Rate of convergence

Summary

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

[1]

The rate of convergence 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 methodsEdit

Q-convergence definitionsEdit

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.[3]

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  .[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   to   for   if

 

for some positive constant   (not necessarily less than 1 if  ). In particular, convergence with order

  •   is called linear convergence (if  ),
  •   is called quadratic convergence,
  •   is called cubic convergence,
  • etc.

Some sources require that   is strictly greater than   since the   case requires   so is best treated separately.[5] 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.[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 estimationEdit

A practical method to calculate the order of convergence for a sequence is to calculate the following sequence, which converges to  

 [7]

R-convergence definitionEdit

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

 

and   converges Q-linearly to zero.[3] The "R-" prefix stands for "root". [6]: 620 

ExamplesEdit

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  .

The sequence

 

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  .)

The sequence

 

converges superlinearly. In fact, it is quadratically convergent.

Finally, the sequence

 

converges sublinearly and logarithmically.

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

Convergence speed for discretization methodsEdit

A similar situation exists for discretization methods designed to approximate a function  , which might be an integral being approximated by numerical quadrature, or the solution of an ordinary differential equation (see example below). The discretization method generates a sequence  , where each successive   is a function of   along with the grid spacing   between successive values of the independent variable  . The important parameter here for the convergence speed to   is the grid spacing  , inversely proportional to the number of grid points, i.e. the number of points in the sequence required to reach a given value of  .

In this case, the sequence   is said to converge to the sequence   with order q if there exists a constant C such that

 

This is written as   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   and   and calculate the resulting errors   and  . The order of convergence is then approximated by the following formula:

 [citation needed]

which comes from writing the truncation error, at the old and new grid spacings, as

 

The error   is, more specifically, a global truncation error (GTE), in that it represents a sum of errors accumulated over all   iterations, as opposed to a local truncation error (LTE) over just one iteration.

Example of discretization methodsEdit

Consider the ordinary differential equation

 

with initial condition  . We can solve this equation using the Forward Euler scheme for numerical discretization:

 

which generates the sequence

 

In terms of  , this sequence is as follows, from the Binomial theorem:

 

The exact solution to this ODE is  , corresponding to the following Taylor expansion in   for  :

 

In this case, the truncation error is

 

so   converges to   with a convergence rate  .

Examples (continued)Edit

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?]

Recurrent sequences and fixed pointsEdit

The case of recurrent sequences   which occurs in dynamical systems and in the context of various fixed point theorems is of particular interest. Assuming that the relevant derivatives of f are continuous, one can (easily) show that for a fixed point   such that  , one has at linear convergence for any starting value   sufficiently close to p. If   and  , then one has at least quadratic convergence, and so on. If  , then one has a repulsive fixed point and no starting value will produce a sequence converging to p (unless one directly jumps to the point p itself).

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. (It should be noted, though, that these methods in general (and in particular Aitken's method) do not increase the order of convergence, and are useful only if initially the convergence is not faster than linear: If   convergences linearly, one gets a sequence   that still converges linearly (except for pathologically designed special cases), but faster in the sense that  . On the other hand, if the convergence is already of order ≥ 2, Aitken's method will bring no improvement.)

ReferencesEdit

  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.

LiteratureEdit

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

  • Nocedal, Jorge; Wright, Stephen J. (2006). Numerical Optimization (2nd ed.). Berlin, New York: Springer-Verlag. pp. 619+620. ISBN 978-0-387-30303-1..