Bicubic+Akima

The Bicubic+Akima algorithm is part of the Estimate Gridded Data option in the Non-Parametric menu. This algorithm combines a bicubic B-Spline with the extrapolation offered in the Akima III procedure.

Author

Carl de Boor/IMSL
Hiroshi Akima
AISN Software

References

Carl de Boor, "A Practical Guide to Splines", Springer-Velag, 1978, p332-346, also SPLI2D p347-349.
IMSL Library, BS2IN
Hiroshi Akima, "Algorithm 760: Rectangular-Grid-Data Surface Fitting that has the Accuracy of a Bicubic Polynomial", ACM Transactions on Mathematical Software, Vol. 22, No. 3, Sept. 1996, p357-361.

Description

This option offers a Bicubic Spline interpolant combined with the Akima III extrapolant. Smoothness of the function and continuity of the first partial derivatives are preserved across the interpolation-extrapolation boundary. Smoothness of the first partial derivatives is not conserved at the bound.

Interpolation

Except at the boundary, both the bicubic (order 3/3) spline interpolant and the Akima III extrapolant offer a smooth surface, smooth first partial derivatives, and continuous second partial derivatives. This option exactly interpolates the data. The x and y knots will equal the number of x and y mesh values.

Extrapolation

The algorithm offers the Akima III extrapolant.

Estimated Partial Derivatives

In the data region, the B-spline algorithm computes exact partial derivatives. In an extrapolated region, analytic partial derivatives of the Akima extrapolants are used. These are used when graphing the partial derivative surfaces and for partial derivatives computed in the Evaluation procedure. Note that the first partial derivatives will be continuous but not smooth at the transition between the two algorithms.

Estimated Volumes

The B-spline algorithm computes exact integrals within the bounds of the data. An analytic integration is also used to compute exact integrals for any portion of the integration outside the data region. As such the integrations will consist of full precision. A repeat evaluation with the same limits will use a numeric double integration procedure. A very fast double Gaussian quadrature procedure is first attempted to 1E-5 precision. If this is unsuccessful, a double adaptive quadrature procedure is then used. This second evaluation is offered as a verification of accuracy.

Considerations

This is the only option in TableCurve 3D where the traditional B-spline is augmented for extrapolation.

Algorithm Adjustments

This algorithm has no user adjustments.