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In [[numerical analysis]], the speed at which a [[limit of a sequence|convergent sequence]] approaches its limit is called the '''rate of convergence'''. Although strictly speaking, a limit does not give information about any finite first part of the sequence, this concept is of practical importance if we deal with a sequence of successive approximations for an [[iterative method]], as then typically fewer iterations are needed to yield a useful approximation if the rate of convergence is higher. This may even make the difference between needing ten or a million iterations. | |||
Similar concepts are used for [[discretization]] methods. The solution of the discretized problem converges to the solution of the continuous problem as the grid size goes to zero, and the speed of convergence is one of the factors of the efficiency of the method. However, the terminology in this case is different from the terminology for iterative methods. | |||
[[Series acceleration]] is a collection of techniques for improving the rate of convergence of a series discretization . Such acceleration is commonly accomplished with [[sequence transformation]]s. | |||
==Convergence speed for iterative methods== | |||
===Basic definition=== | |||
Suppose that the sequence {''x''<sub>''k''</sub>} converges to the number L. | |||
<span id="linear_convergence"> | |||
We say that this sequence '''converges linearly''' to L, if there exists a number μ ∈ (0, 1) such that | |||
:<math> \lim_{k\to \infty} \frac{|x_{k+1}-L|}{|x_k-L|} = \mu.</math> | |||
The number μ is called the ''rate of convergence''. | |||
</span> | |||
If the sequence converges, and | |||
* μ = 0, then the sequence is said to '''converge superlinearly'''. | |||
* μ = 1, then the sequence is said to '''converge sublinearly'''. | |||
If the sequence ''converges sublinearly'' and additionally | |||
:<math> \lim_{k\to \infty} \frac{|x_{k+2} - x_{k+1}|}{|x_{k+1} - x_k|} = 1,</math> | |||
then it is said the sequence {''x''<sub>''k''</sub>} '''converges logarithmically''' to L. | |||
The next definition is used to distinguish superlinear rates of convergence. We say that the sequence '''converges with order ''q'' to L''' for ''q''>1<ref>''q'' may be non-integer. For example, the [[secant method]] has, in the case of convergence to a regular root, convergence order [[golden ratio|φ]]=1.618....</ref> if | |||
:<math> \lim_{k\to \infty} \frac{|x_{k+1}-L|}{|x_k-L|^q} = \mu \,\big|\; \mu > 0.</math> | |||
In particular, convergence with order | |||
* ''q'' = 2 is called '''quadratic convergence''', | |||
* ''q'' = 3 is called '''cubic convergence''', | |||
* etc. | |||
This is sometimes called ''Q-linear convergence'', ''Q-quadratic convergence'', etc., to distinguish it from the definition below. The Q stands for "quotient," because the definition uses the quotient between two successive terms. | |||
===Extended definition=== | |||
The drawback of the above definitions is that these do not catch some sequences which still converge reasonably fast, but whose "speed" is variable, such as the sequence {''b''<sub>''k''</sub>} below. Therefore, the definition of rate of convergence is sometimes extended as follows. | |||
Under the new definition, the sequence {''x''<sub>''k''</sub>} converges with at least order ''q'' if there exists a sequence {ε<sub>''k''</sub>} such that | |||
:<math>|x_k - L|\le\varepsilon_k\quad\mbox{for all }k,</math> | |||
and the sequence {ε<sub>''k''</sub>} converges to zero with order ''q'' according to the above "simple" definition. To distinguish it from that definition, this is sometimes called ''R-linear convergence'', ''R-quadratic convergence'', etc. (with the R standing for "root"). | |||
===Examples=== | |||
Consider the following sequences: | |||
:<math>\begin{align} | |||
a_0 &= 1 ,\, &&a_1 = \frac12 ,\, &&a_2 = \frac14 ,\, &&a_3 = \frac18 ,\, &&a_4 = \frac1{16} ,\, &&a_5 = \frac1{32} ,\, &&\ldots ,\, &&a_k = \frac1{2^k} ,\, &&\ldots \\ | |||
b_0 &= 1 ,\, &&b_1 = 1 ,\, &&b_2 = \frac14 ,\, &&b_3 = \frac14 ,\, &&b_4 = \frac1{16} ,\, &&b_5 = \frac1{16} ,\, &&\ldots ,\, &&b_k = \frac1{4^{\left\lfloor \frac{k}{2} \right\rfloor}} ,\, &&\ldots \\ | |||
c_0 &= \frac12 ,\, &&c_1 = \frac14 ,\, &&c_2 = \frac1{16} ,\, &&c_3 = \frac1{256} ,\, &&c_4 = \frac1{65\,536} ,\, &&&&\ldots ,\, &&c_k = \frac1{2^{2^k}} ,\, &&\ldots \\ | |||
d_0 &= 1 ,\, &&d_1 = \frac12 ,\, &&d_2 = \frac13 ,\, &&d_3 = \frac14 ,\, &&d_4 = \frac15 ,\, &&d_5 = \frac16 ,\, &&\ldots ,\, &&d_k = \frac1{k+1} ,\, &&\ldots | |||
\end{align}</math> | |||
The sequence {''a''<sub>''k''</sub>} converges linearly to 0 with rate 1/2. More generally, the sequence ''Cμ''<sup>''k''</sup> converges linearly with rate μ if |μ| < 1. The sequence {''b''<sub>''k''</sub>} also converges linearly to 0 with rate 1/2 under the extended definition, but not under the simple definition. The sequence {''c''<sub>''k''</sub>} converges superlinearly. In fact, it is quadratically convergent. Finally, the sequence {''d''<sub>''k''</sub>} converges sublinearly. | |||
[[Image:ConvergencePlots.png|thumb|alt=Plot showing the different rates of convergence for the sequences a_k, b_k, c_k and d_k.|Linear, Linear, Superlinear/Quadratic and sublinear rate of convergence.|400px|center]] | |||
==Convergence speed for discretization methods== | |||
A similar situation exists for discretization methods. The important parameter here for the convergence speed is not the iteration number ''k'' but it depends on the number of grid points and grid spacing. In this case, the number of grid points in a discretization process is inversely proportional to the grid spacing here denoted as ''n''. | |||
In this case, a sequence <math>x_n</math> is said to converge to L with order ''p'' if there exists a constant ''C'' such that | |||
:<math> |x_n - L| < Cn^{-p} \text{ for all } n. </math> | |||
This is written as |''x''<sub>''n''</sub> - L| = O(''n''<sup>-''p''</sup>) using the [[big O notation]]. | |||
This is the relevant definition when discussing methods for [[numerical quadrature]] or the [[numerical ordinary differential equations|solution of ordinary differential equations]]. | |||
=== Examples === | |||
The sequence {''d''<sub>''k''</sub>} with ''d''<sub>''k''</sub> = 1 / (''k''+1) was introduced above. This sequence converges with order 1 according to the convention for discretization methods. | |||
The sequence {''a''<sub>''k''</sub>} with ''a''<sub>''k''</sub> = 2<sup>-''k''</sup>, which was also introduced above, converges with order ''p'' for every number ''p''. It is said to converge exponentially using the convention for discretization methods. However, it only converges linearly (that is, with order 1) using the convention for iterative methods. | |||
The order of convergence of a discretization method is related to its [[Truncation error (numerical integration)|Global Truncation Error (GTE)]]. | |||
==Acceleration of convergence== | |||
Many methods exist to increase the rate of convergence of a given sequence, | |||
i.e. to [[sequence transformation|transform a given sequence]] into one converging faster to the same limit. Such techniques are in general known as "[[series acceleration]]". The goal of the transformed sequence is to be much less "expensive" to calculate than the original sequence. One example of series acceleration is [[Aitken's delta-squared process]]. | |||
==References== | |||
<references /> | |||
==Literature== | |||
The simple definition is used in | |||
* Michelle Schatzman (2002), ''Numerical analysis: a mathematical introduction'', Clarendon Press, Oxford. ISBN 0-19-850279-6. | |||
The extended definition is used in | |||
* Kendell A. Atkinson (1988), ''An introduction to numerical analysis'' (2nd ed.), John Wiley and Sons. ISBN 0-471-50023-2. | |||
* http://web.mit.edu/rudin/www/MukherjeeRuSc11COLT.pdf | |||
* Walter Gautschi (1997), ''Numerical analysis: an introduction,'' Birkhäuser, Boston. ISBN 0-8176-3895-4. | |||
* [[Endre Süli]] and David Mayers (2003), ''An introduction to numerical analysis,'' Cambridge University Press. ISBN 0-521-00794-1. | |||
Logarithmic convergence is used in | |||
* {{cite journal | last = Van Tuyl | first = Andrew H. | year = 1994 | title = Acceleration of convergence of a family of logarithmically convergent sequences | journal = Mathematics of Computation | volume = 63 | issue = 207 | pages = 229–246 | url = http://www.ams.org/journals/mcom/1994-63-207/S0025-5718-1994-1234428-2/S0025-5718-1994-1234428-2.pdf}} | |||
The Big O definition is used in | |||
*Richard L. Burden and J. Douglas Faires (2001), ''Numerical Analysis'' (7th ed.), Brooks/Cole. ISBN 0-534-38216-9 | |||
The terms ''Q-linear'' and ''R-linear'' are used in; The Big O definition when using Taylor series is used in | |||
* {{Cite book | last1=Nocedal | first1=Jorge | last2=Wright | first2=Stephen J. | title=Numerical Optimization | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-30303-1 | year=2006 | pages=619+620 | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}. | |||
One may also study the following paper for Q-linear and R-linear: | |||
* {{cite journal | last = Potra | first = F. A. | year = 1989 | title = On Q-order and R-order of convergence | journal = J. Optim. Th. Appl. | volume = 63 | issue = 3 | pages = 415–431 | doi = 10.1007/BF00939805}} | |||
[[Category:Numerical analysis]] | |||
Revision as of 08:56, 16 January 2014
In numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence. Although strictly speaking, a limit does not give information about any finite first part of the sequence, this concept is of practical importance if we deal with a sequence of successive approximations for an iterative method, as then typically fewer iterations are needed to yield a useful approximation if the rate of convergence is higher. This may even make the difference between needing ten or a million iterations.
Similar concepts are used for discretization methods. The solution of the discretized problem converges to the solution of the continuous problem as the grid size goes to zero, and the speed of convergence is one of the factors of the efficiency of the method. However, the terminology in this case is different from the terminology for iterative methods.
Series acceleration is a collection of techniques for improving the rate of convergence of a series discretization . Such acceleration is commonly accomplished with sequence transformations.
Convergence speed for iterative methods
Basic definition
Suppose that the sequence {xk} converges to the number L.
We say that this sequence converges linearly to L, if there exists a number μ ∈ (0, 1) such that
The number μ is called the rate of convergence.
If the sequence converges, and
- μ = 0, then the sequence is said to converge superlinearly.
- μ = 1, then the sequence is said to converge sublinearly.
If the sequence converges sublinearly and additionally
then it is said the sequence {xk} converges logarithmically to L.
The next definition is used to distinguish superlinear rates of convergence. We say that the sequence converges with order q to L for q>1[1] if
In particular, convergence with order
- q = 2 is called quadratic convergence,
- q = 3 is called cubic convergence,
- etc.
This is sometimes called Q-linear convergence, Q-quadratic convergence, etc., to distinguish it from the definition below. The Q stands for "quotient," because the definition uses the quotient between two successive terms.
Extended definition
The drawback of the above definitions is that these do not catch some sequences which still converge reasonably fast, but whose "speed" is variable, such as the sequence {bk} below. Therefore, the definition of rate of convergence is sometimes extended as follows.
Under the new definition, the sequence {xk} converges with at least order q if there exists a sequence {εk} such that
and the sequence {εk} converges to zero with order q according to the above "simple" definition. To distinguish it from that definition, this is sometimes called R-linear convergence, R-quadratic convergence, etc. (with the R standing for "root").
Examples
Consider the following sequences:
The sequence {ak} converges linearly to 0 with rate 1/2. More generally, the sequence Cμk converges linearly with rate μ if |μ| < 1. The sequence {bk} also converges linearly to 0 with rate 1/2 under the extended definition, but not under the simple definition. The sequence {ck} converges superlinearly. In fact, it is quadratically convergent. Finally, the sequence {dk} converges sublinearly.
Convergence speed for discretization methods
A similar situation exists for discretization methods. The important parameter here for the convergence speed is not the iteration number k but it depends on the number of grid points and grid spacing. In this case, the number of grid points in a discretization process is inversely proportional to the grid spacing here denoted as n.
In this case, a sequence is said to converge to L with order p if there exists a constant C such that
This is written as |xn - L| = O(n-p) using the big O notation.
This is the relevant definition when discussing methods for numerical quadrature or the solution of ordinary differential equations.
Examples
The sequence {dk} with dk = 1 / (k+1) was introduced above. This sequence converges with order 1 according to the convention for discretization methods.
The sequence {ak} with ak = 2-k, which was also introduced above, converges with order p for every number p. It is said to converge exponentially using the convention for discretization methods. However, it only converges linearly (that is, with order 1) using the convention for iterative methods.
The order of convergence of a discretization method is related to its Global Truncation Error (GTE).
Acceleration of convergence
Many methods exist to increase the rate of convergence of a given sequence, i.e. to transform a given sequence into one converging faster to the same limit. Such techniques are in general known as "series acceleration". The goal of the transformed sequence is to be much less "expensive" to calculate than the original sequence. One example of series acceleration is Aitken's delta-squared process.
References
- ↑ q may be non-integer. For example, the secant method has, in the case of convergence to a regular root, convergence order φ=1.618....
Literature
The simple definition is used in
- Michelle Schatzman (2002), Numerical analysis: a mathematical introduction, Clarendon Press, Oxford. ISBN 0-19-850279-6.
The extended definition is used in
- Kendell A. Atkinson (1988), An introduction to numerical analysis (2nd ed.), John Wiley and Sons. ISBN 0-471-50023-2.
- http://web.mit.edu/rudin/www/MukherjeeRuSc11COLT.pdf
- Walter Gautschi (1997), Numerical analysis: an introduction, Birkhäuser, Boston. ISBN 0-8176-3895-4.
- Endre Süli and David Mayers (2003), An introduction to numerical analysis, Cambridge University Press. ISBN 0-521-00794-1.
Logarithmic convergence is used in
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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
The Big O definition is used in
- Richard L. Burden and J. Douglas Faires (2001), Numerical Analysis (7th ed.), Brooks/Cole. ISBN 0-534-38216-9
The terms Q-linear and R-linear are used in; The Big O definition when using Taylor series is used in
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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