Relaxed method to certify the solution of a linear system

Hong Diep Nguyen 1 Nathalie Revol 1, *
* Corresponding author
1 ARENAIRE - Computer arithmetic
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : We propose a relaxed method to certify the solution of a linear system Ax=b. To certify this system, we compute an enclosure of the error while simultaneously refining the floating-point solution. We proceed by adapting floating-point iterative refinement methods. To this end, the residual system is first calculated using interval arithmetic instead of floating-point arithmetic. This residual system is preconditioned by an approximate inverse of A. Then we apply an iterative method to this system, namely the interval Gauss-Seidel method, in order to improve the error bound. A problem of this method is that the use of interval arithmetic increases significantly the execution time. To tackle this problem, we introduce a relaxation technique that yields an execution time for one iteration similar to the execution time of a floating-point matrix-vector product. Although it does not avoid the precondition step, this relaxed method helps to make the interval iterative part negligible in terms of excution time, meanwhile still producing very accurate solutions, which are exact to almost the last bit when the matrix of the system is not too ill-conditioned.
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https://hal.archives-ouvertes.fr/hal-00748631
Contributor : Nathalie Revol <>
Submitted on : Monday, November 5, 2012 - 4:24:23 PM
Last modification on : Thursday, January 17, 2019 - 3:16:03 PM

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  • HAL Id : hal-00748631, version 1

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Hong Diep Nguyen, Nathalie Revol. Relaxed method to certify the solution of a linear system. SWIM (Small Workshop on Interval Methods), Jun 2009, Lausanne, France. ⟨hal-00748631⟩

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