Discuss of Error Analysis of Gauss-Jordan Elimination For Linear Algebraic Systems

The paper establishes explicit representations of the errors and residuals of approximate solutions of triangular linear systems by Jordan elimination and of general linear algebraic systems by Gauss- Jordan elimination as functions of the data perturbations and the rounding errors in arithmetic floating-point operations. From these representations strict optimal componentwise error and residual bounds are derived. Further, stability estimates for the solutions are discussed. The error bounds for the solutions of triangular linear systems are compared to the optimal error bounds for the solutions by back substitution and by Gaussian elimination with back substitution, respectively. The results confirm in a very detailed form that the errors of the solutions by Jordan elimination and by Gauss-Jordan elimination cannot be essentially greater than the possible maximal errors of the solutions by back substitution and by Gaussian elimination, respectively. Finally, the theoretical results are illustrated by two numerical examples.