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Compute Cubic Spline Coefficients

Syntax	cubespl(x, d, b1, bn)
See Also	cubeval , interp , lagrange

Description
Returns a matrix, with one less row than x and three columns, that defines a cubic spline through the data pairs (x(i), d(i)).

Let a, b, and c, be the first, second, and third columns of the return value. If z is between x(i) and x(i + 1), the cubic spline's value at z is equal to
               3              2
     a (z - x )  +  b (z - x )  +  c (z - x )  +  d
      i      i       i      i       i      i       i
The column vector x must be either real or double-precision and x(i) < x(i+1) for all i. The vector d must have the same type and dimension as x. The scalars b1 and bn must have the same type as x. The second derivative of the spline at x(1) 2 b1 and at x(n) is 2 bn, where n is the row dimension of x.

The return value has the same type as x.

Example
The following program fits a cubic spline though the function
      3            3             2
     z  = 1 (z - 1)  +  3 (z - 1)  +  3 (z - 1) + d
                                                   1
                   3             2
        = 1 (z - 2)  +  6 (z - 2)  + 12 (z - 2) + d
                                                   2
Note that the second derivative of this function is 6 z. It follows from the equalities above that, the values of a, b, and c, are 1, 3, 3 in the first interval and 1, 6, 12 in the second interval.

clear
#
x   = {1., 2., 3.}
d   = x^3
b1  = 6 * x(1) / 2
bn  = 6 * x(3) / 2
cubespl(x, d, b1, bn)