In numerical analysis

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

, the Shanks transformation is a non-linear series acceleration

Series acceleration

In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration...

 method to increase the rate of convergence

Rate of convergence

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...

 of a sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

. This method is named after Daniel Shanks

Daniel Shanks

Daniel Shanks was an American mathematician who worked primarily in numerical analysis and number theory. He is best known as the first to compute π to 100,000 decimal places, and for his book Solved and Unsolved Problems in Number Theory.-Life and education:Dan Shanks was born on January 17,...

, who rediscovered this sequence transformation in 1955. It was first derived and published by R. Schmidt in 1941.

Formulation

For a sequence the series


is to be determined. First, the partial sum is defined as:


and forms a new sequence . Provided the series converges, will approach in the limit to as
The Shanks transformation of the sequence is defined as


and forms a new sequence. The sequence often converges more rapidly than the sequence
Further speed-up may be obtained by repeated use of the Shanks transformation, by computing etc.

Note that the non-linear transformation as used in the Shanks transformation is of similar form as used in Aitken's delta-squared process

Aitken's delta-squared process

In numerical analysis, Aitken's delta-squared process is a series acceleration method, used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926. Its early form was known to Seki Kōwa and was found for rectification of the...

. But while Aitken's method operates on the coefficients of the original sequence, the Shanks transformation operates on the partial sums

Example

As an example, consider the slowly-convergent series


which has the exact sum π ≈ 3.14159265. The partial sum has only one digit accuracy, while six-figure accuracy requires summing about 400,000 terms.

In the table below, the partial sums , the Shanks transformation on them, as well as the repeated Shanks transformations and are given for up to 12. The figure to the right shows the absolute error for the partial sums and Shanks transformation results, clearly showing the improved accuracy and convergence rate.

		transient Transient (oscillation) A transient event is a short-lived burst of energy in a system caused by a sudden change of state.The source of the transient energy may be an internal event or a nearby event... ly to the series result for So for and the respective partial sums are: These three equations contain three unknowns: and Solving for gives In the (exceptional) case that the denominator is equal to zero: then for all Generalized Shanks transformation The generalized kth-order Shanks transformation is given as the ratio of the determinant Determinant In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well... s: with It is the solution of a model for the convergence behaviour of the partial sums with distinct transients: This model for the convergence behaviour contains unknowns. By evaluating the above equation at the elements and solving for the above expression for the th-order Shanks transformation is obtained. The first-order generalized Shanks transformation is equal to the ordinary Shanks transformation: The generalized Shanks transformation is closely related to Padé approximant Padé approximant Padé approximant is the "best" approximation of a function by a rational function of given order - under this technique, the approximant's power series agrees with the power series of the function it is approximating.... s and Padé table Padé table In complex analysis, a Padé table is an array, possibly of infinite extent, of the rational Padé approximantsto a given complex formal power series... s. See also Aitken's delta-squared process Aitken's delta-squared process In numerical analysis, Aitken's delta-squared process is a series acceleration method, used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926. Its early form was known to Seki Kōwa and was found for rectification of the... rate of convergence Rate of convergence 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... Richardson extrapolation Richardson extrapolation In numerical analysis, Richardson extrapolation is a sequence acceleration method, used to improve the rate of convergence of a sequence. It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century. In the words of Birkhoff and Rota, "..... sequence transformation The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL. Silverdale Interactive © 2025. All Rights Reserved.   x OK