SJSU Singular Matrix Database

Accuracy of calculated singular values and calculated numerical rank

Due to computer arithmetic calculated singular values may be different from the exact singular values of A. Therefore quantities, such as the condition number, the gap, and the numerical rank that are calculated from the singular values may not be exact. For example, if the tolerance used to define the numerical rank is close to a singular value of A then small changes in A could potentially change the calculated rank. Consequently, if the tolerance is in the middle of a cluster of almost equal singular values and potential errors in A are sufficiently large, the numerical rank would not be well defined for that tolerance. On the other hand if the tolerance is near the middle of a large gap in the singular values and if potential errors in A are small, then the numerical rank is well defined for that tolerance. The numerical rank is meaningful if, for example,

If the errors in A are due to computer arithmetic there are bounds for the errors in the calculated singular values: