This function evaluates the integral of a piecewise polynomial.

Function Return Value

PPITG — Value of the integral from A to B of the piecewise polynomial.   (Output)

Required Arguments

A — Lower limit of integration.   (Input)

B — Upper limit of integration.   (Input)

BREAK — Array of length NINTV + 1 containing the breakpoints for the piecewise polynomial.   (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

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

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

FORTRAN 90 Interface

Generic:          PP1TG (A, B, BREAK, PPCOEF [,…])

Specific:         The specific interface names are S_PP1TG and D_PP1TG.

FORTRAN 77 Interface

Single:            PP1TG (A, B, KORDER, NINTV, BREAK, PPCOEF)

Double:          The double precision function name is DPP1TG.

Description

The routine PPITG evaluates the integral of a piecewise polynomial over an interval.

Example

In this example, we compute a quadratic spline interpolant to the function x2 using the IMSL routine BSINT. We then evaluate the integral of the spline interpolant over the intervals [0, 1/2] and [0, 2]. The interpolant reproduces x2, and hence, the values of the integrals are 1/24 and 8/3, respectively.

 

      USE IMSL_LIBRARIES

 

      IMPLICIT   NONE

      INTEGER    KORDER, NDATA, NKNOT

      PARAMETER  (KORDER=3, NDATA=10, NKNOT=NDATA+KORDER)

!

      INTEGER    I, NOUT, NPPCF

      REAL       A, B, BREAK(NDATA), BSCOEF(NDATA), EXACT, F,&

                 FDATA(NDATA), FI, FLOAT, PPCOEF(KORDER,NDATA),&

                 VALUE, X, XDATA(NDATA), XKNOT(NKNOT)

      INTRINSIC  FLOAT

!

      F(X)  = X*X

      FI(X) = X*X*X/3.0

!                                  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)

!                                  Interpolate

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

!                                  Convert to piecewise polynomial

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

!                                  Compute the integral of F over

!                                  [0.0,0.5]

      A     = 0.0

      B     = 0.5

      VALUE = PPITG(A,B,BREAK,PPCOEF,NINTV=NPPCF)

      EXACT = FI(B) - FI(A)

!                                  Get output unit number

      CALL UMACH (2, NOUT)

!                                  Print the result

      WRITE (NOUT,99999) A, B, VALUE, EXACT, EXACT - VALUE

!                                  Compute the integral of F over

!                                  [0.0,2.0]

      A     = 0.0

      B     = 2.0

      VALUE = PPITG(A,B,BREAK,PPCOEF,NINTV=NPPCF)

      EXACT = FI(B) - FI(A)

!                                  Print the result

      WRITE (NOUT,99999) A, B, VALUE, EXACT, EXACT - VALUE

99999 FORMAT (' On the closed interval (', F3.1, ',', F3.1,&

             ') we have :', /, 1X, 'Computed Integral = ', F10.5, /,&

             1X, 'Exact Integral    = ', F10.5, /, 1X, 'Error         '&

             , '    = ', F10.6, /, /)

!

      END

Output

 

On the closed interval (0.0,0.5) we have :
Computed Integral =    0.04167
Exact Integral    =    0.04167
Error             =   0.000000

On the closed interval (0.0,2.0) we have :
Computed Integral =    2.66667
Exact Integral    =    2.66667
Error             =   0.000001

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