Integral of Lagrange polynomial

Consider the Fourier transform from begr to endr of a function defined as a Lagrange polynomial.

The function through these points is

The integral of f is

The Integrals are of the form

Define and let so that the integral becomes

Define

So that

The integral is

The code to integrate from the first point to the last in Fortran is in for\AINT.FR

FUNCTION TAINT(XDAT,FDAT,N)

IMPLICIT REAL*8 (A-H,O-Z)

DIMENSION XDAT(*)

COMPLEX*16 FDAT(*)

C EVALUATES THE INTEGRAL FROM XDAT(1) TO XDAT(N)
C LAGRANGE POLYNOMIAL THROUGH THE NEAREST 4 POINTS

BEGR=XDAT(1)

ENDR=XDAT(3)

IF(N.LE.4)THEN

IF(N.LT.4)THEN

PRINT*,' AT LEAST 4 POINTS ARE NEEDED FOR'
PRINT*,' INTERPOLATION'

READ(*,*)ITEST

STOP

ELSE

ENDR=XDAT(4)

ENDIF

ENDIF

TAINT=AINT2(XDAT(1),XDAT(2),XDAT(3),

A XDAT(4),BEGR,ENDR

2 ,FDAT(1),FDAT(2),FDAT(3),FDAT(4))

IF(N.EQ.4)RETURN

DO I=3,N-3

BEGR=XDAT(I)

ENDR=XDAT(I+1)

TAINT=TAINT+

2 AINT2(XDAT(I-1),XDAT(I),XDAT(I+1),XDAT(I+2),BEGR,ENDR,

3 FDAT(I-1),FDAT(I),FDAT(I+1),FDAT(I+2))

ENDDO

BEGR=XDAT(N-2)

ENDR=XDAT(N)

TAINT=TAINT+

2 AINT2(XDAT(N-3),XDAT(N-2),XDAT(N-1),XDAT(N),BEGR,ENDR,

3 FDAT(N-3),FDAT(N-2),FDAT(N-1),FDAT(N))

RETURN

END

FUNCTION AINT2(SA,SB,SC,SD,BEGR,ENDR,FA,FB,FC,FD)

IMPLICIT REAL*8 (A-H,O-Z)

COMPLEX*16 FA,FB,FC,FD

H=ENDR-BEGR

XMID=(BEGR+ENDR)/2

AINT2=FA*AIHAT2(SB,SC,SD,H,XMID)/((SA-SB)*(SA-SC)*(SA-SD))

2 +FB*AIHAT2(SA,SC,SD,H,XMID)/((SB-SA)*(SB-SC)*(SB-SD))

3 +FC*AIHAT2(SA,SB,SD,H,XMID)/((SC-SA)*(SC-SB)*(SC-SD))

4 +FD*AIHAT2(SA,SB,SC,H,XMID)/((SD-SA)*(SD-SB)*(SD-SC))

RETURN

END

C

FUNCTION AIHAT2(SB,SC,SD,H,XMID)

IMPLICIT REAL*8 (A-H,O-Z)

CAPB=(XMID-SB)

CAPC=(XMID-SC)

CAPD=(XMID-SD)

AIHAT2=H*(CAPB*CAPC*CAPD+H*H*

2(CAPB+CAPC+CAPD)/12)

RETURN

END

Any function can be found using findfun and integrated by the above to yield

In the above let so that

The function G is known on the grid of points found by findfun. In a general integral of the form

make the variable change using a positive definite g

Then