Improvement of Romberg extrapolation
------------------------------------
Key: MATH-325
URL: https://issues.apache.org/jira/browse/MATH-325
Project: Commons Math
Issue Type: Improvement
Affects Versions: 2.0
Reporter: Andreas mueller
Fix For: 2.1
One can use a one-dimensional array (instead of Romberg's tableau) for
extrapolating subsequent values.
Please have a look at following code fragments (which I've taken form the class
RombergExtrapolator of
my MathLibrary). Feel free to use this code.
/**
* Default number of maximal extrapolation steps.
*/
public static int DEF_MAXIMAL_EXTRAPOLATION_COUNT = 8;
/**
* The approximation order. <br>
* Assume that f(h) is approximated by a function a(h), so that f(h) =
a(h) +
* O(h<sup>p</sup>). We say that p is the approximation order.
*/
private int approximationOrder;
private int extrapolationCount = 0;
private double prevResult;
/**
* The estimate and tolerance may be used to deside wether to finalize
the
* iteration process (|estimate| < tolerance).
*/
/** Holds the current estimated error. */
private double estimate;
/** Holds the current reached tolerance. */
private double tolerance;
private double result[] = new double[DEF_MAXIMAL_EXTRAPOLATION_COUNT +
1];;
/**
* Set the maximal number of subsequent extrapolation steps.
*
* @param maximalExtrapolationCount
* maximal extrapolation steps
*/
public void setMaximalExtrapolationCount(int maximalExtrapolationCount)
{
result = new double[maximalExtrapolationCount + 1];
}
/**
* Extrapolate a sequence of values by means of Romberg's algorithm.
* Therefore a polynomial of degree maximalExtraploationCount
* is used. Calculates the current estimate and tolerance using the
* approximation order.
*
* @param value
* value to extrapolate
* @return extrapolated value
*/
public double extrapolate(double value)
{
if (extrapolationCount == 0) {
// first estimate
estimate = value;
tolerance = -1.0;
prevResult = 0;
}
int i, m, m1 = idx(extrapolationCount);
long k = (1 << approximationOrder);
int imin = Math.max(0, extrapolationCount - (result.length -
1));
result[m1] = value;
for (i = extrapolationCount - 1; i >= imin; i--) {
m = idx(i);
m1 = idx(i + 1);
result[m] = (k * result[m1] - result[m]) / (k - 1);
k <<= approximationOrder;
}
m1 = idx(i + 1);
estimate = result[m1] - prevResult;
tolerance = Math.abs(result[m1]) * relativeAccuracy +
absoluteAccuracy;
prevResult = result[m1];
extrapolationCount++;
return result[m1];
}
/**
* Ring buffer index
*/
private int idx(int i)
{
return (i % result.length);
}
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