Rate of convergence

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https://en.wikipedia.org/wiki/Linear_convergence

In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence ${\displaystyle (x_{n})}$ that converges to ${\displaystyle x^{*}}$ is said to have order of convergence ${\displaystyle q\geq 1}$ and rate of convergence ${\displaystyle \mu }$ if

${\displaystyle \lim _{n\rightarrow \infty }{\frac {\left|x_{n+1}-x^{*}\right|}{\left|x_{n}-x^{*}\right|^{q}}}=\mu .}$ [1]

The rate of convergence ${\displaystyle \mu }$ is also called the asymptotic error constant. Note that this terminology is not standardized and some authors will use rate where this article uses order (e.g., [2]).

In practice, the rate and order of convergence provide useful insights when using iterative methods for calculating numerical approximations. If the order of convergence is higher, then typically fewer iterations are necessary to yield a useful approximation. Strictly speaking, however, the asymptotic behavior of a sequence does not give conclusive information about any finite part of the sequence.

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

Q-convergence definitions

Suppose that the sequence ${\displaystyle (x_{k})}$ converges to the number ${\displaystyle L}$. The sequence is said to converge Q-linearly to ${\displaystyle L}$ if there exists a number ${\displaystyle \mu \in (0,1)}$ such that

${\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|}}=\mu .}$

The number ${\displaystyle \mu }$ is called the rate of convergence. [3]

The sequence is said to converge Q-superlinearly to ${\displaystyle L}$ (i.e. faster than linearly) if

${\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|}}=0}$

and it is said to converge Q-sublinearly to ${\displaystyle L}$ (i.e. slower than linearly) if

${\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|}}=1.}$

If the sequence converges sublinearly and additionally

${\displaystyle \lim _{k\to \infty }{\frac {|x_{k+2}-x_{k+1}|}{|x_{k+1}-x_{k}|}}=1,}$

then it is said that the sequence ${\displaystyle (x_{k})}$ converges logarithmically to ${\displaystyle L}$. [4] Note that unlike previous definitions, logarithmic convergence is not called "Q-logarithmic."

In order to further classify convergence, the order of convergence is defined as follows. The sequence is said to converge with order ${\displaystyle q}$ to ${\displaystyle L}$ for ${\displaystyle q\geq 1}$ if

${\displaystyle \lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|^{q}}}

for some positive constant ${\displaystyle M>0}$ (not necessarily less than 1 if ${\displaystyle q>1}$). In particular, convergence with order

• ${\displaystyle q=1}$ is called linear convergence (if ${\displaystyle M<1}$),
• ${\displaystyle q=2}$ is called quadratic convergence,
• ${\displaystyle q=3}$ is called cubic convergence,
• etc.

Some sources require that ${\displaystyle q}$ is strictly greater than ${\displaystyle 1}$ since the ${\displaystyle q=1}$ case requires ${\displaystyle M<1}$ so is best treated separately. [5] It is not necessary, however, that ${\displaystyle q}$ be an integer. For example, the secant method, when converging to a regular, simple root, has an order of φ ≈ 1.618.[ citation needed]

In the definitions above, the "Q-" stands for "quotient" because the terms are defined using the quotient between two successive terms. [6]: 619  Often, however, the "Q-" is dropped and a sequence is simply said to have linear convergence, quadratic convergence, etc.

Order estimation

A practical method to calculate the order of convergence for a sequence is to calculate the following sequence, which converges to ${\displaystyle q}$

${\displaystyle q\approx {\frac {\log \left|{\frac {x_{k+1}-x_{k}}{x_{k}-x_{k-1}}}\right|}{\log \left|{\frac {x_{k}-x_{k-1}}{x_{k-1}-x_{k-2}}}\right|}}.}$ [7]

R-convergence definition

The Q-convergence definitions have a shortcoming in that they do not include some sequences, such as the sequence ${\displaystyle (b_{k})}$ below, which converge reasonably fast, but whose rate is variable. Therefore, the definition of rate of convergence is extended as follows.

Suppose that ${\displaystyle (x_{k})}$ converges to ${\displaystyle L}$. The sequence is said to converge R-linearly to ${\displaystyle L}$ if there exists a sequence ${\displaystyle (\varepsilon _{k})}$ such that

${\displaystyle |x_{k}-L|\leq \varepsilon _{k}\quad {\text{for all }}k\,,}$

and ${\displaystyle (\varepsilon _{k})}$ converges Q-linearly to zero. [3] The "R-" prefix stands for "root". [6]: 620

Examples

Consider the sequence

${\displaystyle (a_{k})=\left\{1,{\frac {1}{2}},{\frac {1}{4}},{\frac {1}{8}},{\frac {1}{16}},{\frac {1}{32}},\ldots ,{\frac {1}{2^{k}}},...\right\}.}$

It can be shown that this sequence converges to ${\displaystyle L=0}$. To determine the type of convergence, we plug the sequence into the definition of Q-linear convergence,

${\displaystyle \lim _{k\to \infty }{\frac {\left|1/2^{k+1}-0\right|}{\left|1/2^{k}-0\right|}}=\lim _{k\to \infty }{\frac {2^{k}}{2^{k+1}}}={\frac {1}{2}}.}$

Thus, we find that ${\displaystyle (a_{k})}$ converges Q-linearly and has a convergence rate of ${\displaystyle \mu =1/2}$. More generally, for any ${\displaystyle c\in \mathbb {R} ,\mu \in (-1,1)}$, the sequence ${\displaystyle (c\mu ^{k})}$ converges linearly with rate ${\displaystyle |\mu |}$.

The sequence

${\displaystyle (b_{k})=\left\{1,1,{\frac {1}{4}},{\frac {1}{4}},{\frac {1}{16}},{\frac {1}{16}},\ldots ,{\frac {1}{4^{\left\lfloor {\frac {k}{2}}\right\rfloor }}},\,\ldots \right\}}$

also converges linearly to 0 with rate 1/2 under the R-convergence definition, but not under the Q-convergence definition. (Note that ${\displaystyle \lfloor x\rfloor }$ is the floor function, which gives the largest integer that is less than or equal to ${\displaystyle x}$.)

The sequence

${\displaystyle (c_{k})=\left\{{\frac {1}{2}},{\frac {1}{4}},{\frac {1}{16}},{\frac {1}{256}},{\frac {1}{65,\!536}},\ldots ,{\frac {1}{2^{2^{k}}}},\ldots \right\}}$

converges superlinearly. In fact, it is quadratically convergent.

Finally, the sequence

${\displaystyle (d_{k})=\left\{1,{\frac {1}{2}},{\frac {1}{3}},{\frac {1}{4}},{\frac {1}{5}},{\frac {1}{6}},\ldots ,{\frac {1}{k+1}},\ldots \right\}}$

converges sublinearly and logarithmically.

Linear, linear, superlinear (quadratic), and sublinear rates of convergence

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 the number of grid points and grid spacing. In this case, the number of grid points n in a discretization process is inversely proportional to the grid spacing.

In this case, a sequence ${\displaystyle (x_{n})}$ is said to converge to L with order q if there exists a constant C such that

${\displaystyle |x_{n}-L|

This is written as ${\displaystyle |x_{n}-L|={\mathcal {O}}(n^{-q})}$ using big O notation.

This is the relevant definition when discussing methods for numerical quadrature or the solution of ordinary differential equations.[ example needed]

A practical method to estimate the order of convergence for a discretization method is pick step sizes ${\displaystyle h_{\text{new}}}$ and ${\displaystyle h_{\text{old}}}$ and calculate the resulting errors ${\displaystyle e_{\text{new}}}$ and ${\displaystyle e_{\text{old}}}$. The order of convergence is then approximated by the following formula:

${\displaystyle q\approx {\frac {\log(e_{\text{new}}/e_{\text{old}})}{\log(h_{\text{new}}/h_{\text{old}})}}.}$[ citation needed]

Examples (continued)

The sequence ${\displaystyle (d_{k})}$ with ${\displaystyle d_{k}=1/(k+1)}$ was introduced above. This sequence converges with order 1 according to the convention for discretization methods.[ why?]

The sequence ${\displaystyle (a_{k})}$ with ${\displaystyle a_{k}=2^{-k}}$, which was also introduced above, converges with order q for every number q. 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.[ why?]

The order of convergence of a discretization method is related to its global truncation error (GTE).[ how?]

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 reduce the computational cost of the calculation. One example of series acceleration is Aitken's delta-squared process.

References

1. ^ Ruye, Wang (2015-02-12). "Order and rate of convergence". hmc.edu. Retrieved 2020-07-31.
2. ^ Senning, Jonathan R. "Computing and Estimating the Rate of Convergence" (PDF). gordon.edu. Retrieved 2020-08-07.
3. ^ a b Bockelman, Brian (2005). "Rates of Convergence". math.unl.edu. Retrieved 2020-07-31.
4. ^ Van Tuyl, Andrew H. (1994). "Acceleration of convergence of a family of logarithmically convergent sequences" (PDF). Mathematics of Computation. 63 (207): 229–246. doi: 10.2307/2153571. JSTOR  2153571. Retrieved 2020-08-02.
5. ^ Porta, F. A. (1989). "On Q-Order and R-Order of Convergence" (PDF). Journal of Optimization Theory and Applications. 63 (3): 415–431. doi: 10.1007/BF00939805. S2CID  116192710. Retrieved 2020-07-31.
6. ^ a b Nocedal, Jorge; Wright, Stephen J. (2006). Numerical Optimization (2nd ed.). Berlin, New York: Springer-Verlag. ISBN  978-0-387-30303-1.
7. ^ Senning, Jonathan R. "Computing and Estimating the Rate of Convergence" (PDF). gordon.edu. Retrieved 2020-08-07.

Literature

The simple definition is used in

The extended definition is used in

• 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.

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