Consistent System Of Linear Equations Research Articles

An algorithm for the solution of ill-posed linear systems arising from the discretization of Fredholm integral equation of the first kind

An algorithm for obtaining approximate solutions of ill-posed systems of linear equations arising from the discretization of Fredholm integral equation of the first kind is described. The ill-posed system is first replaced by an equivalent consistent system of linear equations. The method calculates the minimum length least squares solution of the consistent system. Starting from rank = 1 of the consistent system, the rank is increased by one in succession and a new solution is calculated. This is repeated until a certain simple criterion is satisfied. Linear programming techniques are used for which successive solutions are the basic solutions in the successive simplex tableaux. The algorithm is numerically stable. Numerical results show that this method compares favorably with other direct methods.

Generalized inverse matrices and their applications

There are a number of versatile generalizations of the usual inverse matrix, referred to in this paper as generalized inverse matrices. The definitions and properties of some of the common generalized inverse matrices are described, including methods for constructing them. A number of applications are discussed, including their use in solving consistent systems of linear equations which do not have the same number of equations as variables, or which have a singular coefficient determinant. A certain type of generalized inverse is shown to give the least‐squares solution of an inconsistent system of linear equations. Other applications are to systems of nonlinear equations, to integer solutions of systems of equations and to linear programming.

A combined direct-iterative approach for solving large scale singular and rectangular consistent systems of linear equations

We construct a certain iterative scheme for solving large scale consistent systems of linear equations Ax = b , (*) where A is a complex m X n matrix of rank r , m ⩾ n , and where A is assumed reasonably well conditioned. The iterative method is obtained through a careful exploitation of an LU -decomposition of A , and, disregarding roundoff errors, it converges to a solution to ( * ), though not necessarily the minimal l 2 -norm one, from any starting vector in r iterations. Moreover, once the LU -decomposition of A is complete, only about r 3 /2 arithmetic operations (multiplications and divisions) are needed to execute the r iterations.

Digital image restoration using quadratic programming

The problem of digital image restoration is considered by obtaining an approximate solution to the Fredholm integral equation of the first kind in two variables. The system of linear equations resulting from the discretization of the integral equation is converted to a consistent system of linear equations. The problem is then solved as a quadratic programming problem with bounded variables where the unknown solution is minimized in the L(2) norm. In this method minimum computer storage is needed, and the repeated solutions are obtained in an efficient way. Also the rank of the consistent system which gives a best or near best solution is estimated. Computer simulated examples using spatially separable pointspread functions are presented. Comments and conclusion are given.

Linear Programming Algorithms for the Chebyshev Solution to a System of Consistent Linear Equations

Efficient linear programming algorithms for the solution of a set of overdetermined linear equations in the $l_1 $ norm are employed in order to obtain a minimum $l_\infty $ solution to a set of consistent linear equations. In addition, the solution is proved to be of a particular form in case the set of equations is a Chebyshev system.

Minimum L∞ solution of underdetermined systems of linear equations

The problem of obtaining a minimum L∞ solution of an underdetermined system of consistent linear equations is reduced to a linear programming problem. A modified simplex algorithm is then described. In this algorithm no conditions are imposed on the coefficient matrix, minimum computer storage is required and no artificial variables are needed. The algorithm is a simple and fast one. Numerical results are given.

An Efficient Algorithmic Procedure for Obtaining a Minimum $L_\infty $-Norm Solution to a System of Consistent Linear Equations

Herein is described an algorithmic procedure for determining a minimum $l_\infty $-norm solution to the system of consistent linear equations \[ Ax = y,\] where A is a $m \times n$ matrix of rank m, y is a known $m \times 1$ vector and x is an unknown $n \times 1$ vector. The algorithm’s development is based on some fundamental concepts from functional analysis. Its computational efficiency is shown to easily exceed that of the linear programming formulization of the same problem.

A Finite Algorithm for the Minimum $l_\infty $ Solution to a System of Consistent Linear Equations

A column exchange algorithm is presented for the determination of a minimum $l_\infty $ solution to a system of consistent linear equations. The algorithm is based on first solving the associated dual problem as specified by a well-known theorem from functional analysis and then generating the required solution by means of an alignment criterion. The procedure has been shown to have a rapid speed of convergence on all examples treated to date.

Functional analysis and the optimal control of linear discrete systems

The optimal regulation of a linear discrete system in which one wishes to use either the minimum amount of fuel, energy or amplitude in the control has received much attention. It is readily shown that each of these control problems can be equated to that of finding a specific solution to a system of consistent linear equations in the control variables. Various algorithms have been developed which will yield an optimal control ; however, they tend to be computationally inefficient (for the minimum fuel and amplitude problems) and are therefore of little practical value. In this paper, a number of important properties related to the solution of a consistent system of linear equations will be developed. These properties are proven by appealing to same basic theorems from functional analysis and will be directly applicable to the above control problems. Algorithmic procedures for solving the minimum fuel (minimum l∞ solution) and minimum amplitude (minimum l;∞ solution) are then developed. These algorithmic ...

Absolutely pure modules

A moduleAAis shown to be absolutely pure if and only if every finite consistent system of linear equations overAAhas a solution inAA. Noetherian, semihereditary, regular and Prüfer rings are characterized according to properties of absolutely pure modules over these rings. For example,RRis Noetherian if and only if every absolutely pureRR-module is injective and semihereditary if and only if the class of absolutely pureRR-modules is closed under homomorphic images. IfRRis a Prüfer domain, then the absolutely pureRR-modules are the divisible modules andExtR1⁡(M,A)=0\operatorname {Ext} _R^1(M,A) = 0wheneverAAis divisible andMMis a countably generated torsion-freeRR-module.