A direct method for the general solution of a system of linear equations |
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Authors: | H Y Huang |
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Institution: | (1) Rice University, Houston, Texas;(2) Present address: Exxon Production Research Company, Houston, Texas |
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Abstract: | A computationally stable method for the general solution of a system of linear equations is given. The system isA
Tx–B=0, where then-vectorx is unknown and then×q matrixA and theq-vectorB are known. It is assumed that the matrixA
T and the augmented matrix A
T,B] are of the same rankm, wherem n, so that the system is consistent and solvable. Whenm<n, the method yields the minimum modulus solutionx
m and a symmetricn ×n matrixH
m of rankn–m, so thatx=x
m+H
my satisfies the system for ally, ann-vector. Whenm=n, the matrixH
m reduces to zero andx
m becomes the unique solution of the system.The method is also suitable for the solution of a determined system ofn linear equations. When then×n coefficient matrix is ill-conditioned, the method can produce a good solution, while the commonly used elimination method fails.This research was supported by the National Science Foundation, Grant No. GP-41158. |
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Keywords: | Mathematical programming conjugate-gradient methods variable-metric methods linear equations numerical methods computing methods |
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