Akima III

The Akima III algorithm is part of the Estimate Gridded Data option in the Non-Parametric menu.

Author
Hiroshi Akima

References
"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
For the Akima III algorithm, the late Hiroshi Akima’s ACM 760 procedure is implemented. In this algorithm, the interpolating function is a piecewise function composed of a set of bicubic (bivariate third-degree) polynomials, each applicable to a rectangle of the input grid in the x-y plane. Each polynomial is determined locally. The procedure has the accuracy of a bicubic polynomial (it interpolates accurately when all data points lie on a surface of a bicubic polynomial). The algorithm also uses three different extrapolants.

Revisions to Algorithm
Routines were added which compute exact analytic partial derivatives. An exact integration procedure was also added. There were written for both the interpolant and the extrapolation functions used by the algorithm.

Interpolation
The algorithm estimates the first three partial derivatives (with respect to x, y, and to x and y) or each rectangle in the grid. These are used to compute the coefficients of the interpolating polynomial. The interpolant produces smooth continuous first partial derivatives and continuous second partial derivatives.

Extrapolation
The algorithm directly provides for extrapolation. The first partial derivatives will be continuous but not smooth at the data bounds.

Estimated Partial Derivatives
Full analytic derivatives are available (once with respect to x, to y, twice with respect to x, to y, and with respect to both x and y). These are used for 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 data boundary where the transition between interpolation and extrapolation occurs.The higher order partials will be continuous but not smooth.

Estimated Volumes
The primary method is a direct analytic integration of the interpolating and/or extrapolation functions. this offers 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 algorithm is the fastest of the gridded procedures.

Algorithm Adjustments
This algorithm has no user adjustments.