Editing Rate of convergence (section)

=== Definitions ===
In [[asymptotic analysis]] in general, one sequence <math>(a_k)_{k \in \mathbb{N}}</math> that converges to a [[Limit (mathematics)|limit]] <math>L</math> is said to asymptotically converge to <math>L</math> with a faster order of convergence than another sequence <math>(b_k)_{k \in \mathbb{N}}</math> that converges to <math>L</math> in a shared [[metric space]] with [[distance metric]] <math>|\cdot|,</math> such as the [[Real number|real numbers]] or [[Complex number|complex numbers]] with the ordinary [[absolute difference]] metrics, if

<math display="block">\lim _{k \rightarrow \infty} \frac{\left|a_k - L\right|}{|b_k - L|} = 0,</math>

the two are said to asymptotically converge to <math>L</math> with the same order of convergence if 

<math display="block">\lim_{k \rightarrow \infty} \frac{\left|a_k - L\right|}{|b_k - L|} = \mu</math>

for some positive finite constant <math>\mu,</math> and the two are said to asymptotically converge to <math>L</math> with the same rate and order of convergence if 

<math display="block">\lim_{k \rightarrow \infty} \frac{\left|a_k - L\right|}{|b_k - L|} = 1.</math>

These comparative definitions of rate and order of asymptotic convergence are fundamental in [[asymptotic analysis]].<ref name="Balcázar">{{cite journal |last1=Balcázar |first1=José L. |last2=Gabarró |first2=Joaquim |title=Nonuniform complexity classes specified by lower and upper bounds |url=http://archive.numdam.org/article/ITA_1989__23_2_177_0.pdf |url-status=live |journal=RAIRO – Theoretical Informatics and Applications – Informatique Théorique et Applications |language=en |volume=23 |issue=2 |page=180 |issn=0988-3754 |archive-url=https://web.archive.org/web/20170314153158/http://archive.numdam.org/article/ITA_1989__23_2_177_0.pdf |archive-date=14 March 2017 |access-date=14 March 2017 |via=Numdam}}</ref><ref name="Cucker">{{cite book |last1=Cucker |first1=Felipe |title=Condition: The Geometry of Numerical Algorithms |last2=Bürgisser |first2=Peter |publisher=Springer |year=2013 |isbn=978-3-642-38896-5 |location=Berlin, Heidelberg |pages=467–468 |chapter=A.1 Big Oh, Little Oh, and Other Comparisons |doi=10.1007/978-3-642-38896-5 |chapter-url=https://books.google.com/books?id=SNu4BAAAQBAJ&pg=PA467}}</ref> For the first two of these there are associated expressions in [[Big O notation|asymptotic O notation]]: the first is that <math>a_k - L = o(b_k - L)</math> in small o notation<ref name=":22">{{Cite book |last=Apostol |first=Tom M. |author-link=Tom M. Apostol |title=Calculus |date=1967 |publisher=John Wiley & Sons |isbn=0-471-00005-1 |edition=2nd |volume=1 |location=USA |pages=286}}</ref> and the second is that <math>a_k - L = \Theta(b_k - L) </math> in Knuth notation.<ref name="knuth">{{cite journal |last=Knuth |first=Donald |date=April–June 1976 |title=Big Omicron and big Omega and big Theta |journal=SIGACT News |volume=8 |issue=2 |pages=18–24 |doi=10.1145/1008328.1008329 |s2cid=5230246 |doi-access=free |s2cid-access=free}}</ref> The third is also called asymptotic equivalence, expressed <math>a_k - L \sim b_k - L.</math><ref name=":2">{{Cite book |last=Apostol |first=Tom M. |title=Calculus |date=1967 |publisher=John Wiley & Sons |isbn=0-471-00005-1 |edition=2nd |volume=1 |location=USA |pages=396}}</ref><ref>{{SpringerEOM|id=Asymptotic_equality|title=Asymptotic equality}}</ref>