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!
! Copyright (C) 2002-2009 Quantum ESPRESSO group
! This file is distributed under the terms of the
! GNU General Public License. See the file `License'
! in the root directory of the present distribution,
! or http://www.gnu.org/copyleft/gpl.txt .
!
!---------------------------------------------------------------------
function qe_erf (x)
!---------------------------------------------------------------------
!
! Error function - computed from the rational approximations of
! W. J. Cody, Math. Comp. 22 (1969), pages 631-637.
!
! for abs(x) le 0.47 erf is calculated directly
! for abs(x) gt 0.47 erf is calculated via erf(x)=1-erfc(x)
!
use kinds, only : DP
implicit none
real(DP), intent(in) :: x
real(DP) :: x2, p1 (4), q1 (4)
real(DP), external :: qe_erfc
real(DP) :: qe_erf
data p1 / 2.426679552305318E2_DP, 2.197926161829415E1_DP, &
6.996383488619136_DP, -3.560984370181538E-2_DP /
data q1 / 2.150588758698612E2_DP, 9.116490540451490E1_DP, &
1.508279763040779E1_DP, 1.000000000000000_DP /
!
if (abs (x) > 6.0_DP) then
!
! erf(6)=1-10^(-17) cannot be distinguished from 1
!
qe_erf = sign (1.0_DP, x)
else
if (abs (x) <= 0.47_DP) then
x2 = x**2
qe_erf=x *(p1 (1) + x2 * (p1 (2) + x2 * (p1 (3) + x2 * p1 (4) ) ) ) &
/ (q1 (1) + x2 * (q1 (2) + x2 * (q1 (3) + x2 * q1 (4) ) ) )
else
qe_erf = 1.0_DP - qe_erfc (x)
endif
endif
!
return
end function qe_erf
!
!---------------------------------------------------------------------
function qe_erfc (x)
!---------------------------------------------------------------------
!
! erfc(x) = 1-erf(x) - See comments in erf
!
use kinds, only : DP
implicit none
real(DP),intent(in) :: x
real(DP) :: qe_erfc
real(DP) :: ax, x2, xm2, p2 (8), q2 (8), p3 (5), q3 (5), pim1
real(DP), external :: qe_erf
data p2 / 3.004592610201616E2_DP, 4.519189537118719E2_DP, &
3.393208167343437E2_DP, 1.529892850469404E2_DP, &
4.316222722205674E1_DP, 7.211758250883094_DP, &
5.641955174789740E-1_DP,-1.368648573827167E-7_DP /
data q2 / 3.004592609569833E2_DP, 7.909509253278980E2_DP, &
9.313540948506096E2_DP, 6.389802644656312E2_DP, &
2.775854447439876E2_DP, 7.700015293522947E1_DP, &
1.278272731962942E1_DP, 1.000000000000000_DP /
data p3 /-2.996107077035422E-3_DP,-4.947309106232507E-2_DP, &
-2.269565935396869E-1_DP,-2.786613086096478E-1_DP, &
-2.231924597341847E-2_DP /
data q3 / 1.062092305284679E-2_DP, 1.913089261078298E-1_DP, &
1.051675107067932_DP, 1.987332018171353_DP, &
1.000000000000000_DP /
data pim1 / 0.56418958354775629_DP /
! ( pim1= sqrt(1/pi) )
ax = abs (x)
if (ax > 26.0_DP) then
!
! erfc(26.0)=10^(-296); erfc( 9.0)=10^(-37);
!
qe_erfc = 0.0_DP
elseif (ax > 4.0_DP) then
x2 = x**2
xm2 = (1.0_DP / ax) **2
qe_erfc = (1.0_DP / ax) * exp ( - x2) * (pim1 + xm2 * (p3 (1) &
+ xm2 * (p3 (2) + xm2 * (p3 (3) + xm2 * (p3 (4) + xm2 * p3 (5) &
) ) ) ) / (q3 (1) + xm2 * (q3 (2) + xm2 * (q3 (3) + xm2 * &
(q3 (4) + xm2 * q3 (5) ) ) ) ) )
elseif (ax > 0.47_DP) then
x2 = x**2
qe_erfc = exp ( - x2) * (p2 (1) + ax * (p2 (2) + ax * (p2 (3) &
+ ax * (p2 (4) + ax * (p2 (5) + ax * (p2 (6) + ax * (p2 (7) &
+ ax * p2 (8) ) ) ) ) ) ) ) / (q2 (1) + ax * (q2 (2) + ax * &
(q2 (3) + ax * (q2 (4) + ax * (q2 (5) + ax * (q2 (6) + ax * &
(q2 (7) + ax * q2 (8) ) ) ) ) ) ) )
else
qe_erfc = 1.0_DP - qe_erf (ax)
endif
!
! erf(-x)=-erf(x) => erfc(-x) = 2-erfc(x)
!
if (x < 0.0_DP) qe_erfc = 2.0_DP - qe_erfc
!
return
end function qe_erfc
!
!---------------------------------------------------------------------
function gauss_freq (x)
!---------------------------------------------------------------------
!
! gauss_freq(x) = (1+erf(x/sqrt(2)))/2 = erfc(-x/sqrt(2))/2
! - See comments in erf
!
use kinds, only : DP
implicit none
real(DP),intent(in) :: x
real(DP) :: gauss_freq
real(DP), parameter :: c = 0.7071067811865475_DP
! ( c= sqrt(1/2) )
real(DP), external :: qe_erfc
!
gauss_freq = 0.5_DP * qe_erfc ( - x * c)
!
return
end function gauss_freq
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