This function evaluates the integral of a cubic spline.

Function Return Value

CSITG — Value of the integral of the spline from A to B.   (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 cubic representation.   (Input)
BREAK must be strictly increasing.

CSCOEF — Matrix of size 4 by NINTV + 1 containing the local coefficients of the cubic pieces.   (Input)

Optional Arguments

NINTV — Number of polynomial pieces.   (Input)
Default: NINTV = size (BREAK,1) – 1.

FORTRAN 90 Interface

Generic:          CSITG (A, B, BREAK, CSCOEF [,…])

Specific:         The specific interface names are S_CSITG and D_CSITG.

FORTRAN 77 Interface

Single:            CSITG(A, B, NINTV, BREAK, CSCOEF)

Double:          The double precision function name is DCSITG.

Description

The function CSITG evaluates the integral of a cubic spline over an interval. It is a special case of the routine PPITG, which evaluates the integral of a piecewise polynomial. (A cubic spline is a piecewise polynomial of order 4.)

Example

This example computes a cubic spline interpolant to the function x2 using CSINT and evaluates its integral over the intervals [0., .5] and [0., 2.]. Since CSINT uses the not-a knot condition, the interpolant reproduces x2, hence the integral values are 1/24 and 8/3, respectively.

 

      USE CSITG_INT

      USE UMACH_INT

      USE CSINT_INT

 

      IMPLICIT   NONE

      INTEGER    NDATA

      PARAMETER  (NDATA=10)

!

      INTEGER    I, NINTV, NOUT

      REAL       A, B, BREAK(NDATA), CSCOEF(4,NDATA), ERROR,&

                 EXACT, F, FDATA(NDATA), FI, FLOAT, VALUE, X,&

                 XDATA(NDATA)

      INTRINSIC  FLOAT

!                                  Define function and integral

      F(X)  = X*X

      FI(X) = X*X*X/3.0

!                                  Set up a grid

      DO 10  I=1, NDATA

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

         FDATA(I) = F(XDATA(I))

   10 CONTINUE

!                                  Compute cubic spline interpolant

      CALL CSINT (XDATA, FDATA, BREAK, CSCOEF)

!                                  Compute the integral of F over

!                                  [0.0,0.5]

      A     = 0.0

      B     = 0.5

      NINTV = NDATA - 1

      VALUE = CSITG(A,B,BREAK,CSCOEF)

      EXACT = FI(B) - FI(A)

      ERROR = EXACT - VALUE

!                                  Get output unit number

      CALL UMACH (2, NOUT)

!                                  Print the result

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

!                                  Compute the integral of F over

!                                  [0.0,2.0]

      A     = 0.0

      B     = 2.0

      VALUE = CSITG(A,B,BREAK,CSCOEF)

      EXACT = FI(B) - FI(A)

      ERROR = EXACT - VALUE

!                                  Print the result

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

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 =    2.66666
Exact Integral    =    2.66667
Error             =   0.000006

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