Institute for Reliable Computing

Head: Prof. Dr. Siegfried M. Rump

INTLAB Version 9

INTLAB includes a linear system solver to treat extremely ill-conditioned linear systems.
It is based on a method I designed in about 1984 and uses error-free transformations for extra-precise
residual calculations. Usually the method works until condition numbers of about
10^{30}, occasionally
for much larger condition numbers. The following test uses Matlab's invhilb and the right hand side
ones(n,1), where the displayed
condition number is the true 2-norm condition number.

Ill-coonditioned linear systems, time [sec]

true | linear system | maximum | ||||

dimension | cond. number | verified [sec] | rel.error | |||

10 | 1.6e+013 | 0.13 | 2.0e-015 | |||

20 | 5.5e+027 | 0.22 | 5.0e-013 | |||

30 | 4.3e+042 | 0.40 | 1.1e-011 | |||

40 | 3.2e+057 | 0.7 | 1.7e-008 | |||

50 | 4.3e+072 | 1.1 | 2.8e-009 | |||

100 | 2.9e+148 | 4.5 | 4.4e-002 |

If you are interested in extremely ill-conditioned examples, consider

A = [ | -5046135670319638, | -3871391041510136, | -5206336348183639, | -6745986988231149 ; |

-640032173419322, | 8694411469684959, | -564323984386760, | -2807912511823001 ; | |

-16935782447203334, | -18752427538303772, | -8188807358110413, | -14820968618548534 ; | |

-1069537498856711, | -14079150289610606, | 7074216604373039, | 7257960283978710 ]; |

For details, see randmat.m. With a condition number of about 10^{65}, this is too much for the
algorithm implemented in INTLAB.

Institute for Reliable Computing

Hamburg University of Technology

Schwarzenbergstr. 95

21071 Hamburg

Germany