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|
! Re-Parametrization of the level set function
! $Header$
#include "cctk.h"
#include "cctk_Parameters.h"
#include "cctk_Arguments.h"
! Control routine for reparametrization of the level set function.
subroutine EHFinder_ReParametrizeControl(CCTK_ARGUMENTS)
use EHFinder_mod
implicit none
DECLARE_CCTK_PARAMETERS
DECLARE_CCTK_ARGUMENTS
DECLARE_CCTK_FUNCTIONS
CCTK_REAL, dimension(eh_number_level_sets) :: fmin, fmax, fminloc, fmaxloc
CCTK_INT :: i, j, k, l
! print*,'EHFinder_ReParametrize1 Entered'
! Initialize the control variables. The default means no re-parametrization.
re_param_control_pde = 0
re_param_control_approx = 0
! If the method is 'approx' or 'mixed' check if it is time for an
! approximation re-parametrization and set the control variable to 1.
if ( ( CCTK_EQUALS ( re_param_method, 'approx' ) ) .or. &
( CCTK_EQUALS ( re_param_method, 'mixed' ) ) ) then
if ( reparametrize_every_approx .gt. 0 ) then
if ( mod(cctk_iteration,reparametrize_every_approx) .eq. 0 ) then
re_param_control_approx = 1
end if
end if
end if
! If the method is 'pde' or 'mixed' check if it is time for a pde
! re-parametrization and set the control variable to 1. Note it overrides
! the approximation re-parametrization.
if ( ( CCTK_EQUALS ( re_param_method, 'pde' ) ) .or. &
( CCTK_EQUALS ( re_param_method, 'mixed' ) ) ) then
if ( reparametrize_every_pde .gt. 0 ) then
if ( mod(cctk_iteration,reparametrize_every_pde) .eq. 0 ) then
re_param_control_approx = 0
re_param_control_pde = 1
end if
end if
end if
! If it is time for a re-parametrization get reduction handles for
! minimum and maximum reductions.
if ( ( re_param_control_approx .gt. 0 ) .or. &
( re_param_control_pde .gt. 0 ) ) then
call CCTK_ReductionArrayHandle ( max_handle, 'maximum' )
if ( max_handle .lt. 0 ) then
call CCTK_WARN(0,'Could not obtain a handle for maximum reduction')
end if
call CCTK_ReductionArrayHandle ( min_handle, 'minimum' )
if ( min_handle .lt. 0 ) then
call CCTK_WARN(0,'Could not obtain a handle for minimum reduction')
end if
! if ( cctk_bbox(1) .eq. 0 ) eh_mask(1:ngx,:,:) = -2
! if ( cctk_bbox(2) .eq. 0 ) eh_mask(nx-ngx+1:nx,:,:) = -2
! if ( cctk_bbox(3) .eq. 0 ) eh_mask(:,1:ngy,:) = -2
! if ( cctk_bbox(4) .eq. 0 ) eh_mask(:,ny-ngy+1:ny,:) = -2
! if ( cctk_bbox(5) .eq. 0 ) eh_mask(:,:,1:ngz) = -2
! if ( cctk_bbox(6) .eq. 0 ) eh_mask(:,:,nz-ngz+1:nz) = -2
! Set the courant factor for the 'Euler' re-parametrization scheme.
if ( CCTK_EQUALS ( re_param_int_method, 'Euler' ) ) then
! hfac = one / eight
hfac = one / four
end if
! Set the courant factor for the 'rk2' re-parametrization scheme.
if ( CCTK_EQUALS ( re_param_int_method, 'rk2' ) ) then
! hfac = half
hfac = one / four
end if
! Set up counter for the number of iterations. Copy f into ftmp and
! fbak and eh_mask into eh_mask_bak (in case the re-parametrization has
! to be undone) and find the min and max values of f.
ncalls = 0
ftmp = f
fbak = f
eh_mask_bak = eh_mask
! rep_mask = 0
do l = 1, eh_number_level_sets
fminloc(l) = minval(f(:,:,:,l))
fmaxloc(l) = maxval(f(:,:,:,l))
end do
call CCTK_ReduceLocArrayToArray1D ( ierr, cctkGH, -1, max_handle, &
fmaxloc, fmax, eh_number_level_sets, &
CCTK_VARIABLE_REAL )
if ( ierr .ne. 0 ) then
call CCTK_WARN(0,'Reduction of fmax failed')
end if
call CCTK_ReduceLocArrayToArray1D ( ierr, cctkGH, -1, min_handle, &
fminloc, fmin, eh_number_level_sets, &
CCTK_VARIABLE_REAL )
if ( ierr .ne. 0 ) then
call CCTK_WARN(0,'Reduction of fmin failed')
end if
reparam_this_level_set = .true.
! If fmin and fmax have the same sign, there is no surface present and
! re-parametrization will not be performed for the given surface.
do l = 1, eh_number_level_sets
if ( fmin(l) * fmax(l) .gt. zero ) then
reparam_this_level_set(l) = .false.
end if
end do
! If none of the surfaces should be reparametrized set the control variables
! to zero.
if ( all ( .not. reparam_this_level_set ) ) then
re_param_control_approx = 0
re_param_control_pde = 0
call CCTK_INFO ( 'No zero-points of the level set functions. No re-parametrization performed.' )
end if
end if
! print*,'EHFinder_ReParametrizeControl Exited'
end subroutine EHFinder_ReParametrizeControl
!subroutine EHFinder_ReParametrizeRK2_1(CCTK_ARGUMENTS)
!
! use EHFinder_mod
!
! implicit none
!
! DECLARE_CCTK_PARAMETERS
! DECLARE_CCTK_ARGUMENTS
! DECLARE_CCTK_FUNCTIONS
!
! CCTK_INT :: i, j, k
! CCTK_REAL :: idx, idy, idz
! CCTK_REAL :: al, ar, bl, br, cl, cr
! CCTK_REAL :: alminus, alplus, blminus, blplus, clminus, clplus
! CCTK_REAL :: arminus, arplus, brminus, brplus, crminus, crplus
!
!! print*,'EHFinder_ReParametrizeRK2_1 Entered'
! h = hfac*minval(cctk_delta_space)
!
! if ( CCTK_EQUALS ( pde_differences, 'centered' ) ) then
!
!# include "include/centered_second.h"
!
! end if
!
! if ( CCTK_EQUALS ( pde_differences, 'upwind' ) ) then
!
!# include "include/upwind_first.h"
!
! end if
!
! if ( CCTK_EQUALS ( pde_differences, 'upwind2' ) ) then
!
!# include "include/upwind_second.h"
!
! end if
!
! where ( eh_mask .ge. 0 )
! sftmp = sqrt( dfx**2 + dfy**2 + dfz**2 )
! sftmp = - half * h * f / sqrt(f**2+one) * ( sftmp - one )
! f = f + sftmp
! elsewhere
! sftmp = one
! end where
!! print*,'EHFinder_ReParametrizeRK2_1 Exited'
!
!end subroutine EHFinder_ReParametrizeRK2_1
!subroutine EHFinder_ReParametrizeRK2_2(CCTK_ARGUMENTS)
!
!
! use EHFinder_mod
!
! implicit none
!
! DECLARE_CCTK_PARAMETERS
! DECLARE_CCTK_ARGUMENTS
! DECLARE_CCTK_FUNCTIONS
!
! CCTK_INT :: i, j, k
! CCTK_REAL :: idx, idy, idz, maxf, maxdfloc, maxdf, mindfloc, mindf
! CCTK_REAL :: al, ar, bl, br, cl, cr
! CCTK_REAL :: alminus, alplus, blminus, blplus, clminus, clplus
! CCTK_REAL :: arminus, arplus, brminus, brplus, crminus, crplus
! character(len=128) :: info_message
!
!! print*,'EHFinder_ReParametrizeRK2_2 Entered'
!
! if ( CCTK_EQUALS ( pde_differences, 'centered' ) ) then
!
!# include "include/centered_second.h"
!
! end if
!
! if ( CCTK_EQUALS ( pde_differences, 'upwind' ) ) then
!
!# include "include/upwind_first.h"
!
! end if
!
! if ( CCTK_EQUALS ( pde_differences, 'upwind2' ) ) then
!
!# include "include/upwind_second.h"
!
! end if
!
! where ( eh_mask .ge. 0 )
!
! sftmp = sqrt( dfx**2 + dfy**2 + dfz**2 )
! sftmp = - h * f / sqrt(f**2+one) * ( sftmp - one )
! f = ftmp + sftmp
! elsewhere
! sftmp = one
! end where
!
! maxdfloc = zero
! mindfloc = 1.0d23
! do k = kzl, kzr
! do j = jyl, jyr
! do i =ixl, ixr
! if ( eh_mask(i,j,k) .ge. 0 ) then
! maxdfloc = max ( maxdfloc, abs(sftmp(i,j,k)) )
! mindfloc = min ( mindfloc, abs(sftmp(i,j,k)) )
! end if
! end do
! end do
! end do
!
! call CCTK_ReduceLocScalar ( ierr, cctkGH, -1, max_handle, &
! maxdfloc, maxdf, CCTK_VARIABLE_REAL )
! if ( ierr .ne. 0 ) then
! call CCTK_WARN(0,'Reduction of maxdf failed')
! end if
! call CCTK_ReduceLocScalar ( ierr, cctkGH, -1, min_handle, &
! mindfloc, mindf, CCTK_VARIABLE_REAL )
! if ( ierr .ne. 0 ) then
! call CCTK_WARN(0,'Reduction of mindf failed')
! end if
!
! if ( maxdf .lt. h*minval(cctk_delta_space)**2 ) re_param_control_pde = 0
!
! ncalls = ncalls + 1
!
! if ( re_param_control_pde .eq. 0 ) then
! write(info_message,'(a35,i5,a12)') &
! 'PDE re-parametrization complete in ',ncalls,' iterations.'
! call CCTK_INFO(info_message)
! end if
!
! if ( ncalls .gt. re_param_max_iter ) then
! call CCTK_WARN(1,'Re-parametrization failed to converge')
! re_param_control_pde = 0
! end if
!
! ftmp = f
!! print*,ncalls,maxdf,mindf
!! print*,mx1,my1,mz2,mx2,my2,mz2
!! print*,f(mx1,my1,mz1),f(mx2,my2,mz2),eh_mask(mx1,my1,mz1),eh_mask(mx2,my2,mz2)
!! print*,f(27,53,27),dfsq(27,53,27),ftmp(27,53,27)
!!
!! call CCTK_INFO('I was called')
!! print*,'EHFinder_ReParametrizeRK2_2 Exited'
!
!end subroutine EHFinder_ReParametrizeRK2_2
subroutine EHFinder_ReParametrizeEuler(CCTK_ARGUMENTS)
use EHFinder_mod
implicit none
DECLARE_CCTK_PARAMETERS
DECLARE_CCTK_ARGUMENTS
DECLARE_CCTK_FUNCTIONS
CCTK_INT :: i, j, k, l
CCTK_REAL :: idx, idy, idz, maxf
CCTK_REAL, dimension(eh_number_level_sets) :: maxdfloc, maxdf, &
mindfloc, mindf
CCTK_REAL :: al, ar, bl, br, cl, cr
CCTK_REAL :: alminus, alplus, blminus, blplus, clminus, clplus
CCTK_REAL :: arminus, arplus, brminus, brplus, crminus, crplus
character(len=128) :: info_message
! print*,'EHFinder_ReParametrizeEuler Entered'
h = hfac*minval(cctk_delta_space)
if ( CCTK_EQUALS ( pde_differences, 'centered' ) ) then
# include "include/centered_second.h"
end if
if ( CCTK_EQUALS ( pde_differences, 'upwind' ) ) then
# include "include/upwind_first.h"
end if
if ( CCTK_EQUALS ( pde_differences, 'upwind2' ) ) then
# include "include/upwind_second.h"
end if
where ( eh_mask .ge. 0 )
sftmp = sqrt( dfx**2 + dfy**2 + dfz**2 )
sftmp = - h * f / sqrt(f**2+one) * ( sftmp - one )
f = f + sftmp
elsewhere
sftmp = one
end where
! print*,f(2:4,2:4,2:4,5)
! print*,sftmp(3,3,3,5)
! stop
! do l = 1, eh_number_level_sets
! maxdfloc = zero
! mindfloc = 1.0d23
! do k = kzl, kzr
! do j = jyl, jyr
! do i = ixl, ixr
! if ( eh_mask(i,j,k,l) .ge. 0 ) then
! maxdfloc = max ( maxdfloc, abs(sftmp(i,j,k,l)) )
! mindfloc = min ( mindfloc, abs(sftmp(i,j,k,l)) )
! end if
! end do
! end do
! end do
! end do
do l = 1, eh_number_level_sets
maxdfloc(l) = maxval ( abs(sftmp(ixl:ixr,jyl:jyr,kzl:kzr,l)), &
mask = eh_mask(ixl:ixr,jyl:jyr,kzl:kzr,l) .ge. 0)
mindfloc(l) = minval ( abs(sftmp(ixl:ixr,jyl:jyr,kzl:kzr,l)), &
mask = eh_mask(ixl:ixr,jyl:jyr,kzl:kzr,l) .ge. 0)
end do
call CCTK_ReduceLocArrayToArray1D ( ierr, cctkGH, -1, max_handle, &
maxdfloc, maxdf, eh_number_level_sets, &
CCTK_VARIABLE_REAL )
if ( ierr .ne. 0 ) then
call CCTK_WARN(0,'Reduction of maxdf failed')
end if
call CCTK_ReduceLocArrayToArray1D ( ierr, cctkGH, -1, min_handle, &
mindfloc, mindf, eh_number_level_sets, &
CCTK_VARIABLE_REAL )
if ( ierr .ne. 0 ) then
call CCTK_WARN(0,'Reduction of mindf failed')
end if
if ( all ( maxdf .lt. h*minval(cctk_delta_space)**2 ) ) then
re_param_control_pde = 0
endif
ncalls = ncalls + 1
if ( re_param_control_pde .eq. 0 ) then
write(info_message,'(a35,i5,a12)') &
'PDE re-parametrization complete in ',ncalls,' iterations.'
call CCTK_INFO(info_message)
end if
if ( ncalls .gt. re_param_max_iter ) then
call CCTK_WARN(1,'Re-parametrization failed to converge')
re_param_control_pde = 0
end if
! print*,ncalls
! print*,maxdf
! print*,mindf
! print*,'EHFinder_ReParametrizeEuler Exited'
end subroutine EHFinder_ReParametrizeEuler
!subroutine EHFinder_ReParametrize5(CCTK_ARGUMENTS)
!
! use EHFinder_mod
!
! implicit none
!
! DECLARE_CCTK_PARAMETERS
! DECLARE_CCTK_ARGUMENTS
! DECLARE_CCTK_FUNCTIONS
!
! CCTK_INT :: i, j, k
! CCTK_INT :: rep_countloc, rep_totalloc
! CCTK_REAL :: idx2, idy2, idz2, idx, idy, idz
! CCTK_REAL :: d2fdx2, d2fdy2, d2fdz2, d2fdxdy, d2fdxdz, d2fdydz
! CCTK_REAL :: n_x, n_y, n_z
! CCTK_REAL :: a, b, c, lambda
!
! interface
! CCTK_REAL function solve_quadratic ( a, b, c )
! CCTK_REAL, intent(in) :: a, b, c
! end function solve_quadratic
! end interface
!
! if ( re_param_control_approx .eq. 1 ) then
! if ( reparametrize_every_approx .gt. 0 ) then
! if ( mod(cctk_iteration,reparametrize_every_approx) .eq. 0 ) then
! call CCTK_ReductionArrayHandle ( sum_handle, 'sum' )
! if ( sum_handle .lt. 0 ) then
! call CCTK_WARN(0,'Could not obtain a handle for sum reduction')
! end if
!
!# include "include/centered_second.h"
!
! ncalls = 0
!
! rep_totalloc = count ( eh_mask(ixl:ixr,jyl:jyr,kzl:kzr) .ge. 0 )
!
! rep_countloc = 0
!
! ftmp = f
!
! ixr = min ( ixr, nx-1 )
! jyr = min ( jyr, ny-1 )
! kzr = min ( kzr, nz-1 )
!
! do k = kzl, kzr
! do j = jyl, jyr
! do i = ixl, ixr
! if ( eh_mask(i,j,k) .eq. 0 ) then
! if ( ( f(i,j,k)*f(i-1,j,k) .le. zero ) .or. &
! ( f(i,j,k)*f(i+1,j,k) .le. zero ) .or. &
! ( f(i,j,k)*f(i,j-1,k) .le. zero ) .or. &
! ( f(i,j,k)*f(i,j+1,k) .le. zero ) .or. &
! ( f(i,j,k)*f(i,j,k-1) .le. zero ) .or. &
! ( f(i,j,k)*f(i,j,k+1) .le. zero ) ) then
!
! rep_mask(i,j,k) = 1
! rep_countloc = rep_countloc + 1
!
! d2fdx2 = idx2 * ( f(i-1,j,k) - two * f(i,j,k) + f(i+1,j,k) )
! d2fdy2 = idy2 * ( f(i,j-1,k) - two * f(i,j,k) + f(i,j+1,k) )
! d2fdz2 = idz2 * ( f(i,j,k-1) - two * f(i,j,k) + f(i,j,k+1) )
!
! d2fdxdy = idx * idy * ( f(i-1,j-1,k) - f(i+1,j-1,k) - &
! f(i-1,j+1,k) + f(i+1,j+1,k) )
! d2fdxdz = idx * idz * ( f(i-1,j,k-1) - f(i+1,j,k-1) - &
! f(i-1,j,k+1) + f(i+1,j,k+1) )
! d2fdydz = idy * idz * ( f(i,j-1,k-1) - f(i,j+1,k-1) - &
! f(i,j-1,k+1) + f(i,j+1,k+1) )
!
! c = f(i,j,k)
! b = sqrt ( dfx(i,j,k)**2 + dfy(i,j,k)**2 + dfz(i,j,k)**2 )
! n_x = dfx(i,j,k) / b
! n_y = dfy(i,j,k) / b
! n_z = dfz(i,j,k) / b
! a = half*( n_x**2*d2fdx2 + n_y**2*d2fdy2 + n_z**2*d2fdz2 ) +&
! n_x*n_y*d2fdxdy + n_x*n_z*d2fdxdz + n_y*n_z*d2fdydz
!
! if ( a .ne. 0 ) then
! lambda = solve_quadratic ( a, b, c )
! else
! lambda = abs ( c / b )
! endif
! if ( lambda .eq. huge ) then
! lambda = abs ( c / b )
! end if
!
! if ( lambda .gt. two*delta ) then
! print*,i,j,k
! print*,f(i-1:i+1,j-1:j+1,k-1:k+1)
! print*,eh_mask(i-1:i+1,j-1:j+1,k-1:k+1)
! print*,a,b,c,lambda
! call CCTK_WARN(0,'Re-parametrization failed')
! end if
!
! ftmp(i,j,k) = sign ( lambda, f(i,j,k) )
!! print*,i,j,k
!! print*,lambda,f(i,j,k), ftmp(i,j,k)
!! print*,f(i-1,j,k),f(i+1,j,k)
!! print*,f(i,j-1,k),f(i,j+1,k)
!! print*,f(i,j,k-1),f(i,j,k+1)
! end if
! end if
! end do
! end do
! end do
!
! where ( eh_mask .ge. 0 ) f = ftmp
!
!! print*,rep_countloc,rep_totalloc
! call CCTK_ReduceLocScalar ( ierr, cctkGH, -1, sum_handle, &
! rep_countloc, rep_count, CCTK_VARIABLE_INT )
! if ( ierr .ne. 0 ) then
! call CCTK_WARN(0,'Sum of rep_count failed')
! end if
!
! call CCTK_ReduceLocScalar ( ierr, cctkGH, -1, sum_handle, &
! rep_totalloc, rep_total, CCTK_VARIABLE_INT )
! if ( ierr .ne. 0 ) then
! call CCTK_WARN(0,'Sum of rep_total failed')
! end if
!! print*,'First step'
!! print*,rep_count,rep_total
!! call CCTK_WARN(0,'Finished')
!
!! call CartSymGN(ierr,cctkGH,'ehfinder::level_set')
!! call CartSymGN(ierr,cctkGH,'ehfinder::rep_mask')
!
! end if
! end if
!
! end if
!
!end subroutine EHFinder_ReParametrize5
!subroutine EHFinder_ReParametrize6(CCTK_ARGUMENTS)
!
! use EHFinder_mod
!
! implicit none
!
! DECLARE_CCTK_PARAMETERS
! DECLARE_CCTK_ARGUMENTS
! DECLARE_CCTK_FUNCTIONS
!
! CCTK_INT :: i, j, k, l, nneigh, rep_loc1, rep_loc2
! CCTK_INT, dimension(1) :: lmax
! CCTK_REAL, dimension(3) :: n0, n1, vec
! CCTK_REAL, dimension(0:5) :: dp1, dp2, fnew
! CCTK_REAL :: dfs1, dfs2
!
!# include "include/physical_part.h"
!
! ncalls = ncalls + 1
!
! rep_loc1 = 0
!
! do k = kzl, kzr
! do j = jyl, jyr
! do i = ixl, ixr
! if ( ( eh_mask(i,j,k) .ge. 0 ) .and. ( rep_mask(i,j,k) .eq. 0 ) ) then
! nneigh = 0
! fnew = zero
! dfs1 = one / sqrt ( dfx(i,j,k)**2 + dfy(i,j,k)**2 + dfz(i,j,k)**2 )
! n0(1) = dfx(i,j,k) * dfs1
! n0(2) = dfy(i,j,k) * dfs1
! n0(3) = dfz(i,j,k) * dfs1
! do l = 0, 5
! if ( .not. btest(eh_mask(i,j,k),l) ) then
! if ( rep_mask(i+ix(l),j+jy(l),k+kz(l)) .eq. 1 ) then
! vec(1) = -ix(l) * cctk_delta_space(1)
! vec(2) = -jy(l) * cctk_delta_space(2)
! vec(3) = -kz(l) * cctk_delta_space(3)
! dfs2 = one / sqrt ( dfx(i+ix(l),j+jy(l),k+kz(l))**2 + &
! dfy(i+ix(l),j+jy(l),k+kz(l))**2 + &
! dfz(i+ix(l),j+jy(l),k+kz(l))**2 )
! n1(1) = dfx(i+ix(l),j+jy(l),k+kz(l)) * dfs2
! n1(2) = dfy(i+ix(l),j+jy(l),k+kz(l)) * dfs2
! n1(3) = dfz(i+ix(l),j+jy(l),k+kz(l)) * dfs2
! dp1(nneigh) = vec(1) * n0(1) + vec(2) * n0(2) + vec(3) * n0(3)
! dp2(nneigh) = vec(1) * n1(1) + vec(2) * n1(2) + vec(3) * n1(3)
! fnew(nneigh) = f(i+ix(l),j+jy(l),k+kz(l)) + &
! half * ( dp1(nneigh) + dp2(nneigh) )
! nneigh = nneigh + 1
!
! end if
! end if
! end do
!
! if ( nneigh .gt. 0 ) then
! f(i,j,k) = sum(fnew(0:nneigh-1)) / nneigh
! rep_mask(i,j,k) = 2
! rep_loc1 = rep_loc1 + 1
! end if
! end if
! end do
! end do
! end do
!
! call CCTK_ReduceLocScalar ( ierr, cctkGH, -1, sum_handle, &
! rep_loc1, rep_loc2, CCTK_VARIABLE_INT )
! if ( ierr .ne. 0 ) then
! call CCTK_WARN(0,'Sum of rep_count failed')
! end if
!! print*,rep_loc1,rep_loc2
!
! rep_count = rep_count + rep_loc2
!! print*,ncalls,rep_count,rep_total
!
! if ( rep_count .ge. rep_total ) then
! re_param_control_approx = 0
! call CCTK_INFO('Approximation re-parametrization complete')
! end if
!
!! call CCTK_WARN(0,'Testing')
!! print*,'after'
!! print*,rep_mask(24,21,1:4),eh_mask(24,21,1:4)
!! print*,f(24,21,1:4)
!! print*
!end subroutine EHFinder_ReParametrize6
!subroutine EHFinder_ReParametrize7(CCTK_ARGUMENTS)
!
! use EHFinder_mod
!
! implicit none
!
! DECLARE_CCTK_PARAMETERS
! DECLARE_CCTK_ARGUMENTS
! DECLARE_CCTK_FUNCTIONS
!
! where ( rep_mask .eq. 2 ) rep_mask = 1
!
!end subroutine EHFinder_ReParametrize7
!CCTK_REAL function solve_quadratic ( a, b, c )
!
! use EHFinder_Constants
!
! CCTK_REAL, intent(in) :: a, b, c
! CCTK_REAL :: d2, q
!
! d2 = b**2 - four * a * c
!
! if ( d2 .ge. zero ) then
! q = - half * ( b + sign(one,b) * sqrt(d2) )
!
! solve_quadratic = min ( abs(q/a), abs(c/q) )
! else
! solve_quadratic = huge
! end if
!
!end function solve_quadratic
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