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.


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Basic Examples  (1)

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

Scope  (4)

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:

Wolfram Research (1991), Tolerance, Wolfram Language function,


Wolfram Research (1991), Tolerance, Wolfram Language function,


@misc{reference.wolfram_2020_tolerance, author="Wolfram Research", title="{Tolerance}", year="1991", howpublished="\url{}", note=[Accessed: 02-March-2021 ]}


@online{reference.wolfram_2020_tolerance, organization={Wolfram Research}, title={Tolerance}, year={1991}, url={}, note=[Accessed: 02-March-2021 ]}


Wolfram Language. 1991. "Tolerance." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (1991). Tolerance. Wolfram Language & System Documentation Center. Retrieved from