Editing Round-off error (section)

{{short description|Computational error due to rounding numbers}}
{{For|the acrobatic movement, roundoff|Roundoff}}
{{Use dmy dates|date=August 2019|cs1-dates=y}}

In [[computing]], a '''roundoff error''',<ref>{{citation |title=Introduction to Numerical Analysis Using MATLAB |author-first=Rizwan |author-last=Butt |publisher=Jones & Bartlett Learning |date=2009 |isbn=978-0-76377376-2 |pages=11–18 |url=https://books.google.com/books?id=QWub-UVGxqkC&pg=PA11}}</ref> also called '''rounding error''',<ref>{{citation |title=Numerical Computation 1: Methods, Software, and Analysis |author-first=Christoph W. |author-last=Ueberhuber |publisher=Springer |date=1997 |isbn=978-3-54062058-7 |url=https://books.google.com/books?id=JH9I7EJh3JUC&pg=PA139 |pages=139–146}}</ref> 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, [[Rounding|rounded]] arithmetic.<ref name="Forrester_2018">{{cite book |title= Math/Comp241 Numerical Methods (lecture notes) |author-first=Dick |author-last=Forrester |publisher=[[Dickinson College]] |date=2018}}</ref> Rounding errors are due to inexactness in the representation of [[real number]]s and the arithmetic operations done with them. This is a form of [[quantization error]].<ref>{{citation |title=Information Technology in Theory |author-first1=Pelin |author-last1=Aksoy |author-first2=Laura |author-last2=DeNardis |publisher=Cengage Learning |date=2007 |isbn=978-1-42390140-2 |page=134 |url=https://books.google.com/books?id=KGS5IcixljwC&pg=PA134}}</ref> When using approximation [[equation]]s 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 [[error analysis (mathematics)|estimate]] computation errors.<ref>{{citation |title=A First Course in Numerical Analysis |edition=2nd |series=Dover Books on Mathematics |author-first1=Anthony |author-last1=Ralston |author-first2=Philip |author-last2=Rabinowitz |publisher=Courier Dover Publications |date=2012 |isbn=978-0-48614029-2 |url=https://books.google.com/books?id=TVq8AQAAQBAJ&pg=PA2 |pages=2–4}}</ref> Computation errors, also called [[numerical error]]s, include both [[truncation error]]s 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.<ref>{{citation |title=MATLAB Programming with Applications for Engineers |author-first=Stephen |author-last=Chapman |publisher=Cengage Learning |date=2012 |isbn=978-1-28540279-6 |url=https://books.google.com/books?id=of8KAAAAQBAJ&pg=PA454 |page=454}}</ref>

In short, there are two major facets of roundoff errors involved in numerical calculations:<ref name="Chapra_2012">{{cite book |author-last=Chapra |author-first=Steven |title=Applied Numerical Methods with MATLAB for Engineers and Scientists |publisher=[[McGraw Hill Education|McGraw-Hill]] |date=2012 |isbn=9780073401102 |edition=3rd}}</ref>
# 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.