/
MakeGravGridDH.f95
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MakeGravGridDH.f95
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subroutine MakeGravGridDH(cilm, lmax, gm, r0, a, f, rad_grid, theta_grid, &
phi_grid, total_grid, n, sampling, lmax_calc, omega, &
normal_gravity, pot_grid, extend, exitstatus)
!------------------------------------------------------------------------------
!
! Given the spherical harmonic coefficients CILM of the gravitational
! potential, this subroutine will compute 2D Driscol and Healy sampled grids
! of the three vector components of the gravity vector (gravitational force +
! centrifugal force), and the magnitude of the gravity vector in SI units.
! The output grids are calculated on a flattened ellipsoid with semi-major
! axis A and flattening F. In order to calculate the entire gravity vector,
! it is necessary that the degree-0 term be set equal to 1. Radial gravity is
! positive when directed UPWARDS.
!
! If the optional parameter OMEGA is specified, the gravity vector will be
! calculated in the reference frame of a rotating body and will include the
! contribution of the centrifugal force. If the parameter NORMAL_GRAVITY is
! set to 1, the normal gravity predicted for a flattened ellipsoid with A, F,
! and OMEGA will be removed from the magnitude of the total gravity, yielding
! the gravity disturbance.
!
! The gravitational potential is defined as
!
! V = GM/r Sum_{l=0}^LMAX (R0/r)^l Sum_{m=-l}^l C_{lm} Y_{lm},
!
! and the gravitational acceleration is
!
! B = Grad V.
!
! The output grids contain N samples in latitude and longitude by default,
! but if the optional parameter SAMPLING is set to 2, the grids will contain
! N samples in latitude and 2N samples in longitude. When SAMPLING = 1, the
! output grids contain N samples in latitude from 90 to -90 + interval and N
! samples in longitude from 0 to 360-2*interval, where N=2*(LMAX+1) and
! interval=180/N. When SAMPLING = 2, the grids are equally spaced in degrees
! latitude and longitude with dimension (N x 2N). If the optional parameter
! EXTEND is set to 1, the output grids will contain an extra column
! corresponding to 360 E and an extra row corresponding to 90 S, which
! increases each of the dimensions of the grid by one.
!
! This routine assumes that the spherical harmonic coefficients are
! 4-pi normalized and exclude the Condon-Shortley phase factor.
!
! Calling Parameters
!
! IN
! cilm Gravitational spherical harmonic coefficients.
! lmax The maximum spherical harmonic degree of the function,
! used to determine the number of samples N.
! GM Product of the gravitatonal constant and the planet's
! mass.
! r0 Reference radius of the potential coefficients.
! a The semimajor axis of the flattened ellipsoid.
! f Flattening of the planet.
!
! IN, OPTIONAL
! sampling (1) The output grids are N by N (default).
! (2) The output grids are N by 2N.
! lmax_calc The maximum spherical harmonic degree to evaluate
! the coefficients up to.
! omega Angular rotation rate of the planet.
! normal_gravity
! If 1, the magnitude of the normal gravity on the
! ellipsoid will be removed from the magnitude of the
! total gravity vector. This is the "gravity
! disturbance."
! extend If 1, return a grid that contains an additional column
! and row corresponding to 360 E longitude and 90 S
! latitude, respectively.
!
! OUT
! rad_grid Gridded expansion of the radial component of the
! gravity field.
! theta_grid Gridded expansaion of the theta component of the
! gravity field.
! phi_grid Gridded expansaion of the phi component of the
! gravity field.
! total_grid Gridded expansaion of the the magnitude of the
! gravity field.
! N Number of samples in latitude. Number of samples in
! longitude is N when sampling is 1 (default), and is 2N
! when sampling is 2.
!
! OUT, OPTIONAL
! pot_grid Gravity potential on the ellipsoid in SI units.
! exitstatus If present, instead of executing a STOP when an error
! is encountered, the variable exitstatus will be
! returned describing the error.
! 0 = No errors;
! 1 = Improper dimensions of input array;
! 2 = Improper bounds for input variable;
! 3 = Error allocating memory;
! 4 = File IO error.
!
! Notes:
! 1. If lmax is greater than the the maximum spherical harmonic
! degree of the input coefficients, then the coefficients will be
! zero padded.
! 2. Latitude is geocentric latitude.
!
! Copyright (c) 2005-2019, SHTOOLS
! All rights reserved.
!
!------------------------------------------------------------------------------
use FFTW3
use SHTOOLS, only: normalgravity
use ftypes
use, intrinsic :: iso_c_binding
implicit none
real(dp), intent(in) :: cilm(:,:,:), gm, r0, a, f
real(dp), intent(out) :: rad_grid(:,:), theta_grid(:,:), phi_grid(:,:), &
total_grid(:,:)
real(dp), intent(in), optional :: omega
real(dp), intent(out), optional :: pot_grid(:,:)
integer(int32), intent(in) :: lmax
integer(int32), intent(out) :: n
integer(int32), intent(in), optional :: sampling, lmax_calc, &
normal_gravity, extend
integer(int32), intent(out), optional :: exitstatus
integer(int32) :: l, m, i, l1, m1, lmax_comp, i_eq, i_s, astat(4), nlong, &
nlat_out, nlong_out, extend_grid
real(dp) :: grid(4*lmax+4), pi, theta, scalef, rescalem, u, p, dpl, pmm, &
pm1, pm2, z, tempr, r_ex, lat, prefactor(lmax), coefr0, &
coefu0, coefrs0, coeft0, coefts0, coefp0, coefps0, coefus0, b
complex(dp) :: coef(2*lmax+3), coefr(2*lmax+3), coefrs(2*lmax+3), &
coeft(2*lmax+3), coefts(2*lmax+3), coefp(2*lmax+3), &
coefps(2*lmax+3), coefu(2*lmax+3), coefus(2*lmax+3), tempc
type(C_PTR) :: plan
real(dp), save, allocatable :: ff1(:,:), ff2(:,:), sqr(:)
integer(int8), save, allocatable :: fsymsign(:,:)
integer(int32), save :: lmax_old = 0
logical :: calcu
!$OMP threadprivate(ff1, ff2, sqr, fsymsign, lmax_old)
if (present(exitstatus)) exitstatus = 0
n = 2 * lmax + 2
if (present(sampling)) then
if (sampling == 1) then
nlong = n
else if (sampling == 2) then
nlong = 2 * n
else
print*, "Error --- MakeGravGridDH"
print*, "Optional parameter SAMPLING must be 1 (N by N) " // &
"or 2 (N by 2N)."
print*, "Input value is ", sampling
if (present(exitstatus)) then
exitstatus = 2
return
else
stop
end if
end if
else
nlong = n
end if
if (present(extend)) then
if (extend == 0) then
extend_grid = 0
nlat_out = n
nlong_out = nlong
else if (extend == 1) then
extend_grid = 1
nlat_out = n + 1
nlong_out = nlong + 1
else
print*, "Error --- MakeGravGridDH"
print*, "Optional parameter EXTEND must be 0 or 1."
print*, "Input value is ", extend
if (present(exitstatus)) then
exitstatus = 2
return
else
stop
end if
end if
else
extend_grid = 0
nlat_out = n
nlong_out = nlong
end if
if (size(cilm(:,1,1)) < 2) then
print*, "Error --- MakeGravGridDH"
print*, "CILM must be dimensioned as (2, *, *)."
print*, "Input dimension is ", size(cilm(:,1,1)), size(cilm(1,:,1)), &
size(cilm(1,1,:))
if (present(exitstatus)) then
exitstatus = 1
return
else
stop
end if
end if
if (size(rad_grid(:,1)) < nlat_out .or. size(rad_grid(1,:)) < nlong_out &
.or. size(theta_grid(:,1)) < nlat_out .or. size(theta_grid(1,:)) < &
nlong_out .or. size(phi_grid(:,1)) < nlat_out .or. size(phi_grid(1,:)) &
< nlong_out .or. size(total_grid(:,1)) < nlat_out .or. &
size(total_grid(1,:)) < nlong_out) then
print*, "Error --- MakeGravGridDH"
print*, "RAD_GRID, THETA_GRID, PHI_GRID, and TOTAL_GRID " // &
"must be dimensioned as: ", nlat_out, nlong_out
print*, "Input dimensions are ", size(rad_grid(:,1)), &
size(rad_grid(1,:)), size(theta_grid(:,1)), &
size(theta_grid(1,:)), size(phi_grid(:,1)), &
size(phi_grid(1,:)), size(total_grid(:,1)), &
size(total_grid(1,:))
if (present(exitstatus)) then
exitstatus = 1
return
else
stop
end if
end if
if (present(normal_gravity)) then
if (normal_gravity /= 0 .and. normal_gravity /= 1) then
print*, "Error --- MakeGravGridDH"
print*, "NORMAL_GRAVITY must be either 1 (remove normal gravity)"
print*, "or 0 (do not remove normal gravity)."
print*, "Input value of NORMAL_GRAVITY is ", normal_gravity
if (present(exitstatus)) then
exitstatus = 2
return
else
stop
end if
else if (.not. present(omega) .and. normal_gravity == 1) then
print*, "Error --- MakeGravGridDH"
print*, "OMEGA must be specified when removing the normal gravity."
if (present(exitstatus)) then
exitstatus = 5
return
else
stop
end if
end if
end if
if (cilm(1,1,1) == 0.0_dp .and. f /= 0.0_dp) then
print*, "Warning --- MakeGravGridDH"
print*, "The degree-0 term of the spherical harmonic " // &
"coefficients is equal to zero."
print*, "The variation in gravity resulting from variations in " // &
"radius of the flattened ellipsoid will not be " // &
"taken into account."
print*, "C00 = ", cilm(1,1,1)
print*, "F = ", f
end if
if (present(pot_grid)) then
calcu = .true.
if (size(pot_grid(:,1)) < nlat_out .or. size(pot_grid(1,:)) &
< nlong_out) then
print*, "Error --- MakeGravGridDH"
print*, "POT_GRID must be dimensioned as: ", nlat_out, nlong_out
print*, "Input dimensions are ", size(pot_grid(:,1)), &
size(pot_grid(1,:))
if (present(exitstatus)) then
exitstatus = 1
return
else
stop
end if
end if
else
calcu = .false.
end if
pi = acos(-1.0_dp)
scalef = 1.0e-280_dp
if (present(lmax_calc)) then
if (lmax_calc > lmax) then
print*, "Error --- MakeGravGridDH"
print*, "LMAX_CALC must be less than or equal to LMAX."
print*, "LMAX = ", lmax
print*, "LMAX_CALC = ", lmax_calc
if (present(exitstatus)) then
exitstatus = 2
return
else
stop
end if
else
lmax_comp = min(lmax, size(cilm(1,1,:))-1, size(cilm(1,:,1))-1, &
lmax_calc)
end if
else
lmax_comp = min(lmax, size(cilm(1,1,:))-1, size(cilm(1,:,1))-1)
end if
!--------------------------------------------------------------------------
!
! Calculate recursion constants used in computing Legendre functions.
!
!--------------------------------------------------------------------------
if (lmax_comp /= lmax_old) then
if (allocated (sqr)) deallocate (sqr)
if (allocated (ff1)) deallocate (ff1)
if (allocated (ff2)) deallocate (ff2)
if (allocated (fsymsign)) deallocate (fsymsign)
allocate (sqr(2*lmax_comp+1), stat=astat(1))
allocate (ff1(lmax_comp+1,lmax_comp+1), stat=astat(2))
allocate (ff2(lmax_comp+1,lmax_comp+1), stat=astat(3))
allocate (fsymsign(lmax_comp+1,lmax_comp+1), stat=astat(4))
if (sum(astat(1:4)) /= 0) then
print*, "Error --- MakeGravGridDH"
print*, "Problem allocating arrays SQR, FF1, FF2, or FSYMSIGN", &
astat(1), astat(2), astat(3), astat(4)
if (present(exitstatus)) then
exitstatus = 3
return
else
stop
end if
end if
!----------------------------------------------------------------------
!
! Calculate signs used for symmetry of Legendre functions about
! equator. For the first derivative in theta, these signs are
! reversed.
!
!----------------------------------------------------------------------
do l = 0, lmax_comp, 1
do m = 0, l, 1
if (mod(l-m,2) == 0) then
fsymsign(l+1,m+1) = 1
else
fsymsign(l+1,m+1) = -1
end if
end do
end do
!----------------------------------------------------------------------
!
! Precompute square roots of integers that are used several times.
!
!----------------------------------------------------------------------
do l = 1, 2*lmax_comp+1
sqr(l) = sqrt(dble(l))
end do
!----------------------------------------------------------------------
!
! Precompute multiplicative factors used in recursion relationships
! P(l,m) = x*f1(l,m)*P(l-1,m) - P(l-2,m)*f2(l,m)
! k = l*(l+1)/2 + m + 1
! Note that prefactors are not used for the case when m=l as a
! different recursion is used. Furthermore, for m=l-1, Plmbar(l-2,m)
! is assumed to be zero.
!
!----------------------------------------------------------------------
if (lmax_comp /= 0) then
ff1(2,1) = sqr(3)
ff2(2,1) = 0.0_dp
end if
do l = 2, lmax_comp, 1
ff1(l+1,1) = sqr(2*l-1) * sqr(2*l+1) / dble(l)
ff2(l+1,1) = dble(l-1) * sqr(2*l+1) / sqr(2*l-3) / dble(l)
do m = 1, l-2, 1
ff1(l+1,m+1) = sqr(2*l+1) * sqr(2*l-1) / sqr(l+m) / sqr(l-m)
ff2(l+1,m+1) = sqr(2*l+1) * sqr(l-m-1) * sqr(l+m-1) &
/ sqr(2*l-3) / sqr(l+m) / sqr(l-m)
end do
ff1(l+1,l) = sqr(2*l+1) * sqr(2*l-1) / sqr(l+m) / sqr(l-m)
ff2(l+1,l) = 0.0_dp
end do
lmax_old = lmax_comp
end if
!--------------------------------------------------------------------------
!
! Create generic plan for grid.
!
!--------------------------------------------------------------------------
plan = fftw_plan_dft_c2r_1d(nlong, coef(1:nlong/2+1), grid(1:nlong), &
FFTW_MEASURE)
!--------------------------------------------------------------------------
!
! Determine Clms one l at a time by intergrating over latitude.
!
!--------------------------------------------------------------------------
i_eq = n / 2 + 1 ! Index correspondong to zero latitude
! First do equator
r_ex = a
theta = pi / 2.0_dp
z = 0.0_dp
u = 1.0_dp
coefr(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefr0 = 0.0_dp
if (calcu) then
coefu(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefu0 = 0.0_dp
end if
coeft(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coeft0 = 0.0_dp
coefp(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefp0 = 0.0_dp
pm2 = 1.0_dp
coefr0 = coefr0 - cilm(1,1,1) * pm2
if (calcu) coefu0 = coefu0 + cilm(1,1,1) * pm2
! derivative of l=0 term is 0, so no need to calculate this
if (lmax_comp /= 0) then ! l = 1
prefactor(1) = r0 / r_ex
do l = 2, lmax_comp,1
prefactor(l) = prefactor(l-1) * r0 / r_ex
end do
pm1 = 0.0_dp
if (calcu) coefu0 = coefu0 + cilm(1,2,1) * pm1 * prefactor(1)
dpl = ff1(2,1)
coeft0 = coeft0 + cilm(1,2,1) * dpl * prefactor(1)
end if
do l = 2, lmax_comp, 1
l1 = l + 1
p = - ff2(l1,1) * pm2
coefr0 = coefr0 + cilm(1,l1,1) * p * (-l1) * prefactor(l)
if (calcu) coefu0 = coefu0 + cilm(1,l1,1) * p * prefactor(l)
dpl = l * (sqr(2*l+1) / sqr(2*l-1) * pm1)
coeft0 = coeft0 + cilm(1,l1,1) * dpl * prefactor(l)
pm2 = pm1
pm1 = p
end do
pmm = sqr(2) * scalef
rescalem = 1.0_dp / scalef
do m = 1, lmax_comp-1, 1
m1 = m + 1
pmm = pmm * sqr(2*m+1) / sqr(2*m)
pm2 = pmm
coefr(m1) = coefr(m1) + cmplx(cilm(1,m1,m1), - cilm(2,m1,m1), dp) &
* pm2 * (-m-1) * prefactor(m)
if (calcu) coefu(m1) = coefu(m1) + cmplx(cilm(1,m1,m1), &
- cilm(2,m1,m1), dp) * pm2 * prefactor(m)
coefp(m1) = coefp(m1) + cmplx(cilm(2,m1,m1), cilm(1,m1,m1), dp) &
* pm2 * prefactor(m) * m
pm1 = 0.0_dp
dpl = (sqr(2*m+3) * pmm)
coeft(m1) = coeft(m1) + cmplx(cilm(1,m1+1,m1), &
- cilm(2,m1+1,m1), dp) * dpl * prefactor(m+1)
do l = m+2, lmax_comp, 1
l1 = l + 1
p = - ff2(l1,m1) * pm2
coefr(m1) = coefr(m1) + cmplx(cilm(1,l1,m1), - cilm(2,l1,m1), dp) &
* p * (-l1) * prefactor(l)
if (calcu) coefu(m1) = coefu(m1) + cmplx(cilm(1,l1,m1), &
- cilm(2,l1,m1), dp) * p * prefactor(l)
coefp(m1) = coefp(m1) + cmplx(cilm(2,l1,m1), cilm(1,l1,m1), dp) &
* p * prefactor(l) * m
dpl = ( sqr(2*l+1) * sqr(l-m) * sqr(l+m) / sqr(2*l-1) * pm1)
coeft(m1) = coeft(m1) + cmplx(cilm(1,l1,m1), - cilm(2,l1,m1), dp) &
* dpl * prefactor(l)
pm2 = pm1
pm1 = p
end do
coefr(m1) = coefr(m1) * rescalem
if (calcu) coefu(m1) = coefu(m1) * rescalem
coefp(m1) = coefp(m1) * rescalem
coeft(m1) = coeft(m1) * rescalem
end do
if (lmax_comp /= 0) then
pmm = pmm * sqr(2*lmax_comp+1) / sqr(2*lmax_comp) * rescalem
coefr(lmax_comp+1) = coefr(lmax_comp+1) + &
cmplx(cilm(1,lmax_comp+1,lmax_comp+1), &
- cilm(2,lmax_comp+1,lmax_comp+1), dp) &
* pmm * (-lmax_comp-1) * prefactor(lmax_comp)
if (calcu) coefu(lmax_comp+1) = coefu(lmax_comp+1) + &
cmplx(cilm(1,lmax_comp+1,lmax_comp+1), &
- cilm(2,lmax_comp+1,lmax_comp+1), dp) &
* pmm * prefactor(lmax_comp)
coefp(lmax_comp+1) = coefp(lmax_comp+1) &
+ cmplx(cilm(2,lmax_comp+1,lmax_comp+1), &
cilm(1,lmax_comp+1,lmax_comp+1), dp) &
* pmm * prefactor(lmax_comp) * lmax_comp
dpl = -lmax_comp * z * pmm / u**2
coeft(lmax_comp+1) = coeft(lmax_comp+1) + &
cmplx(cilm(1,lmax_comp+1,lmax_comp+1), &
- cilm(2,lmax_comp+1,lmax_comp+1), dp) &
* dpl * prefactor(lmax_comp)
end if
coef(1) = cmplx(coefr0, 0.0_dp, dp)
coef(2:lmax+1) = coefr(2:lmax+1) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
rad_grid(i_eq,1:nlong) = grid(1:nlong) * gm / r_ex**2
if (calcu) then
coef(1) = cmplx(coefu0, 0.0_dp, dp)
coef(2:lmax+1) = coefu(2:lmax+1) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
pot_grid(i_eq,1:nlong) = grid(1:nlong) * gm / r_ex
end if
coef(1) = cmplx(coeft0, 0.0_dp, dp)
coef(2:lmax+1) = coeft(2:lmax+1) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
theta_grid(i_eq,1:nlong) = -sin(theta) * grid(1:nlong) * gm / r_ex**2
coef(1) = cmplx(coefp0, 0.0_dp, dp)
coef(2:lmax+1) = coefp(2:lmax+1) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
phi_grid(i_eq,1:nlong) = grid(1:nlong) * (gm / r_ex**2) / sin(theta)
do i=1, i_eq-1, 1
i_s = 2 * i_eq - i
theta = pi * dble(i-1) / dble(n)
z = cos(theta)
u = sqrt( (1.0_dp-z) * (1.0_dp+z) )
lat = pi / 2.0_dp - theta
if (i == 1) then ! Reference ellipsoid radius
r_ex = a * (1.0_dp - f)
else
r_ex = cos(lat)**2 + sin(lat)**2 / (1.0_dp - f)**2
r_ex = a * sqrt(1.0_dp / r_ex)
end if
coefr(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefr0 = 0.0_dp
coefrs(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefrs0 = 0.0_dp
coeft(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coeft0 = 0.0_dp
coefts(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefts0 = 0.0_dp
coefp(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefp0 = 0.0_dp
coefps(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefps0 = 0.0_dp
if (calcu) then
coefu(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefu0 = 0.0_dp
coefus(1:lmax+2) = cmplx(0.0_dp, 0.0_dp, dp)
coefus0 = 0.0_dp
end if
pm2 = 1.0_dp
tempr = -cilm(1,1,1) * pm2 ! l = 0
coefr0 = coefr0 + tempr
coefrs0 = coefrs0 + tempr ! fsymsign is always 1 for l=m=0
if (calcu) then
tempr = cilm(1,1,1) * pm2 ! l = 0
coefu0 = coefu0 + tempr
coefus0 = coefus0 + tempr ! fsymsign is always 1 for l=m=0
end if
! derivative of l=0 term is 0, so no need to calculate this
if (lmax_comp /= 0) then ! l = 1
prefactor(1) = r0 / r_ex
do l = 2, lmax_comp, 1
prefactor(l) = prefactor(l-1) * r0 / r_ex
end do
pm1 = ff1(2,1) * z
! -2 = (l+1) prefactor
tempr = cilm(1,2,1) * pm1 * (-2) * prefactor(1)
coefr0 = coefr0 + tempr
coefrs0 = coefrs0 - tempr ! fsymsign = -1
if (calcu) then
tempr = cilm(1,2,1) * pm1 * prefactor(1)
coefu0 = coefu0 + tempr
coefus0 = coefus0 - tempr ! fsymsign = -1
end if
dpl = ff1(2,1)
tempr = cilm(1,2,1) * dpl * prefactor(1)
coeft0 = coeft0 + tempr
coefts0 = coefts0 + tempr ! reverse fsymsign
end if
do l = 2, lmax_comp, 1
l1 = l + 1
p = ff1(l1,1) * z * pm1 - ff2(l1,1) * pm2
tempr = cilm(1,l1,1) * p * (-l1) * prefactor(l)
coefr0 = coefr0 + tempr
coefrs0 = coefrs0 + tempr * fsymsign(l1,1)
if (calcu) then
tempr = cilm(1,l1,1) * p * prefactor(l)
coefu0 = coefu0 + tempr
coefus0 = coefus0 + tempr * fsymsign(l1,1)
end if
dpl = l * ( sqr(2*l+1) / sqr(2*l-1) * pm1 - z * p ) / u**2
tempr = cilm(1,l1,1) * dpl * prefactor(l)
coeft0 = coeft0 + tempr
coefts0 = coefts0 - tempr * fsymsign(l1,1) ! reverse fsymsign
pm2 = pm1
pm1 = p
end do
pmm = sqr(2) * scalef
rescalem = 1.0_dp / scalef
do m = 1, lmax_comp-1, 1
m1 = m + 1
rescalem = rescalem * u
pmm = pmm * sqr(2*m+1) / sqr(2*m)
pm2 = pmm
tempc = cmplx(cilm(1,m1,m1), - cilm(2,m1,m1), dp) * pm2 * (-m-1) &
* prefactor(m) ! (m,m)
coefr(m1) = coefr(m1) + tempc
coefrs(m1) = coefrs(m1) + tempc
! fsymsign = 1
if (calcu) then
tempc = cmplx(cilm(1,m1,m1), - cilm(2,m1,m1), dp) * pm2 &
* prefactor(m) ! (m,m)
coefu(m1) = coefu(m1) + tempc
coefus(m1) = coefus(m1) + tempc
! fsymsign = 1
end if
tempc = cmplx(cilm(2,m1,m1), cilm(1,m1,m1), dp) * pm2 &
* prefactor(m) * m ! (m,m)
coefp(m1) = coefp(m1) + tempc
coefps(m1) = coefps(m1) + tempc
! fsymsign = 1
dpl = -m * z * pm2 / u**2
tempc = cmplx(cilm(1,m1,m1), - cilm(2,m1,m1), dp) * dpl &
* prefactor(m) ! (m,m)
coeft(m1) = coeft(m1) + tempc
coefts(m1) = coefts(m1) - tempc ! reverse fsymsign
pm1 = z * ff1(m1+1,m1) * pm2
tempc = cmplx(cilm(1,m1+1,m1), - cilm(2,m1+1,m1), dp) * pm1 &
* (-m-2) * prefactor(m+1) ! (m+1,m)
coefr(m1) = coefr(m1) + tempc
coefrs(m1) = coefrs(m1) - tempc
! fsymsign = -1
if (calcu) then
tempc = cmplx(cilm(1,m1+1,m1), - cilm(2,m1+1,m1), dp) * pm1 &
* prefactor(m+1) ! (m+1,m)
coefu(m1) = coefu(m1) + tempc
coefus(m1) = coefus(m1) - tempc
end if
tempc = cmplx(cilm(2,m1+1,m1), cilm(1,m1+1,m1), dp) * pm1 &
* prefactor(m+1) * m ! (m+1,m)
coefp(m1) = coefp(m1) + tempc
coefps(m1) = coefps(m1) - tempc
! fsymsign = -1
dpl = ( sqr(2*m+3) * pmm - z * (m+1) * pm1) / u**2
tempc = cmplx(cilm(1,m1+1,m1), - cilm(2,m1+1,m1), dp) * dpl &
* prefactor(m+1) ! (m+1,m)
coeft(m1) = coeft(m1) + tempc
coefts(m1) = coefts(m1) + tempc ! reverse fsymsign
do l=m+2, lmax_comp, 1
l1 = l + 1
p = z * ff1(l1,m1) * pm1 - ff2(l1,m1) * pm2
tempc = cmplx(cilm(1,l1,m1), - cilm(2,l1,m1), dp) * p &
* (-l1) * prefactor(l)
coefr(m1) = coefr(m1) + tempc
coefrs(m1) = coefrs(m1) + tempc * fsymsign(l1,m1)
if (calcu) then
tempc = cmplx(cilm(1,l1,m1), - cilm(2,l1,m1), dp) * p &
* prefactor(l)
coefu(m1) = coefu(m1) + tempc
coefus(m1) = coefus(m1) + tempc * fsymsign(l1,m1)
end if
tempc = cmplx(cilm(2,l1,m1), cilm(1,l1,m1), dp) * p &
* prefactor(l) * m
coefp(m1) = coefp(m1) + tempc
coefps(m1) = coefps(m1) + tempc * fsymsign(l1,m1)
dpl = ( sqr(2*l+1) * sqr(l-m) * sqr(l+m) / sqr(2*l-1) * pm1 &
- z * l * p) / u**2
tempc = cmplx(cilm(1,l1,m1), - cilm(2,l1,m1), dp) * dpl &
* prefactor(l)
coeft(m1) = coeft(m1) + tempc
! reverse fsymsign
coefts(m1) = coefts(m1) - tempc * fsymsign(l1,m1)
pm2 = pm1
pm1 = p
end do
coefr(m1) = coefr(m1) * rescalem
coefrs(m1) = coefrs(m1) * rescalem
if (calcu) then
coefu(m1) = coefu(m1) * rescalem
coefus(m1) = coefus(m1) * rescalem
end if
coeft(m1) = coeft(m1) * rescalem
coefts(m1) = coefts(m1) * rescalem
coefp(m1) = coefp(m1) * rescalem
coefps(m1) = coefps(m1) * rescalem
end do
if (lmax_comp /= 0) then
rescalem = rescalem * u
pmm = pmm * sqr(2*lmax_comp+1) / sqr(2*lmax_comp) * rescalem
tempc = cmplx(cilm(1,lmax_comp+1,lmax_comp+1), &
- cilm(2,lmax_comp+1,lmax_comp+1), dp) * pmm &
* (-lmax_comp-1) * prefactor(lmax_comp)
coefr(lmax_comp+1) = coefr(lmax_comp+1) + tempc
coefrs(lmax_comp+1) = coefrs(lmax_comp+1) + tempc
! fsymsign = 1
if (calcu) then
tempc = cmplx(cilm(1,lmax_comp+1,lmax_comp+1), &
- cilm(2,lmax_comp+1,lmax_comp+1), dp) * pmm &
* prefactor(lmax_comp)
coefu(lmax_comp+1) = coefu(lmax_comp+1) + tempc
coefus(lmax_comp+1) = coefus(lmax_comp+1) + tempc
end if
tempc = cmplx(cilm(2,lmax_comp+1,lmax_comp+1), &
cilm(1,lmax_comp+1,lmax_comp+1), dp) * pmm &
* prefactor(lmax_comp) * lmax_comp
coefp(lmax_comp+1) = coefp(lmax_comp+1) + tempc
coefps(lmax_comp+1) = coefps(lmax_comp+1) + tempc
! fsymsign = 1
dpl = -lmax_comp * z * pmm / u**2
tempc = cmplx(cilm(1,lmax_comp+1,lmax_comp+1), &
- cilm(2,lmax_comp+1,lmax_comp+1), dp) * dpl &
* prefactor(lmax_comp)
coeft(lmax_comp+1) = coeft(lmax_comp+1) + tempc
! reverse fsymsign
coefts(lmax_comp+1) = coefts(lmax_comp+1) - tempc
end if
coef(1) = cmplx(coefr0, 0.0_dp, dp)
coef(2:lmax+1) = coefr(2:lmax+1) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
rad_grid(i,1:nlong) = grid(1:nlong) * gm / r_ex**2
if (calcu) then
coef(1) = cmplx(coefu0, 0.0_dp, dp)
coef(2:lmax+1) = coefu(2:lmax+1) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
pot_grid(i,1:nlong) = grid(1:nlong) * gm / r_ex
end if
if (i == 1) then
! These two derivatives are undefined at the pole
theta_grid(1,1:nlong) = 0.0_dp
phi_grid(1,1:nlong) = 0.0_dp
else
coef(1) = cmplx(coeft0, 0.0_dp, dp)
coef(2:lmax+1) = coeft(2:lmax+1) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
theta_grid(i,1:nlong) = -sin(theta) * grid(1:nlong) * gm / r_ex**2
coef(1) = cmplx(coefp0, 0.0_dp, dp)
coef(2:lmax+1) = coefp(2:lmax+1) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
phi_grid(i,1:nlong) = grid(1:nlong) * (gm/r_ex**2) / sin(theta)
end if
! don't compute value for south pole when extend = 0.
if (.not. (i == 1 .and. extend_grid == 0) ) then
coef(1) = cmplx(coefrs0, 0.0_dp, dp)
coef(2:lmax+1) = coefrs(2:lmax+1) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
rad_grid(i_s,1:nlong) = grid(1:nlong) * gm / r_ex**2
if (calcu) then
coef(1) = cmplx(coefus0, 0.0_dp, dp)
coef(2:lmax+1) = coefus(2:lmax+1) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
pot_grid(i_s,1:nlong) = grid(1:nlong) * gm / r_ex
end if
if (i == 1) then
! These two derivatives are undefined at the pole
theta_grid(i_s,1:nlong) = 0.0_dp
phi_grid(i_s,1:nlong) = 0.0_dp
else
coef(1) = cmplx(coefts0, 0.0_dp, dp)
coef(2:lmax+1) = coefts(2:lmax+1) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
theta_grid(i_s,1:nlong) = -sin(theta) * grid(1:nlong) &
* gm / r_ex**2
coef(1) = cmplx(coefps0, 0.0_dp, dp)
coef(2:lmax+1) = coefps(2:lmax+1) / 2.0_dp
if (present(sampling)) then
if (sampling == 2) then
coef(lmax+2:2*lmax+3) = cmplx(0.0_dp, 0.0_dp, dp)
end if
end if
call fftw_execute_dft_c2r(plan, coef, grid)
phi_grid(i_s,1:nlong) = grid(1:nlong) * (gm/r_ex**2) / &
sin(theta)
end if
end if
end do
call fftw_destroy_plan(plan)
!--------------------------------------------------------------------------
!
! Add rotational effects
!
!--------------------------------------------------------------------------
if (present(omega)) then