scipy.interpolate.krogh_interpolate

scipy.interpolate.krogh_interpolate(xi, yi, x, der=0)

Convenience function for polynomial interpolation.

Constructs a polynomial that passes through a given set of points, optionally with specified derivatives at those points. Evaluates the polynomial or some of its derivatives. For reasons of numerical stability, this function does not compute the coefficients of the polynomial, although they can be obtained by evaluating all the derivatives.

Be aware that the algorithms implemented here are not necessarily the most numerically stable known. Moreover, even in a world of exact computation, unless the x coordinates are chosen very carefully - Chebyshev zeros (e.g. cos(i*pi/n)) are a good choice - polynomial interpolation itself is a very ill-conditioned process due to the Runge phenomenon. In general, even with well-chosen x values, degrees higher than about thirty cause problems with numerical instability in this code.

Based on Krogh 1970, “Efficient Algorithms for Polynomial Interpolation and Numerical Differentiation”

The polynomial passes through all the pairs (xi,yi). One may additionally specify a number of derivatives at each point xi; this is done by repeating the value xi and specifying the derivatives as successive yi values.

Parameters :

xi : array_like, length N

known x-coordinates

yi : array_like, N by R

known y-coordinates, interpreted as vectors of length R, or scalars if R=1

x : scalar or array_like of length N

Point or points at which to evaluate the derivatives

der : integer or list

How many derivatives to extract; None for all potentially nonzero derivatives (that is a number equal to the number of points), or a list of derivatives to extract. This number includes the function value as 0th derivative.

Returns :

d : ndarray

If the interpolator’s values are R-dimensional then the returned array will be the number of derivatives by N by R. If x is a scalar, the middle dimension will be dropped; if the yi are scalars then the last dimension will be dropped.

Notes

Construction of the interpolating polynomial is a relatively expensive process. If you want to evaluate it repeatedly consider using the class KroghInterpolator (which is what this function uses).