%0 Journal Article %1 axb-itref-toms %A Demmel, James W. %A Hida, Yozo %A Kahan, W. %A Li, Xiaoye S. %A Mukherjee, Sonil %A Riedy, E. Jason %D 2006 %J ACM Transactions on Mathematical Software %K acm floatingpoint lapack linearalgebra toms %N 2 %P 325--351 %R 10.1145/1141885.1141894 %T Error bounds from extra-precise iterative refinement %V 32 %X We present the design and testing of an algorithm for iterative refinement of the solution of linear equations where the residual is computed with extra precision. This algorithm was originally proposed in 1948 and analyzed in the 1960s as a means to compute very accurate solutions to all but the most ill-conditioned linear systems. However, two obstacles have until now prevented its adoption in standard subroutine libraries like LAPACK: (1) There was no standard way to access the higher precision arithmetic needed to compute residuals, and (2) it was unclear how to compute a reliable error bound for the computed solution. The completion of the new BLAS Technical Forum Standard has essentially removed the first obstacle. To overcome the second obstacle, we show how the application of iterative refinement can be used to compute an error bound in any norm at small cost and use this to compute both an error bound in the usual infinity norm, and a componentwise relative error bound.

@article{axb-itref-toms, abstract = {We present the design and testing of an algorithm for iterative refinement of the solution of linear equations where the residual is computed with extra precision. This algorithm was originally proposed in 1948 and analyzed in the 1960s as a means to compute very accurate solutions to all but the most ill-conditioned linear systems. However, two obstacles have until now prevented its adoption in standard subroutine libraries like LAPACK: (1) There was no standard way to access the higher precision arithmetic needed to compute residuals, and (2) it was unclear how to compute a reliable error bound for the computed solution. The completion of the new BLAS Technical Forum Standard has essentially removed the first obstacle. To overcome the second obstacle, we show how the application of iterative refinement can be used to compute an error bound in any norm at small cost and use this to compute both an error bound in the usual infinity norm, and a componentwise relative error bound.}, added-at = {2007-10-09T06:57:46.000+0200}, author = {Demmel, James W. and Hida, Yozo and Kahan, W. and Li, Xiaoye S. and Mukherjee, Sonil and Riedy, E. Jason}, biburl = {https://www.bibsonomy.org/bibtex/2768d3556a52aa3b08e35f66ecee39609/ejr}, doi = {10.1145/1141885.1141894}, interhash = {462217af6d05a18ad137bf556e12de3e}, intrahash = {768d3556a52aa3b08e35f66ecee39609}, issn = {0098-3500}, journal = {ACM Transactions on Mathematical Software}, keywords = {acm floatingpoint lapack linearalgebra toms}, month = {June}, mrclass = {65F10}, mrnumber = {MR2272365}, number = 2, pages = {325--351}, timestamp = {2007-10-09T06:57:46.000+0200}, title = {Error bounds from extra-precise iterative refinement}, volume = 32, year = 2006 }