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findiffmod.f90
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module findiffmod
use, intrinsic :: iso_fortran_env, only: wp=>real128
implicit none
contains
function findiffhes(j, xstar,eps,order) result(res)
implicit none
interface
function j(x)
import
! This function should accept a state vector x
! and emit the jacobian matrix of a vector-valued function
real(wp), intent(in) :: x(:)
real(wp) :: j(size(x),size(x))
end function j
end interface
real(wp), intent(in) :: xstar(:), eps
integer, intent(in) :: order
real(wp) :: res(size(xstar,1),size(xstar,1), size(xstar,1))
integer :: i, dims
! Perturb initial conditions for deriv calculation
real(wp) :: per(size(xstar), size(xstar)), &
& bigpertmatrix(size(xstar),size(xstar)), &
& pstep(size(xstar),size(xstar)), &
& mstep(size(xstar),size(xstar)), &
& pdf(size(xstar),size(xstar),size(xstar)), &
& mdf(size(xstar),size(xstar),size(xstar))
real(wp), allocatable:: &
& twopstep(:,:), &
& twomstep(:,:), &
& threepstep(:,:), &
& threemstep(:,:), &
& fourpstep(:,:), &
& fourmstep(:,:), &
& twopdf(:,:,:), &
& twomdf(:,:,:), &
& threepdf(:,:,:), &
& threemdf(:,:,:), &
& fourpdf(:,:,:), &
& fourmdf(:,:,:)
! Perturb initial conditions for deriv calculation
dims = size(xstar)
per = 0._wp
do i=1,dims
bigpertmatrix(:,i) = xstar
per(i,i) = eps
end do
pstep = bigpertmatrix + per
mstep = bigpertmatrix - per
if (order > 3 ) then
allocate(twopstep, twomstep, mold=per)
allocate(twopdf, twomdf, mold=pdf)
twopstep = pstep + per
twomstep = mstep - per
end if
if (order > 5) then
allocate(threepstep, threemstep, mold=per)
allocate(threepdf, threemdf, mold=pdf)
threepstep = twopstep + per
threemstep = twomstep - per
end if
if (order > 7) then
allocate(fourpstep, fourmstep, mold=per)
allocate(fourpdf, fourmdf, mold=pdf)
fourpstep = threepstep + per
fourmstep = threemstep - per
end if
do i=1,dims
pdf(:,:,i) = j(pstep(:,i))
mdf(:,:,i) = j(mstep(:,i))
end do
if (order > 3 ) then
do i=1,dims
twopdf(:,:,i) = j(twopstep(:,i))
twomdf(:,:,i) = j(twomstep(:,i))
end do
end if
if (order > 5) then
do i=1,dims
threepdf(:,:,i) = j(threepstep(:,i))
threemdf(:,:,i) = j(threemstep(:,i))
end do
end if
if (order > 7) then
do i=1,dims
fourpdf(:,:,i) = j(fourpstep(:,i))
fourmdf(:,:,i) = j(fourmstep(:,i))
end do
end if
select case (order)
case(3)
res = (pdf - mdf) / (2._wp*eps)
case(5)
res = (8._wp*pdf - twopdf - 8._wp*mdf + twomdf)/(12._wp*eps)
case(7)
res = (45._wp*pdf - 9._wp*twopdf + threepdf - 45._wp*mdf + &
& 9._wp*twomdf -threemdf)/(60._wp*eps)
case(9)
res = (3._wp*(fourmdf - fourpdf) + 32._wp*(threepdf-threemdf) + &
& 168._wp*(twomdf-twopdf) + 672._wp*(pdf-mdf))/(840._wp*eps)
case default
print *, 'Warning: order not implemented. Using 3 point stencil'
res = (pdf - mdf) /(2._wp*eps)
end select
end function findiffhes
function findiff(f, xstar,eps,order) result(res)
implicit none
interface
function f(x)
import
real(wp), intent(in) :: x(:)
real(wp) :: f(size(x))
end function f
end interface
real(wp), intent(in) :: xstar(:), eps
integer, intent(in) :: order
real(wp) :: res(size(xstar),size(xstar)), &
& per(size(xstar), size(xstar)), &
& bigpertmatrix(size(xstar),size(xstar)), &
& pstep(size(xstar),size(xstar)), &
& mstep(size(xstar),size(xstar)), &
& pdf(size(xstar),size(xstar)), &
& mdf(size(xstar),size(xstar))
real(wp), allocatable:: &
& twopstep(:,:), &
& twomstep(:,:), &
& twopdf(:,:), &
& twomdf(:,:), &
& threepstep(:,:), &
& threemstep(:,:), &
& threepdf(:,:), &
& threemdf(:,:), &
& fourpstep(:,:), &
& fourmstep(:,:), &
& fourpdf(:,:), &
& fourmdf(:,:)
integer :: i, dims
! Perturb initial conditions for deriv calculation
dims = size(xstar)
per = 0._wp
do i=1,dims
bigpertmatrix(:,i) = xstar
per(i,i) = eps
end do
pstep = bigpertmatrix + per
mstep = bigpertmatrix - per
if (order > 3 ) then
allocate(twopstep, twomstep, twopdf, twomdf, mold=per)
twopstep = pstep + per
twomstep = mstep - per
end if
if (order > 5) then
allocate(threepstep, threemstep, threepdf, threemdf, mold=per)
threepstep = twopstep + per
threemstep = twomstep - per
end if
if (order > 7) then
allocate(fourpstep, fourmstep, fourpdf, fourmdf, mold=per)
fourpstep = threepstep + per
fourmstep = threemstep - per
end if
do i=1,dims
pdf(:,i) = f(pstep(:,i))
mdf(:,i) = f(mstep(:,i))
end do
if (order > 3 ) then
do i=1,dims
twopdf(:,i) = f(twopstep(:,i))
twomdf(:,i) = f(twomstep(:,i))
end do
end if
if (order > 5) then
do i=1,dims
threepdf(:,i) = f(threepstep(:,i))
threemdf(:,i) = f(threemstep(:,i))
end do
end if
if (order > 7) then
do i=1,dims
fourpdf(:,i) = f(fourpstep(:,i))
fourmdf(:,i) = f(fourmstep(:,i))
end do
end if
select case (order)
case(3)
res = (pdf - mdf) / (2._wp*eps)
case(5)
res = (8._wp*pdf - twopdf - 8._wp*mdf + twomdf)/(12._wp*eps)
case(7)
res = (45._wp*pdf - 9._wp*twopdf + threepdf - 45._wp*mdf + &
& 9._wp*twomdf -threemdf)/(60._wp*eps)
case(9)
res = (3._wp*(fourmdf - fourpdf) + 32._wp*(threepdf-threemdf) + &
& 168._wp*(twomdf-twopdf) + 672._wp*(pdf-mdf))/(840._wp*eps)
case default
print *, 'Warning: order not implemented. Using 3 point stencil'
res = (pdf - mdf) /(2._wp*eps)
end select
end function findiff
function findiffmat_td(f, xstar,eps,order,shape_template) result(res)
implicit none
interface
function f(x)
import
real(wp), intent(in) :: x
real(wp), allocatable :: f(:,:)
end function f
end interface
real(wp), intent(in) :: xstar, eps, shape_template(:,:)
integer, intent(in) :: order
real(wp) :: res(size(shape_template,1),size(shape_template,2)), &
& refpoint, &
& pstep, &
& mstep, &
& twopstep, &
& twomstep, &
& threepstep, &
& threemstep, &
& fourpstep, &
& fourmstep, &
& pdf(size(shape_template,1),size(shape_template,2)), &
& mdf(size(shape_template,1),size(shape_template,2))
real(wp) :: &
& twopdf(size(shape_template,1),size(shape_template,2)), &
& twomdf(size(shape_template,1),size(shape_template,2)), &
& threepdf(size(shape_template,1),size(shape_template,2)), &
& threemdf(size(shape_template,1),size(shape_template,2)), &
& fourpdf(size(shape_template,1),size(shape_template,2)), &
& fourmdf(size(shape_template,1),size(shape_template,2))
! Perturb initial conditions for deriv calculation
refpoint = xstar
pstep = refpoint + eps
mstep = refpoint - eps
if (order > 3 ) then
twopstep = pstep + eps
twomstep = mstep - eps
end if
if (order > 5) then
threepstep = twopstep + eps
threemstep = twomstep - eps
end if
if (order > 7) then
fourpstep = threepstep + eps
fourmstep = threemstep - eps
end if
pdf = f(pstep)
mdf = f(mstep)
if (order > 3 ) then
twopdf = f(twopstep)
twomdf = f(twomstep)
end if
if (order > 5) then
threepdf = f(threepstep)
threemdf = f(threemstep)
end if
if (order > 7) then
fourpdf = f(fourpstep)
fourmdf = f(fourmstep)
end if
select case (order)
case(3)
res = (pdf - mdf) / (2._wp*eps)
case(5)
res = (8._wp*pdf - twopdf - 8._wp*mdf + twomdf)/(12._wp*eps)
case(7)
res = (45._wp*pdf - 9._wp*twopdf + threepdf - 45._wp*mdf + &
& 9._wp*twomdf -threemdf)/(60._wp*eps)
case(9)
res = (3._wp*(fourmdf - fourpdf) + 32._wp*(threepdf-threemdf) + &
& 168._wp*(twomdf-twopdf) + 672._wp*(pdf-mdf))/(840._wp*eps)
case default
print *, 'Warning: order not implemented. Using 3 point stencil'
res = (pdf - mdf) /(2._wp*eps)
end select
end function findiffmat_td
function findiff_multiscale(f, xstar, epsvec, order) result(res)
implicit none
interface
function f(x)
import
real(wp), intent(in) :: x(:)
real(wp) :: f(size(x))
end function f
end interface
real(wp), intent(in) :: xstar(:), epsvec(size(xstar))
integer, intent(in) :: order
real(wp) :: res(size(xstar),size(xstar)), &
& per(size(xstar), size(xstar)), &
& bigpertmatrix(size(xstar),size(xstar)), &
& pstep(size(xstar),size(xstar)), &
& mstep(size(xstar),size(xstar)), &
& pdf(size(xstar),size(xstar)), &
& mdf(size(xstar),size(xstar))
real(wp), allocatable:: &
& twopstep(:,:), &
& twomstep(:,:), &
& twopdf(:,:), &
& twomdf(:,:), &
& threepstep(:,:), &
& threemstep(:,:), &
& threepdf(:,:), &
& threemdf(:,:), &
& fourpstep(:,:), &
& fourmstep(:,:), &
& fourpdf(:,:), &
& fourmdf(:,:)
integer :: i, dims
! Perturb initial conditions for deriv calculation
res = 0._wp
dims = size(xstar)
per = 0._wp
do i=1,dims
bigpertmatrix(:,i) = xstar
per(i,i) = epsvec(i)
end do
pstep = bigpertmatrix + per
mstep = bigpertmatrix - per
if (order > 3 ) then
allocate(twopstep, twomstep, twopdf, twomdf, mold=per)
twopstep = pstep + per
twomstep = mstep - per
end if
if (order > 5) then
allocate(threepstep, threemstep, threepdf, threemdf, mold=per)
threepstep = twopstep + per
threemstep = twomstep - per
end if
if (order > 7) then
allocate(fourpstep, fourmstep, fourpdf, fourmdf, mold=per)
fourpstep = threepstep + per
fourmstep = threemstep - per
end if
do i=1,dims
pdf(:,i) = f(pstep(:,i))
mdf(:,i) = f(mstep(:,i))
end do
if (order > 3 ) then
do i=1,dims
twopdf(:,i) = f(twopstep(:,i))
twomdf(:,i) = f(twomstep(:,i))
end do
end if
if (order > 5) then
do i=1,dims
threepdf(:,i) = f(threepstep(:,i))
threemdf(:,i) = f(threemstep(:,i))
end do
end if
if (order > 7) then
do i=1,dims
fourpdf(:,i) = f(fourpstep(:,i))
fourmdf(:,i) = f(fourmstep(:,i))
end do
end if
select case (order)
case(3)
do i=1,dims
res(:,i) = (pdf(:,i) - mdf(:,i)) / (2._wp*epsvec(i))
end do
case(5)
do i=1,dims
res(:,i) = (8._wp*pdf(:,i) - twopdf(:,i) - 8._wp*mdf(:,i) + twomdf(:,i))/(12._wp*epsvec(i))
end do
case(7)
do i=1,dims
res(:,i) = (45._wp*pdf(:,i) - 9._wp*twopdf(:,i) + threepdf(:,i) - 45._wp*mdf(:,i) + &
& 9._wp*twomdf(:,i) -threemdf(:,i))/(60._wp*epsvec(i))
end do
case(9)
do i=1,dims
res(:,i) = (3._wp*(fourmdf(:,i) - fourpdf(:,i)) + 32._wp*(threepdf(:,i)-threemdf(:,i)) + &
& 168._wp*(twomdf(:,i)-twopdf(:,i)) + 672._wp*(pdf(:,i)-mdf(:,i)))/(840._wp*epsvec(i))
end do
case default
print *, 'Warning: order not implemented. Using 3 point stencil'
do i=1,dims
res(:,i) = (pdf(:,i) - mdf(:,i)) /(2._wp*epsvec(i))
end do
end select
end function findiff_multiscale
function findiffhes_multiscale(j, xstar,epsvec,order) result(res)
implicit none
interface
function j(x)
import
! This function should accept a state vector x
! and emit the jacobian matrix of a vector-valued function
real(wp), intent(in) :: x(:)
real(wp) :: j(size(x),size(x))
end function j
end interface
real(wp), intent(in) :: xstar(:), epsvec(size(xstar))
integer, intent(in) :: order
real(wp) :: res(size(xstar,1),size(xstar,1), size(xstar,1))
integer :: i, dims
! Perturb initial conditions for deriv calculation
real(wp) :: per(size(xstar), size(xstar)), &
& bigpertmatrix(size(xstar),size(xstar)), &
& pstep(size(xstar),size(xstar)), &
& mstep(size(xstar),size(xstar)), &
& pdf(size(xstar),size(xstar),size(xstar)), &
& mdf(size(xstar),size(xstar),size(xstar))
real(wp), allocatable:: &
& twopstep(:,:), &
& twomstep(:,:), &
& threepstep(:,:), &
& threemstep(:,:), &
& fourpstep(:,:), &
& fourmstep(:,:), &
& twopdf(:,:,:), &
& twomdf(:,:,:), &
& threepdf(:,:,:), &
& threemdf(:,:,:), &
& fourpdf(:,:,:), &
& fourmdf(:,:,:)
! Perturb initial conditions for deriv calculation
dims = size(xstar)
per = 0._wp
res = 0._wp
do i=1,dims
bigpertmatrix(:,i) = xstar
per(i,i) = epsvec(i)
end do
pstep = bigpertmatrix + per
mstep = bigpertmatrix - per
if (order > 3 ) then
allocate(twopstep, twomstep, mold=per)
allocate(twopdf, twomdf, mold=pdf)
twopstep = pstep + per
twomstep = mstep - per
end if
if (order > 5) then
allocate(threepstep, threemstep, mold=per)
allocate(threepdf, threemdf, mold=pdf)
threepstep = twopstep + per
threemstep = twomstep - per
end if
if (order > 7) then
allocate(fourpstep, fourmstep, mold=per)
allocate(fourpdf, fourmdf, mold=pdf)
fourpstep = threepstep + per
fourmstep = threemstep - per
end if
do i=1,dims
pdf(:,:,i) = j(pstep(:,i))
mdf(:,:,i) = j(mstep(:,i))
end do
if (order > 3 ) then
do i=1,dims
twopdf(:,:,i) = j(twopstep(:,i))
twomdf(:,:,i) = j(twomstep(:,i))
end do
end if
if (order > 5) then
do i=1,dims
threepdf(:,:,i) = j(threepstep(:,i))
threemdf(:,:,i) = j(threemstep(:,i))
end do
end if
if (order > 7) then
do i=1,dims
fourpdf(:,:,i) = j(fourpstep(:,i))
fourmdf(:,:,i) = j(fourmstep(:,i))
end do
end if
select case (order)
case(3)
do i=1,dims
res(:,:,i) = (pdf(:,:,i) - mdf(:,:,i)) / (2._wp*epsvec(i))
end do
case(5)
do i=1,dims
res(:,:,i) = (8._wp*pdf(:,:,i) - twopdf(:,:,i) - 8._wp*mdf(:,:,i) + twomdf(:,:,i))/(12._wp*epsvec(i))
end do
case(7)
do i=1,dims
res(:,:,i) = (45._wp*pdf(:,:,i) - 9._wp*twopdf(:,:,i) + threepdf(:,:,i) - 45._wp*mdf(:,:,i) + &
& 9._wp*twomdf(:,:,i) -threemdf(:,:,i))/(60._wp*epsvec(i))
end do
case(9)
do i=1,dims
res(:,:,i) = (3._wp*(fourmdf(:,:,i) - fourpdf(:,:,i)) + 32._wp*(threepdf(:,:,i)-threemdf(:,:,i)) + &
& 168._wp*(twomdf(:,:,i)-twopdf(:,:,i)) + 672._wp*(pdf(:,:,i)-mdf(:,:,i)))/(840._wp*epsvec(i))
end do
case default
print *, 'Warning: order not implemented. Using 3 point stencil'
do i=1,dims
res(:,:,i) = (pdf(:,:,i) - mdf(:,:,i)) / (2._wp*epsvec(i))
end do
end select
end function findiffhes_multiscale
end module