intgr.F 10.9 KB
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      MODULE m_intgr

!**********************************************************************
! intgr[0-3]:
!    Integrators of a function y(jri) on a logarithmic mesh with jri 
!    mesh points. The output is either a scalar [z] or a field [z(jri)].
!    Either the first meshpoint [r0,r1] and the increment [h] or the 
!    array of meshpoints [rmsh(jri)] is supplied:
!
! intgz0 & intgz1 :
!     (in)definite integrators on linear mesh. tail=.true. will include
!     a tail correction assuming that the function is a simple
!     decaying exponential between the first mesh point and infinity.
!     y contains the nmz function values tabulated at a spacing of h.
!
!            integrator:      ---- input ----      output
!    intgr0:   definite       y r0        h jri  | z
!    intgr1: indefinite       y r1        h jri  | z(jri)
!    intgr2: indefinite       y rmsh(jri) h jri  | z(jri)
!    intgr3:   definite       y rmsh(jri) h jri  | z
!    intgz0:   definite       y tail      n nmz  ! z
!    intgz1: indefinite       y tail      n nmz  ! z(nmz)
!
!                                                            m. weinert
!**********************************************************************
#include"cpp_double.h"
     
      IMPLICIT NONE

        INTRINSIC exp,log
        INTERFACE 
          REAL FUNCTION CPP_BLAS_sdot(n,f1,is1,f2,is2)
           INTEGER, INTENT (IN) :: n,is1,is2
           REAL,    INTENT (IN) :: f1(n),f2(n)
          END FUNCTION 
        END INTERFACE


      INTEGER, PARAMETER, PRIVATE :: nr  = 7 , nr1 = 6
      REAL,    PARAMETER, PRIVATE :: h0 = 140. , zero = 0.0e0
!
! lagrangian integration coefficients (simpson 7 point rule: error  h**9)
!
      INTEGER, DIMENSION(7),   PARAMETER, PRIVATE :: ih =
     +                                       (/41,216,27,272,27,216,41/)
      REAL,    DIMENSION(7,5), PARAMETER, PRIVATE :: a = RESHAPE(
     +  (/19087.,65112.,-46461., 37504.,-20211., 6312.,-863.,
     +     -863.,25128., 46989.,-16256.,  7299.,-2088., 271.,
     +      271.,-2760., 30819., 37504., -6771., 1608.,-191.,
     +     -191., 1608., -6771., 37504., 30819.,-2760., 271.,
     +      271.,-2088.,  7299.,-16256., 46989.,25128.,-863./),(/7,5/))

      CONTAINS

!**********************************************************************
      SUBROUTINE intgr0(y,r0,h,jri,z)
!**********************************************************************
!     ..
!     .. Arguments ..
      INTEGER,    INTENT (IN)  :: jri
      REAL,       INTENT (IN)  :: h,r0
      REAL,       INTENT (IN)  :: y(jri)
      REAL,       INTENT (OUT) :: z
!     ..
!     .. Locals ..
      INTEGER m,n0,nsteps
      REAL dr, r(7)
      INTEGER :: i, j
      REAL    :: alpha, yr(7)

!
!--->    integral from 0 to r1 approximated by leading term in power
!--->    series expansion of y(r)
!
      z = zero
      IF (y(1)*y(2).GT.zero) THEN
         alpha = 1.0 + log(y(2)/y(1))/h
         IF (alpha.GT.zero) z = r0*y(1)/alpha
      ENDIF
!
!--->    determine steps and starting point for simpson
!
      nsteps = (jri-1)/nr1
      n0 = jri - nr1*nsteps
      dr = exp(h)
      r(1) = r0
      DO i = 2,7
         r(i) = dr*r(i-1)
      ENDDO 
!
!--->    lagrange integration for points 1<j<n0, error: h**9
!
      IF (n0.GT.1) THEN
         DO i = 1,7
            yr(i) = r(i)*y(i)
         ENDDO
         DO j = 1,n0 - 1
            z = z + h*CPP_BLAS_sdot(7,a(1,j),1,yr,1)/60480.
         ENDDO
      ENDIF
      r(1) = r(n0)
!
!--->    simpson integration
!
      DO m = 1,nsteps
         DO i = 2,nr
            r(i) = dr*r(i-1)
         ENDDO
         DO i = 1,nr
            yr(i) = h*ih(i)*r(i)/h0
         ENDDO
         z = z + CPP_BLAS_sdot(nr,yr,1,y(n0),1)
         n0 = n0 + nr1
         r(1) = r(nr)
      ENDDO

      RETURN
      END SUBROUTINE intgr0

!**********************************************************************
      SUBROUTINE intgr1(y,r1,h,jri,z)
!**********************************************************************
!     .. 
!     .. Arguments ..
      INTEGER, INTENT (IN) :: jri
      REAL,    INTENT (IN) :: h,r1
      REAL,    INTENT (IN) :: y(jri)
      REAL,    INTENT (OUT):: z(jri)
!     ..
!     .. Locals ..
      REAL dr, rr, r(7)
      INTEGER :: i, j
      REAL    :: alpha, yr(7)
!
!--->    integral from 0 to r1 approximated by leading term in power
!--->    series expansion of y(r)
!
      z(1) = zero
      IF (y(1)*y(2).GT.zero) THEN
         alpha = 1.0 + log(y(2)/y(1))/h
         IF (alpha.GT.zero) z(1) = r1*y(1)/alpha
      ENDIF
!
!--->    lagrange integration for points 1<j<nr, error: h**9
!
      dr = exp(h)
      rr = r1
      DO i = 1,7
         yr(i) = rr*y(i)
         r(i) = rr
         rr = dr*rr
      ENDDO
      DO j = 1,nr - 2
         z(j+1) = z(j) + h*CPP_BLAS_sdot(7,a(1,j),1,yr,1)/60480.
      ENDDO
!
!--->    simpson integration, j>nr-1
!
      DO i = 1,nr
         r(i) = h*ih(i)*r(i)/h0
      ENDDO
      DO j = nr,jri
         z(j) = z(j-nr1) + CPP_BLAS_sdot(nr,r,1,y(j-nr1),1)
         DO i = 1,7
            r(i) = dr*r(i)
         ENDDO
      ENDDO

      RETURN
      END SUBROUTINE intgr1

!**********************************************************************
      SUBROUTINE intgr2(y,rmsh,h,jri,z)
!**********************************************************************
!     ..
!     .. Arguments ..
      INTEGER, INTENT (IN) :: jri
      REAL,    INTENT (IN) :: h
      REAL,    INTENT (IN) ::  rmsh(jri),y(jri)
      REAL,    INTENT (OUT)::  z(jri)
!     ..
!     .. Locals ..
      REAL dr, r(7)
      INTEGER :: i, j
      REAL    :: alpha, yr(7)
!
!--->    integral from 0 to r1 approximated by leading term in power
!--->    series expansion of y(r)
!
      z(1) = zero
      IF (y(1)*y(2).GT.zero) THEN 
         alpha = 1.0 + log(y(2)/y(1))/h
         IF (alpha.GT.zero) z(1) = rmsh(1)*y(1)/alpha
      ENDIF
!
!--->    lagrange integration for points 1<j<nr, error: h**9
!
      dr = exp(h)
      DO i = 1,7
         r(i) = rmsh(i)
         yr(i) = rmsh(i)*y(i)
      ENDDO
      DO j = 1,nr - 2
         z(j+1) = z(j) + h*CPP_BLAS_sdot(7,a(1,j),1,yr,1)/60480.
      ENDDO
!
!--->    simpson integration, j>nr-1
!
      DO i = 1,nr
         r(i) = h*ih(i)*r(i)/h0
      ENDDO
      DO j = nr,jri
         z(j) = z(j-nr1) + CPP_BLAS_sdot(nr,r,1,y(j-nr1),1)
         DO i = 1,7
            r(i) = dr*r(i)
         ENDDO
      ENDDO

      RETURN
      END SUBROUTINE intgr2

!**********************************************************************
      SUBROUTINE intgr3(y,r,h,jri,z)
!**********************************************************************
!     ..
!     .. Arguments ..
      INTEGER, INTENT (IN) :: jri
      REAL,    INTENT (IN) :: h
      REAL,    INTENT (IN) :: r(jri)
      REAL,    INTENT (IN) :: y(jri)
      REAL,    INTENT (OUT):: z
!     ..
!     .. Locals ..
      INTEGER m,n0,nsteps
      REAL tiny, yr(nr), h1, z1, ih1(nr)
      INTEGER :: i, j
      REAL    :: alpha
!
!--->    integral from 0 to r1 approximated by leading term in power
!--->    series expansion of y(r)
!
!      DO i=1,jri
!        IF (abs(y(i)).LT.tiny) y(i) = tiny
!      ENDDO
!
      z = zero
      IF (y(1)*y(2).GT.zero) THEN
         alpha = 1.0 + log(y(2)/y(1))/h
         IF (alpha.GT.zero) z = r(1)*y(1)/alpha
      ENDIF
!
!--->    determine steps and starting point for simpson
!
      nsteps = (jri-1)/nr1
      n0 = jri - nr1*nsteps
!
!--->    lagrange integration for points 1<j<n0, error: h**9
!
      IF (n0.GT.1) THEN
         DO i = 1,7
            yr(i) = r(i)*y(i)
         ENDDO
         z1 = 0.
         DO j = 1,n0 - 1
            z1 = z1 + CPP_BLAS_sdot(7,a(1,j),1,yr,1)
         ENDDO
         z = z + z1 * h / 60480.
      END IF
!
!--->    simpson integration
!
      h1 = h / h0
      DO i = 1,nr
        ih1(i) = h1 * ih(i)
      ENDDO
      DO m = 1,nsteps
         DO i = 1,nr
            yr(i) = ih1(i)*r(i+n0-1)
         ENDDO
         z = z + CPP_BLAS_sdot(nr,yr,1,y(n0),1)
         n0 = n0 + nr1
      ENDDO

      RETURN
      END SUBROUTINE intgr3

!**********************************************************************
      SUBROUTINE intgz0(y,h,nmz,z,tail)
!**********************************************************************
!     ..
!     .. Arguments ..
      INTEGER, INTENT (IN)  :: nmz
      REAL,    INTENT (IN)  :: h
      REAL,    INTENT (IN)  :: y(nmz)
      LOGICAL, INTENT (IN)  :: tail
      REAL,    INTENT (OUT) :: z
!     ..
!     .. Locals ..
      REAL yl,ys
      INTEGER m,n0,nsteps
      INTEGER :: i, j
      REAL    :: alpha, yr(7)
!
!--->    integral from minus infinity to the first mesh point assuming
!--->    exponential decay in this region. this contribution is limited
!--->    to alpha>0.1 which corresponds to square of wavefunctions
!--->    of energy > 0.00125 a.u. below the vacuum level
!
      z = zero
      IF (tail) THEN
         IF (y(1)*y(2).GT.zero) THEN
           alpha = log(y(2)/y(1))/h
           IF (alpha.GT.0.1) z = y(1)/alpha
         ENDIF
      ENDIF
!
!--->    determine steps and starting point for simpson
!
      nsteps = (nmz-1)/nr1
      n0 = nmz - nr1*nsteps
!
!--->    lagrange integration for points 1<j<n0, error: h**9
!
      yl = zero
      IF (n0.GT.1) THEN
         DO j = 1, n0 - 1
            yl = yl + CPP_BLAS_sdot(7,a(1,j),1,y,1)
         ENDDO
         yl = h*yl/60480.
       END IF
!
!--->    simpson integration
!
      ys = zero
      n0 = n0 - 1
      DO m = 1,nsteps
         DO i = 1,nr
            ys = ys + ih(i)*y(n0+i)
         ENDDO
         n0 = n0 + nr1
      ENDDO
      ys = h*ys/h0
      z = z + yl + ys

      RETURN
      END SUBROUTINE intgz0

!**********************************************************************
      SUBROUTINE intgz1(y,h,nmz,z,tail)
!**********************************************************************
!     .. 
!     .. Arguments ..
      INTEGER, INTENT (IN)  :: nmz
      LOGICAL, INTENT (IN)  :: tail
      REAL,    INTENT (IN)  :: h
      REAL,    INTENT (IN)  :: y(nmz)
      REAL,    INTENT (OUT) :: z(nmz)
!     ..
!     .. Locals ..
      REAL yl,ys
      REAL, PARAMETER :: eps = 1.e-38
      INTEGER :: i, j
      REAL    :: alpha, yr(7)
!
!--->    integral from minus infinity to the first mesh point assuming
!--->    exponential decay in this region. this contribution is limited
!--->    to alpha>0.1 which corresponds to square of wavefunctions
!--->    of energy > 0.00125 a.u. below the vacuum level
!
      z(1) = zero
      IF (tail) THEN
         IF (abs(y(1)).GT.eps) THEN
            IF (y(1)*y(2).GT.zero) THEN
              alpha = log(y(2)/y(1))/h
              IF (alpha.GT.0.1) z(1) = y(1)/alpha
            ENDIF
         ENDIF
      ENDIF
!
!--->    lagrange integration for points 1<j<n0, error: h**9
!
      DO j = 1,nr - 2
         yl = 0
         yl = yl + CPP_BLAS_sdot(7,a(1,j),1,y,1)
         z(j+1) = z(j) + h*yl/60480.
       ENDDO
!
!--->    simpson integration
!
      DO j = nr,nmz
         ys = zero
         DO i = 1,nr
            ys = ys + ih(i)*y(j-nr+i)
         ENDDO
         z(j) = z(j-nr1) + h*ys/h0
      ENDDO

      RETURN
      END SUBROUTINE intgz1

      END MODULE m_intgr