Built-in Mathematica Symbol

Tolerance

Tolerance
is an option for various numerical options which specifies the tolerance that should be allowed in computing results.

Compute the singular values larger then

of the largest singular value:

Compute the singular values larger then

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: