A Program to Compute the Condition Numbers of Matrix Eigenvalues without Computing Eigenvectors.

Abstract : The condition number of an eigenvalue measures the sensitivity of that eigenvalue to small changes in the matrix elements. Such extra information is nice, sometimes useful, but how much does it cost. A program is presented here for the most difficult case of a real square matrix whose eigenvalues are wanted without their corresponding eigenvectors. The program requires no extra storage space and the running time is about 50% longer than for the fastest reliable program which only computes eigenvalues. There are many industrial applications in which the matrix elements are known to only two or three decimal figures. Each condition number will indicate how accurately such a matrix determines the associated eigenvalue. When no digits in an eigenvalue are reliable the suspect eigenvalue should be tagged and this information passed on to a higher level in the whole computation. A number of programming devices keep the code, storage, and running time down to a minimum. An interesting case study is included.