# JSXGraph 1.10.0 Reference

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# Namespace JXG.Math.Extrapolate

↳ JXG.Math.Extrapolate

Defined in: extrapolate.js.

Namespace Summary
Constructor Attributes | Constructor Name and Description
--- | ---
| JXG.Math.Extrapolate Functions for extrapolation of sequences.

Method Summary
Method Attributes | Method Name and Description
--- | ---
<static> | JXG.Math.Extrapolate.aitken(s_n, n, a) Aitken transformation.
<static> | JXG.Math.Extrapolate.brezinski(s_n, n, a) Iterated Brezinski transformation.
<static> | JXG.Math.Extrapolate.iteration(x0, h0, f, method, step_type) Extrapolated iteration to approximate the value f(x_0).
<static> | JXG.Math.Extrapolate.levin(s_n, n, numer, denom, numer, denom) Levin transformation.
<static> | JXG.Math.Extrapolate.limit(x0, h0, f)
<static> | JXG.Math.Extrapolate.wynnEps(s_n, n, e) Wynn's epsilon algorithm.

Namespace Detail
JXG.Math.Extrapolate
Functions for extrapolation of sequences. Used for finding limits of sequences which is used for curve plotting.
Method Detail
<static> {Number} JXG.Math.Extrapolate.aitken(s_n, n, a)
Aitken transformation. Ported from the FORTRAN version in Ernst Joachim Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series", Computer Physics Reports Vol. 10, 189-371 (1989).

Parameters:
{Number} s_n
  next value of sequence, i.e. n-th element of sequence
{Number} n
  index of s_n in the sequence
{Array} a
  One-dimensional array containing the extrapolation data. Has to be supplied by the calling routine.

Returns:
  {Number} New estimate of the limit of the sequence.

<static> {Number} JXG.Math.Extrapolate.brezinski(s_n, n, a)
Iterated Brezinski transformation. Ported from the FORTRAN version in Ernst Joachim Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series", Computer Physics Reports Vol. 10, 189-371 (1989).

Parameters:
{Number} s_n
  next value of sequence, i.e. n-th element of sequence
{Number} n
  index of s_n in the sequence
{Array} a
  One-dimensional array containing the extrapolation data. Has to be supplied by the calling routine.

Returns:
  {Number} New estimate of the limit of the sequence.

<static> {Array} JXG.Math.Extrapolate.iteration(x0, h0, f, method, step_type)
Extrapolated iteration to approximate the value f(x_0).

Parameters:
{Number} x0
  Value for which the limit of f is to be determined. f(x0) may or may not exist.
{Number} h0
  Initial (signed) distance from x0.
{Function} f
  Function for which the limit at x0 is to be determined
{String} method
  String to choose the method. Available values: "wynnEps", "aitken", "brezinski"
{Number} step_type
  Approximation method. step_type = 0 uses the sequence x0 + h0/n; step_type = 1 uses the sequence x0 + h0 * 2^(-n)

Returns:
  {Array} Array of length 3. Position 0: estimated value for f(x0), position 1: 'finite', 'infinite', or 'NaN'. Position 2: value between 0 and 1 judging the reliability of the result (1: high, 0: not successful).

See:
  JXG.Math.Extrapolate.limit
  JXG.Math.Extrapolate.wynnEps
  JXG.Math.Extrapolate.aitken
  JXG.Math.Extrapolate.brezinski

<static> JXG.Math.Extrapolate.levin(s_n, n, numer, denom, numer, denom)
Levin transformation. See Numerical Recipes, ed. 3. Not yet ready for use.

Parameters:
{Number} s_n
  next value of sequence, i.e. n-th element of sequence
{Number} n
  index of s_n in the sequence
{Array} numer
  One-dimensional array containing the extrapolation data for the numerator. Has to be supplied by the calling routine.
{Array} denom
  One-dimensional array containing the extrapolation data for the denominator. Has to be supplied by the calling routine.
numer
denom

<static> {Array} JXG.Math.Extrapolate.limit(x0, h0, f)

Parameters:
{Number} x0
  Value for which the limit of f is to be determined. f(x0) may or may not exist.
{Number} h0
  Initial (signed) distance from x0.
{Function} f
  Function for which the limit at x0 is to be determined

Returns:
  {Array} Array of length 3. Position 0: estimated value for f(x0), position 1: 'finite', 'infinite', or 'NaN'. Position 2: value between 0 and 1 judging the reliability of the result (1: high, 0: not successful). In case that the extrapolation fails, position 1 and 2 contain 'direct' and 0.

See:
  JXG.Math.Extrapolate.iteration

Examples:

    var f1 = (x) => Math.log(x),
        f2 = (x) => Math.tan(x - Math.PI * 0.5),
        f3 = (x) => 4 / x;

    var x0 = 0.0000001;
    var h = 0.1;
    for (let f of [f1, f2, f3]) {
        console.log("x0=", x0, f.toString());
        console.log(JXG.Math.Extrapolate.limit(x0, h, f));
     }

<static> {Number} JXG.Math.Extrapolate.wynnEps(s_n, n, e)
Wynn's epsilon algorithm. Ported from the FORTRAN version in Ernst Joachim Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series", Computer Physics Reports Vol. 10, 189-371 (1989).

Parameters:
{Number} s_n
  next value of sequence, i.e. n-th element of sequence
{Number} n
  index of s_n in the sequence
{Array} e
  One-dimensional array containing the extrapolation data. Has to be supplied by the calling routine.

Returns:
  {Number} New estimate of the limit of the sequence.

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