PPDER

This function evaluates the derivative of a piecewise polynomial.

Function Return Value

PPDER — Value of the IDERIV-th derivative of the piecewise polynomial at X. (Output)

Required Arguments

X — Point at which the polynomial is to be evaluated. (Input)

BREAK — Array of length NINTV + 1 containing the breakpoints of the piecewise polynomial representation. (Input)
BREAK must be strictly increasing.

PPCOEF — Array of size KORDER * NINTV containing the local coefficients of the piecewise polynomial pieces. (Input)
PPCOEF is treated internally as a matrix of size KORDER by NINTV.

Optional Arguments

IDERIV — Order of the derivative to be evaluated. (Input)
In particular, IDERIV = 0 returns the value of the polynomial.
Default: IDERIV = 1.

KORDER — Order of the polynomial. (Input)
Default: KORDER = size (PPCOEF,1).

NINTV — Number of polynomial pieces. (Input)
Default: NINTV = size (PPCOEF,2).

FORTRAN 90 Interface

Generic: PPDER (X, BREAK, PPCOEF [, …])

Specific: The specific interface names are S_PPDER and D_PPDER.

FORTRAN 77 Interface

Single: PPDER (IDERIV, X, KORDER, NINTV, BREAK, PPCOEF)

Double: The double precision function name is DPPDER.

Description

The routine PPDER evaluates the derivative of a piecewise polynomial function f at a given point. This routine is based on the subroutine PPVALU by de Boor (1978, page 89). In particular, if the breakpoint sequence is stored in ξ (a vector of length N = NINTV + 1), and if the coefficients of the piecewise polynomial representation are stored in c, then the value of the j-th derivative of f at x in[ξi, ξi + 1) is

 

when j = 0 to k − 1 and zero otherwise. Notice that this representation forces the function to be right continuous. If x is less than ξ1, then i is set to 1 in the above formula; if x is greater than or equal to ξN , then i is set to N − 1. This has the effect of extending the piecewise polynomial representation to the real axis by extrapolation of the first and last pieces.

Example

In this example, a spline interpolant to a function f is computed using the IMSL routine BSINT. This routine represents the interpolant as a linear combination of B-splines. This representation is then converted to piecewise polynomial representation by calling the IMSL routine BSCPP. The piecewise polynomial’s zero-th and first derivative are evaluated using PPDER. These values are compared to the corresponding values of f.

 

USE IMSL_LIBRARIES

 

IMPLICIT NONE

INTEGER KORDER, NCOEF, NDATA, NKNOT

PARAMETER (KORDER=4, NCOEF=20, NDATA=20, NKNOT=NDATA+KORDER)

!

INTEGER I, NOUT, NPPCF

REAL BREAK(NCOEF), BSCOEF(NCOEF), DF, DS, EXP, F,&

FDATA(NDATA), FLOAT, PPCOEF(KORDER,NCOEF), S,&

X, XDATA(NDATA), XKNOT(NKNOT)

INTRINSIC EXP, FLOAT

!

F(X) = X*EXP(X)

DF(X) = (X+1.)*EXP(X)

! Set up interpolation points

DO 10 I=1, NDATA

XDATA(I) = FLOAT(I-1)/FLOAT(NDATA-1)

FDATA(I) = F(XDATA(I))

10 CONTINUE

! Generate knot sequence

CALL BSNAK (NDATA, XDATA, KORDER, XKNOT)

! Compute the B-spline interpolant

CALL BSINT (NCOEF, XDATA, FDATA, KORDER, XKNOT, BSCOEF)

! Convert to piecewise polynomial

CALL BSCPP (KORDER, XKNOT, NCOEF, BSCOEF, NPPCF, BREAK, PPCOEF)

! Get output unit number

CALL UMACH (2, NOUT)

! Write heading

WRITE (NOUT,99999)

! Print the interpolant on a uniform

! grid

DO 20 I=1, NDATA

X = FLOAT(I-1)/FLOAT(NDATA-1)

! Compute value of the piecewise

! polynomial

S = PPDER(X,BREAK,PPCOEF, IDERIV=0, NINTV=NPPCF)

! Compute derivative of the piecewise

! polynomial

DS = PPDER(X,BREAK,PPCOEF, IDERIV=1, NINTV=NPPCF)

WRITE (NOUT,'(2F12.3,F12.6,F12.3,F12.6)') X, S, F(X) - S, DS,&

DF(X), DS

20 CONTINUE

99999 FORMAT (11X, 'X', 8X, 'S(X)', 7X, 'Error', 7X, 'S''(X)', 7X,&

'Error')

END

Output

 

X S(X) Error S’(X) Error

0.000 0.000 0.000000 1.000 -0.000112

0.053 0.055 0.000000 1.109 0.000030

0.105 0.117 0.000000 1.228 -0.000008

0.158 0.185 0.000000 1.356 0.000002

0.211 0.260 0.000000 1.494 0.000000

0.263 0.342 0.000000 1.643 0.000000

0.316 0.433 0.000000 1.804 -0.000001

0.368 0.533 0.000000 1.978 0.000002

0.421 0.642 0.000000 2.165 0.000001

0.474 0.761 0.000000 2.367 0.000000

0.526 0.891 0.000000 2.584 -0.000001

0.579 1.033 0.000000 2.817 0.000001

0.632 1.188 0.000000 3.068 0.000001

0.684 1.356 0.000000 3.338 0.000001

0.737 1.540 0.000000 3.629 0.000001

0.789 1.739 0.000000 3.941 0.000000

0.842 1.955 0.000000 4.276 -0.000006

0.895 2.189 0.000000 4.636 0.000024

0.947 2.443 0.000000 5.022 -0.000090

1.000 2.718 0.000000 5.436 0.000341