Linear Systems Revisited

Article Abstract:

An elegant and well-developed theory exists for solving systems of linear equations. It has been applied successfully for decades in industrial and scientific applications and it forms the basis for linear programming. However many systems of linear equations for industrial applications may require all of the variables to be nonnegative or in an even more restricted area, and-or to be whole numbers. These conditions can create difficulties for the standard approach. So presented here is a new absolute value transformation and a multi-stage Monte Carlo simulation solution technique for dealing with the nonnegativity solution requirement and other conditions. The new approach has the additional advantage of working on nonlinear problems. The system of equations is transformed into a statistic (where the minimum of the statistic is the solution of the system). Then solving the system of equations becomes a problem of finding a way across the sampling distribution to the minimum, which will solve the system of equations. Regular Monte Carlo simulation to solve a system or optimization problem suffers from only being able to roughly approximate the answer. Multi-stage Monte Carlo simulation overcomes this obstacle by using a more sophisticated sampling scheme and additionally exploiting the well-behaved nature of the sampling distributions of transformed systems of equations. (Reprinted by Permission of Publisher.)

Product introduction, Simulation, New Technique, Theory, Methods

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A concise algorithm to solve over-/under-determined linear systems

Article Abstract:

An O(mn2) direct algorithm to compute a solution of a system of m linear equations Ax=b with n variables is presented. It is concise and matrix inversion-free. It provides an in-built consistency check and also produces the rank of the matrix A. Further, if necessary, it can prune the redundant rows of A and convert A into a full row rank matrix thus preserving the complete information of the system. In addition, the algorithm produces the unique projection operator that projects the real (n)-dimensional space orthogonally onto the null space of A and that provides a means of computing a relative error bound for the solution vector as well as a nonnegative solution. (Reprinted by permission of the publisher)

Mathematical logic, Programming Instruction, Algorithm Analysis

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Comments on 'linear systems revisited'

Article Abstract:

Conley (1985) uses a modified Monte Carlo method to solve systems of simultaneous linear algebraic equations and linear programming problems. We have used Conley's two sample problems to compare his proposed method with the classical methods of Gauss-Jordan elimination and the revised simplex algorithm. The classical methods are about three orders of magnitude faster; therefore, Conley's proposed method does not warrant serious consideration as a viable solution procedure. (Reprinted by permission of the publisher.)

Gaussian Elimination

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