RANDOM  INTERPOLATION
Higher Dimensionality
(A Sample Case, 4D)

The sample function has the analytic form:

f(x1, x2, x3, x4 ) = x12 + x22 + 2 x32 + x42- 5 x1 - 5 x2 - 21 x3 + 7 x4 + 50.

(This is the so-called Rosen-Suzuki function.)

where   -10. < x1 < +10. and   -10. < x2 < +10.
            -10. < x3 < +10. and   -10. < x4 < +10.

For this sample case, we have used 36 random locations together with the associated function values to construct the interpolant. Also, delta width values of X1, X2, X3 and X4 are chosen to be 1.

INPUT AREA

Input Location Number (M'):

Delta width for x1 (1) :

Delta width for x2 (2) :

Delta width for x3 (3) :

Delta width for x4 (4) :

Location NumberInput x1Input x2Input x3Input x4Input Function
f(x1, x2, x3, x4)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36

INTERPOLATION RESULT

Provide the desired location ( x1, x2, x3, x4) where the interpolated function value fINT( x1, x2, x3, x4) is calculated.

x1:     ( -10. < x1 < +10. )

x2:    ( -10. < x2 < +10. )

x3:     ( -10. < x3 < +10. )

x4:    ( -10. < x4 < +10. )

Click "COMPUTE" button to find fINT(x1, x2, x3, x4) and f(x1, x2, x3, x4):

         fINT(x1, x2, x3, x4):

     The true value of  f(x1, x2, x3, x4) is :

Note that the more input locations are given, the better the interpolation results will become. As a general, practical rule, delta width value is chosen between 0.5 to 1.5 for a domain interval value 20. The rationale behind this is because of the fact that the delta width essentially controls the weighting factor in the interpolation computation. A slightly larger delta value will give more smoothness of the interpolated function. In addition, for each dimension added, the input locations need to be increased by 6 to 12. For example, for one-dimension interpolation, the input locations can be 6 or 8. For two-dimension, the value can be 12 to 18. There is no definite rule to follow and it depends on the interpolated function at hand. If the function is fairly smooth, then less number of input locations can be adequate. Conversely, if the function is not smooth, more input locations are definitely required.

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