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Matrix Processor

[Category : - Computers and computer accessories ]
[Viewed 860 times]

Overview

A new method and algorithms is presented for solving a system of linear equations with complex coefficients (including underdetermined and overdetermined systems), involving only multiplication of vectors. The matrix processor (specially designed for this problem) is significantly simplified and is able to realize conveyer data processing without important hardware changes.

The method is based on a principle found by the author ? variational optimum principle in linear alternating current electric circuits. This means that for every such circuit there exists a sole optimum of a certain functional. This sole optimum may be found with the aid of a high-speed gradient descent method, which is very attractive for applications. There exists an inverse proportionality between the accuracy and the solution time. On practice it means that the user may quickly look through approximate solutions, and then compute a chosen variant with more accuracy.

The method may be employed not only for computing complex electric circuits, but also for solving linear equations systems with complex coefficients (including underdetermined and overdetermined systems). The main feature of algorithms based on this method is that they involve only summation and multiplication of vectors. Therefore a comparatively simple matrix processor may be developed for these computations.



Typical applications

It is well known that 75% of all numerical mathematical problems are essentially the problems of linear algebra. Among these problems a large share falls on the solution of linear equations system with (generally speaking) complex coefficients. We can literally say that matrix processors owe their very appearance to these problems. But in these problems only the multiplication of matrices harmonizes ideally with the with the possibility of parallel computations in matrix processors. Other operations, and mostly the inversion of matrixes are ill fitted to parallelizing. This problem, along with the high cost of matrix processors, is an obstacle to their expansion.



Key features

? For a proposed matrix processor in the solution of linear equations system with complex coefficient (including underdetermined and overdetermined systems) by the presented method only algebraic addition of complex vectors and multiplication of complex matrices are being used.

? The inverse matrix computation is absent

? Such processor should contain only summators.

? The number of summators S should be in proportion with the processor?s volume and in inverse proportion with the performance time of these operations.

? The solution of linear equations system for an proposed matrix processor is performed iteratively. Here on every iteration the multiplication of a squarte matrix by a vector is being performed.

? The solution time for an proposed matrix processor is proportional to , where N - the vector dimension.



Project Contents

The project includes

? theoretical foundations,

? algorithms,

? programs in the MATLAB system with open codes,

error estimates,

? requirements for matrix processor?s structure,

? calculation time estimate for matrix processor.


The described information is quite sufficient for practical realization of matrix processor.



Contents

Preface

1. The First Method for Solution of Linear Algebraic Equations System

1.1. Description of the Method

1.2. About convergence

1.3. About speed and accuracy

1.4. Underdetermined system

1.5. Overdetermined system

2. The Second Method for Solution of Linear Algebraic Equations System

2.1. Description of the Method

2.2. Underdetermined system

2.3. Overdetermined system

3. About the Matrix Processor

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