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PlON_d1.f95
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PlON_d1.f95
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subroutine PlON_d1(p, dp1, lmax, z, exitstatus)
!------------------------------------------------------------------------------
!
! This function evalutates all of the "ortho-normalized legendre
! polynomials, as well as their first derivatives, up to degree lmax.
!
! Calling Parameters
!
! Out
! p A vector of all normalized Legendgre polynomials evaluated
! at z up to lmax with dimension (lmax+1).
! dp1 A vector of all first derivatives of the normalized
! Legendgre polynomials evaluated at z up to lmax with
! dimension (lmax+1).
!
! IN
! lmax Maximum degree to compute.
! z [-1, 1], cos(colatitude), or sin(latitude).
!
! OPTIONAL (OUT)
! 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. The employed normalization is the "orthonomalization convention."
! 2. The integral of PlON**2 over all space on the sphere is 1.
! 3. The integral of PlON**2 over (-1,1) is 1/2pi.
! 4. The derivative is evaluated with respect to z, and NOT cos(colatitude)
! or sin(latitude).
! 5. Derivatives are calculated according to the geodesy-normalized
! relationships
! P'_0(z) = 0.0, P'_1(z) = 1.0, and
! P'_l(z) = l * (P_{l-1}(z) - z * P_l(z) ) / (1.0_dp - z**2)
! At z = 1, Pl(1) = 1, and P'l(1) = l (l+1) / 2 (Boyd 2001)
! At z = -1 Pl(-1) = (-1)**l, and P'l(-1) = (-1)**(l-1) l (l+1) / 2
!
! Copyright (c) 2005-2019, SHTOOLS
! All rights reserved.
!
!------------------------------------------------------------------------------
use ftypes
implicit none
integer(int32), intent(in) :: lmax
real(dp), intent(out) :: p(:), dp1(:)
real(dp), intent(in) :: z
integer(int32), intent(out), optional :: exitstatus
real(dp) :: pm2, pm1, pl, sinsq, pi
integer(int32) :: l
if (present(exitstatus)) exitstatus = 0
pi = acos(-1.0_dp)
if (size(p) < lmax+1) then
print*, "Error --- PlON_d1"
print*, "P must be dimensioned as (LMAX+1) where LMAX is ", lmax
print*, "Input array is dimensioned ", size(p)
if (present(exitstatus)) then
exitstatus = 1
return
else
stop
end if
else if (size(dp1) < lmax+1) then
print*, "Error --- PlON_d1"
print*, "DP1 must be dimensioned as (LMAX+1) where LMAX is ", lmax
print*, "Input array is dimensioned ", size(dp1)
if (present(exitstatus)) then
exitstatus = 1
return
else
stop
end if
else if (lmax < 0) then
print*, "Error --- PlON_d1"
print*, "LMAX must be greater than or equal to 0."
print*, "Input value is ", lmax
if (present(exitstatus)) then
exitstatus = 2
return
else
stop
end if
else if (abs(z) > 1.0_dp) then
print*, "Error --- PlON_d1"
print*, "ABS(Z) must be less than or equal to 1."
print*, "Input value is ", z
if (present(exitstatus)) then
exitstatus = 2
return
else
stop
end if
end if
if (z == 1.0_dp) then
do l = 0, lmax
p(l+1) = sqrt( dble(2*l+1)) / sqrt(4*pi)
dp1(l+1) = sqrt( dble(2*l+1) ) * dble(l) * dble(l+1) / 2.0_dp &
/ sqrt(4*pi)
end do
else if (z == -1.0_dp) then
do l = 0, lmax
p(l+1) = sqrt( dble(2*l+1)) * dble((-1)**l) / sqrt(4*pi)
dp1(l+1) = sqrt( dble(2*l+1)) * dble(l) * dble(l+1) &
* dble((-1)**(l-1)) / 2.0_dp / sqrt(4*pi)
end do
else
sinsq = (1.0_dp - z**2)
pm2 = 1.0_dp / sqrt(4*pi)
p(1) = pm2
dp1(1) = 0.0_dp
pm1 = sqrt(3.0_dp) * z / sqrt(4*pi)
p(2) = pm1
dp1(2) = sqrt(3.0_dp) / sqrt(4*pi)
do l = 2, lmax, 1
pl = ( sqrt(dble(2*l-1)) * z * pm1 - &
(l-1) * pm2 / sqrt(dble(2*l-3)) ) * &
sqrt(dble(2*l+1)) / dble(l)
p(l+1) = pl
dp1(l+1) = l * ( sqrt( dble(2*l+1) / dble(2*l-1) ) * &
p(l) - z * pl ) / sinsq
pm2 = pm1
pm1 = pl
end do
end if
end subroutine PlON_d1