# Tolerance

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

# Details

• Tolerance->t specifies that a tolerance value t should be allowed.

# Examples

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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, https://reference.wolfram.com/language/ref/Tolerance.html.

#### Text

Wolfram Research (1991), Tolerance, Wolfram Language function, https://reference.wolfram.com/language/ref/Tolerance.html.

#### CMS

Wolfram Language. 1991. "Tolerance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Tolerance.html.

#### APA

Wolfram Language. (1991). Tolerance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Tolerance.html

#### BibTeX

@misc{reference.wolfram_2024_tolerance, author="Wolfram Research", title="{Tolerance}", year="1991", howpublished="\url{https://reference.wolfram.com/language/ref/Tolerance.html}", note=[Accessed: 12-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_tolerance, organization={Wolfram Research}, title={Tolerance}, year={1991}, url={https://reference.wolfram.com/language/ref/Tolerance.html}, note=[Accessed: 12-September-2024 ]}