Round-off error

In computing, a roundoff error,^[1] also called rounding error,^[2] is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic.^[3] Rounding errors are due to inexactness in the representation of real numbers and the arithmetic operations done with them. This is a form of quantization error.^[4] When using approximation equations or algorithms, especially when using finitely many digits to represent real numbers (which in theory have infinitely many digits), one of the goals of numerical analysis is to estimate computation errors.^[5] Computation errors, also called numerical errors, include both truncation errors and roundoff errors.

When a sequence of calculations with an input involving any roundoff error are made, errors may accumulate, sometimes dominating the calculation. In ill-conditioned problems, significant error may accumulate.^[6]

In short, there are two major facets of roundoff errors involved in numerical calculations:^[7]

The ability of computers to represent both magnitude and precision of numbers is inherently limited.
Certain numerical manipulations are highly sensitive to roundoff errors. This can result from both mathematical considerations as well as from the way in which computers perform arithmetic operations.

^ Butt, Rizwan (2009), Introduction to Numerical Analysis Using MATLAB, Jones & Bartlett Learning, pp. 11–18, ISBN 978-0-76377376-2
^ Ueberhuber, Christoph W. (1997), Numerical Computation 1: Methods, Software, and Analysis, Springer, pp. 139–146, ISBN 978-3-54062058-7
^ Forrester, Dick (2018). Math/Comp241 Numerical Methods (lecture notes). Dickinson College.
^ Aksoy, Pelin; DeNardis, Laura (2007), Information Technology in Theory, Cengage Learning, p. 134, ISBN 978-1-42390140-2
^ Ralston, Anthony; Rabinowitz, Philip (2012), A First Course in Numerical Analysis, Dover Books on Mathematics (2nd ed.), Courier Dover Publications, pp. 2–4, ISBN 978-0-48614029-2
^ Chapman, Stephen (2012), MATLAB Programming with Applications for Engineers, Cengage Learning, p. 454, ISBN 978-1-28540279-6
^ Chapra, Steven (2012). Applied Numerical Methods with MATLAB for Engineers and Scientists (3rd ed.). McGraw-Hill. ISBN 9780073401102.

[1] Butt, Rizwan (2009), Introduction to Numerical Analysis Using MATLAB, Jones & Bartlett Learning, pp. 11–18, ISBN 978-0-76377376-2

[2] Ueberhuber, Christoph W. (1997), Numerical Computation 1: Methods, Software, and Analysis, Springer, pp. 139–146, ISBN 978-3-54062058-7

[Forrester_2018-3] Forrester, Dick (2018). Math/Comp241 Numerical Methods (lecture notes). Dickinson College.

[4] Aksoy, Pelin; DeNardis, Laura (2007), Information Technology in Theory, Cengage Learning, p. 134, ISBN 978-1-42390140-2

[5] Ralston, Anthony; Rabinowitz, Philip (2012), A First Course in Numerical Analysis, Dover Books on Mathematics (2nd ed.), Courier Dover Publications, pp. 2–4, ISBN 978-0-48614029-2

[6] Chapman, Stephen (2012), MATLAB Programming with Applications for Engineers, Cengage Learning, p. 454, ISBN 978-1-28540279-6

[Chapra_2012-7] Chapra, Steven (2012). Applied Numerical Methods with MATLAB for Engineers and Scientists (3rd ed.). McGraw-Hill. ISBN 9780073401102.

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