Strict optimal error and residual estimates for the solution of linear algebraic systems by elimination methods in high-accuracy arithmetic
The paper establishes explicit analytical representations of the errors and residuals of the solutions of linear algebraic systems as functions of the data errors and of the rounding errors of a high-accuracy floating-point arithmetic. On this basis, strict, componentwise, and in first order optimal error and residual estimates are obtained. The stability properties of the elimination methods of Doolittle, Crout, and Gauss are compared with each other. The results are applied to three numerical examples arising in difference approximations, boundary and finite element approximations of elliptic boundary value problems. In these examples, only a modest increase of the accuracy of the solutions is achieved by high-accuracy arithmetic.
KeywordsCondition Number Gaussian Elimination Finite Element Approximation Elimination Method Binary Digit
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- 1.Bowdler, H.J., et al.: Solution of real and complex systems of linear equations. Numer. Math. 8, 217–234 (1966).Google Scholar
- 2.Fox, L.: An introduction to numerical linear algebra. Oxford: Clarendon Press 1964.Google Scholar
- 3.Olver, F.W.J., and Wilkinson, J.H.: A posteriori error bounds for Gaussian elimination. IMA J. Num. Analysis 2, 377–406 (1982).Google Scholar
- 4.Stummel, F.: Optimal error estimates for Gaussian elimination in floating-point arithmetic. Z. Angew. Math. Mech. 62, T 355–T 357 (1982).Google Scholar
- 5.Stummel, F.: Forward error analysis of Gaussian elimination. Part I: Error and residual estimates. Numer. Math. (1985). Part II: Stability theorems. Numer. Math. (1985).Google Scholar
- 6.Stummel, F.: Strict optimal error estimates for Gaussian elimination. Z. Angew. Math. Mech. 65, T 396–T 398 (1985).Google Scholar
- 7.Stummel, F.: Strict optimal a posteriori error and residual bounds for Gaussian elimination in floating-point arithmetic. Submitted to Computing.Google Scholar
- 8.Stummel, F.: FORTRAN-Programs for the rounding error analysis of Gaussian elimination. Centre for Mathematical Analysis, The Australian National University, Canberra, CMA-R02-85.Google Scholar
- 9.Wilkinson, J.H.: Rounding errors in algebraic processes. Englewood Cliffs: Prentice Hall 1963.Google Scholar