ppvalu Function

PPVALU calculates the value at X of the JDERIV-th derivative of
the piecewise polynomial function F from its piecewise
polynomial representation.

The interval index I, appropriate for X, is found through a
call to INTERV.  The formula for the JDERIV-th derivative
of F is then evaluated by nested multiplication.

The J-th derivative of F is given by:

  (d**J) F(X) = 
    COEF(J+1,I) + H * (
    COEF(J+2,I) + H * (
    ...
    COEF(K-1,I) + H * (
    COEF(K,  I) / (K-J-1) ) / (K-J-2) ... ) / 2 ) / 1

with

  H = X - BREAK(I)

and

  I = max ( 1, max ( J, BREAK(J) <= X, 1 <= J <= L ) ).

16 February 2007

Carl DeBoor

Carl DeBoor,
A Practical Guide to Splines,
Springer, 2001,
ISBN: 0387953663,
LC: QA1.A647.v27.

Input, real ( kind = 8 ) BREAK(L+1), real COEF(*), integer L, the
piecewise polynomial representation of the function F to be evaluated.

Input, integer ( kind = 4 ) K, the order of the polynomial pieces that 
make up the function F.  The usual value for K is 4, signifying a 
piecewise cubic polynomial.

Input, real ( kind = 8 ) X, the point at which to evaluate F or
of its derivatives.

Input, integer ( kind = 4 ) JDERIV, the order of the derivative to be
evaluated.  If JDERIV is 0, then F itself is evaluated,
which is actually the most common case.  It is assumed
that JDERIV is zero or positive.

Output, real ( kind = 8 ) PPVALU, the value of the JDERIV-th
derivative of F at X.