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From: zosi on 13 Apr 2010 22:39
After having sailed into Mathgroup and Help, we still face the
following problem.

Suppose we have a set of 3D points (x_i, y_j, f[x_i,y_j])
with i=1...N, j=1...M and we need to

a) interpolate the surface by **using**
<<Splines` and SplineFit[...,cubic]
a.1) plot the interpolating (spline) surface
a.2) extract the coefficients of the bicubic polynomials
over each sub-domain
b) calculate the integral of the interpolating spline.

In the following, we show the steps and the difficulties encountered

(* Begin: Suppose we start with the trivial function *)

f[x_, y_] := Sin[x + y]^2

plotfxy =
Plot3D[f[x, y] , { x, 0 , \[Pi] }, {y, 0, 2 \[Pi] }, PlotPoints -> 50 ]

(* For subsequent comparisons we calculate immediately the integral
with 20 digits *)

numintegral =
N[Integrate[f[x, y], {x, 0 , \[Pi]} , {y, 0 , 2 \[Pi]} ], 20]
(* Output 9.8696044010893586188 *)

(* Here we create the grid, very coarse, only 25 points (5x5)
for 16 sub-domains *)

grid3D =
Flatten[Table[{x, y, f[x, y]}, {x, 0, \[Pi], \[Pi]/4}, {y, 0, 2 \[Pi],
\[Pi]/2}], 1]

(* The following use of << Splines` is mandatory *)

<< Splines`

bicubic = SplineFit[grid3D, Cubic];
(* Output: SplineFunction[Cubic, {0., 24.},<>] *)

First question (a.1): How can we plot the spline interpolating surface ?
Several trials using ParametricPlot3D[bicubic[s],....] have failed.

Second question (a.2): How do we extract the coefficients of the bicubic
polynomials
over each of the 16 sub-domains ?
We have, first
coefpoly = InputForm[bicubic] // N;

Even when we try to extract the first List (containing three Lists),
probably relevant to the bicubic polynomial over the
first (0,\Pi/4) x (0,\Pi/2) sub-domain, for example:

coefpoly[[1, 4, 1]]
(* Output {{0., 0.00512628, 0., -0.00512628},
{0., 1.51953, 0., 0.0512628},
{0., 1.72638, 0., -0.726383}}
*)

we are not able to construct the relevant
bicubic(x,y) polynomial and, obviously, the remaining polynomials.

Final question: Is there a single command to evaluate the integral
of "bicubic" over the domain [0,\Pi]x[0,2 \Pi] ?

Many thanks for your patience and help.

Gianfranco Zosi
Dip. Fisica Generale
Universita di Torino
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