Rate of 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 that converges to is said to have order of convergence and rate of convergence if
The rate of convergence is also called the asymptotic error constant.
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.
Convergence speed for iterative methodsEdit
Suppose that the sequence converges to the number . The sequence is said to converge Q-linearly to if there exists a number such that
The number is called the rate of convergence.
The sequence is said to converge Q-superlinearly to (i.e. faster than linearly) if
and it is said to converge Q-sublinearly to (i.e. slower than linearly) if
If the sequence converges sublinearly and additionally
then it is said that the sequence converges logarithmically to . 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 to for if
for some positive constant (not necessarily less than 1). In particular, convergence with order
- is called linear convergence,
- is called quadratic convergence,
- is called cubic convergence,
Some sources require that is strictly greater than . It is not necessary, however, that be an integer. For example, the secant method, when converging to a regular, simple root, has an order of φ ≈ 1.618.
In the definitions above, the "Q-" stands for "quotient" because the terms are defined using the quotient between two successive terms.:619 Often, however, the "Q-" is dropped and a sequence is simply said to have linear convergence, quadratic convergence, etc.
A practical method to calculate the order of convergence for a sequence is to calculate the following sequence, which converges to
The Q-convergence definitions have a shortcoming in that they do not include some sequences, such as the sequence below, which converge reasonably fast, but whose rate is variable. Therefore, the definition of rate of convergence is extended as follows.
Suppose that converges to . The sequence is said to converge R-linearly to if there exists a sequence such that
Consider the sequence
It can be shown that this sequence converges to . To determine the type of convergence, we plug the sequence into the definition of Q-linear convergence,
Thus, we find that converges Q-linearly and has a convergence rate of . More generally, for any , the sequence converges linearly with rate .
also converges linearly to 0 with rate 1/2 under the R-convergence definition, but not under the Q-convergence definition. (Note that is the floor function, which gives the largest integer that is less than or equal to .)
converges superlinearly. In fact, it is quadratically convergent.
Finally, the sequence
converges sublinearly and logarithmically.
Convergence speed for discretization methodsEdit
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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 is said to converge to L with order q if there exists a constant C such that
This is written as using big O notation.
A practical method to estimate the order of convergence for a discretization method is pick step sizes and and calculate the resulting errors and . The order of convergence is then approximated by the following formula:
The sequence with was introduced above. This sequence converges with order 1 according to the convention for discretization methods.[why?]
The sequence with , 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?]
Acceleration of convergenceEdit
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.
- Ruye, Wang (2015-02-12). "Order and rate of convergence". hmc.edu. Retrieved 2020-07-31.
- Bockelman, Brian (2005). "Rates of Convergence". math.unl.edu. Retrieved 2020-07-31.
- 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.
- 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.
- Nocedal, Jorge; Wright, Stephen J. (2006). Numerical Optimization (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-30303-1.
- Senning, Jonathan R. "Computing and Estimating the Rate of Convergence" (PDF). gordon.edu. Retrieved 2020-08-07.
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
- 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