Compute the singular values larger than of the largest singular value:

Numerically approximate all the singular values of a positive definite matrix:

Compare with the numerical values of the exact singular values:

Some values less than the default tolerance are computed poorly due to numerical roundoff:

Get the complete singular value decomposition of a nearly singular matrix:

Reconstruct the matrix:

Without the setting for Tolerance, the matrix is considered effectively singular:

Detect maximum possible numerical rank:

The two rows are only detected as independent because of representation error:

The default tolerance allows for the numerical representation error:

Limit roundoff error at the expense of a larger residual for a least squares problem:

With the default tolerance, numerical roundoff is limited so error is distributed:

Specifying a higher tolerance will limit roundoff errors at the expense of a larger residual:

With Tolerance->0, numerical roundoff can introduce excessive error: