# Analysis of the finite precision s-step biconjugate gradient method

### Erin Carson and James Demmel

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2014-18

March 13, 2014

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-18.pdf

We analyze the $s$-step biconjugate gradient algorithm in finite precision arithmetic and derive a bound for the residual norm in terms of a minimum polynomial of a perturbed matrix multiplied by an amplification factor. Our bound enables comparison of $s$-step and classical biconjugate gradient in terms of amplification factors. Our results show that for $s$-step biconjugate gradient, the amplification factor depends heavily on the quality of $s$-step polynomial bases generated in each outer loop.

BibTeX citation:

@techreport{Carson:EECS-2014-18, Author = {Carson, Erin and Demmel, James}, Title = {Analysis of the finite precision s-step biconjugate gradient method}, Institution = {EECS Department, University of California, Berkeley}, Year = {2014}, Month = {Mar}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-18.html}, Number = {UCB/EECS-2014-18}, Abstract = {We analyze the $s$-step biconjugate gradient algorithm in finite precision arithmetic and derive a bound for the residual norm in terms of a minimum polynomial of a perturbed matrix multiplied by an amplification factor. Our bound enables comparison of $s$-step and classical biconjugate gradient in terms of amplification factors. Our results show that for $s$-step biconjugate gradient, the amplification factor depends heavily on the quality of $s$-step polynomial bases generated in each outer loop.} }

EndNote citation:

%0 Report %A Carson, Erin %A Demmel, James %T Analysis of the finite precision s-step biconjugate gradient method %I EECS Department, University of California, Berkeley %D 2014 %8 March 13 %@ UCB/EECS-2014-18 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2014/EECS-2014-18.html %F Carson:EECS-2014-18