Project Title: Investigations on the Condition Number of Matrix Methods

Summary:

The condition number is a well-known measure of matrices. Large condition numbers cause inaccuracies in the matrix solvers. One of the most useful matrix solvers for high condition numbers is based on Givens plane rotations. Although this algorithm is included in LINPACK, it has been discarded in LAPACK because many scientists think that one should avoid techniques that lead to large condition numbers. A detailed study of the MMP code for computational electromagnetics has shown that it is more likely to obtain accurate results when the condition number is high - provided that the matrix solver does not cause a significant loss of the accuracy. Since this surprising result might be unique for MMP, generalized Fourier analysis and other techniques for the approximation of functions (based on non-orthogonal basis functions) have been studied. This study supports the surprising observation: The probability of obtaining accurate results is higher when the condition number of the corresponding matrix is higher. Therefore, one should not try to reduce the condition number when one is designing a numerical method and one should develop matrix solvers that allow to efficiently handle ill-conditioned matrices. It has also been observed that methods based on ill-conditioned matrices are powerful for the extrapolation of functions known in a finite interval and for the recognition of the type of a measured function.

For more information, you can download an unpublished paper on this topic (Word document cond.doc 376k, compressed with WinZip cond.zip 72k).

Contacts: Ch.Hafner, IFH, ETZG95, ETH Zentrum, CH-8092 Zurich

Electronic Contacts: hafner@ifh.ee.ethz.ch

Index Terms: Condition Number, Matrix Methods, Accuracy