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# Tolerance

 Tolerance is an option for various numerical options which specifies the tolerance that should be allowed in computing results.
• Tolerance->t specifies that a tolerance value t should be allowed.
Compute the singular values larger then of the largest singular value:
Compute the singular values larger then of the largest singular value:
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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, numerical roundoff can introduce excessive error:
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