other procedures have been used for such problems, the present one is believed to be more efficient.

(Image Omitted)

This algorithm provides an efficient procedure for finding the solution for a system of equations: (***) V = b + Beta Av, where V is the M-dimensional

unknown vector.

The Algorithm
1. Gamma(n) = (1 + Beta Av/n/ over ev/n/)/a/.

1.1. a = 1 over 1 - Beta.

1.2. e = (1,1,...,1) is an M-dimensional vector.
2. V/n+1/ = b + Gamma(n) Av/n/Beta.

Notes.

The algorithm when programmed on the APL yielded very efficient convergency to the solution of (***). The number of iterations, needed to obtain a solution, is comparable with the two other algorithms appearing in the IBM Technical Disclosure Bulletin, Vol. 18, No. 10, March 1976, pages 3464, 3465 and 3466, 3467.

For example, for M = 50,Beta = .9, the algorithm gives the solution V less than for 12 iterations with: Epsilon = max 3b + Beta Av/n/ - v/n/3.

1 </- i </- M.

The algorithm works even when A is a double-stochastic matrix and Beta is very close to 1.

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