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A direct method for the general solution of a system of linear equations

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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, wheremn, 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 ranknm, 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.

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Additional information

This research was supported by the National Science Foundation, Grant No. GP-41158.

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Huang, H.Y. A direct method for the general solution of a system of linear equations. J Optim Theory Appl 16, 429–445 (1975). https://doi.org/10.1007/BF00933852

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Key Words

  • Mathematical programming
  • conjugate-gradient methods
  • variable-metric methods
  • linear equations
  • numerical methods
  • computing methods