Interpolation.CubicSpline2ndDifferential Method 

Calculates the second derivatives of the cubic spline interpolation polynomial at the given functions tabulation points when the first derivative at the boundary (equivalently the end points) is given.

Parameters

tabulationPointsInX

An array of doubles which are the values at which the function is tabulated, i.e. functionValuesAtTabulationPoints[i] = f(tabulationPointsInX[i]).

functionValuesAtTabulationPoints

An array of doubles which are the values of the function evaluated at the interpolation points, i.e. functionValuesAtTabulationPoints[i] = f(tabulationPointsInX[i]).

derivativeInterpolationAt0

The first derivative of the interpolation function at the point tabulationPointsInX[0].

derivativeInterpolationAtn_1

The first derivative of the interpolation function at the point tabulatedPointInX[n-1].

Return Value

An array where the k-th term corresponds to the value of the second derivative of the cubic spline interpolation polynomial of the k-th tabulation point. Note that the length of the returned array will be equal to the number of points at which the function is tabulated.

Remarks

The values of the second derivatives can be used within CubicSplinePointwisePreEvaluation in order to evaluate the interpolation function at a given point.

Details

Given arrays tabulationPointsInX[0..n-1] and tabulationPointsInY[0..n-1] containing a tabulated function, i.e. tabulationPointsInY[i] = f(tabulationPointsInY[i]), with tabulationPointsInX[0] < tabulationPointsInX[1] < ... < tabulationPointsInX[n-1], and given values derivativeInterpolationAt0 and derivativeInterpolationAtn_1 for the first derivative of the interpolating function at the points tabulationPointsInX[0] and tabulationPointsInX[n-1], respectively. This method returns an array of length n, that contains the second derivatives of the interpolation function at the tabulation points tabulationPointsInX[i]. If derivativeInterpolationAt0 and/or derivativeInterpolationAtn_1 are equal to 10³⁰ or larger, then the method sets the second derivative at the boundary to be zero.

Remarks of this methods application

• If the three parameters arrays used within this method have different lengths then the length of the shortest array will be used as reference and only the corresponding terms of the other arrays will be used.
• All of the three arrays must have a length equal to or greater than 2.

Exceptions

See Also

Interpolation Class | WebCab.Libraries.Math.Interpolation Namespace