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quadrature_lib.f90.erb
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<%
TS = [:Real, :Dual]
KS = [32, 64, 128]
# These values may be large enough.
ITER_MAXES = [15, 30, 45]
OS = [1, 2]
NS = [1, 2, 5, 7]
K_ITER_MAXES = KS.zip(ITER_MAXES)
PARAMS \
= K_ITER_MAXES.map{|k, iter_max| [:Real, k, iter_max, nil, nil]} \
+ K_ITER_MAXES.product(OS, NS).map{|(k, iter_max), o, n| [:Dual, k, iter_max, o, n]}
ARGS_OR_NONE = [", args", nil]
def _suffix(t, k, o, n, args)
ret = if t == :Real
"#{t}#{k}"
else
"#{t}#{k}_#{o}_#{n}"
end
if !(args.nil?)
ret = ret + '_args'
end
ret
end
def _declare(t, k, o, n)
if t == :Real
"#{t}(kind=real#{k})"
else
"type(#{t}#{k}_#{o}_#{n})"
end
end
def _declare_args(t, k, o, n, args)
ret = _declare(t, k, o, n)
if args.nil?
ret = '! ' + ret
end
ret
end
%>
#include "fortran_lib.h"
module quadrature_lib
use, intrinsic:: iso_fortran_env, only: input_unit, output_unit, error_unit
use, intrinsic:: iso_fortran_env, only: int64
<% KS.each{|k| %>
use, intrinsic:: iso_fortran_env, only: real<%= k %>
<% } %>
USE_FORTRAN_LIB_H
use, non_intrinsic:: ad_lib
implicit none
private
public:: romberg
! Interface
<% PARAMS.product(ARGS_OR_NONE).each{|(t, k, _, o, n), args| %>
<% suffix = _suffix(t, k, o, n, args) %>
interface romberg
module procedure romberg<%= suffix %>
end interface romberg
<% } %>
contains
<% PARAMS.product(ARGS_OR_NONE).each{|(t, k, iter_max, o, n), args| %>
<% suffix = _suffix(t, k, o, n, args) %>
<% decl = _declare(t, k, o, n) %>
<% decl_args = _declare_args(t, k, o, n, args) %>
function romberg<%= suffix %>(f, a, b <%= args %>, rtol, atol, abs_err, err, n_eval) result(ret)
interface
function f(x <%= args %>) result(ret)
use, intrinsic:: iso_fortran_env, only: real<%= k %>
<% if t == :Dual %>
use, non_intrinsic:: ad_lib
<% end %>
<%= decl %>:: ret
Real(kind=real<%= k %>), intent(in):: x
<%= decl_args %>, intent(in):: args(:)
end function f
end interface
<%= decl %>:: ret
Real(kind=real<%= k %>), intent(in):: a, b
<%= decl_args %>, intent(in):: args(:)
Real(kind=real<%= k %>), intent(in), optional:: rtol, atol
<%= decl %>, intent(out), optional:: abs_err
Logical, intent(out), optional:: err
Integer(kind=int64), optional:: n_eval
Real(kind=kind(rtol)):: rtol_, atol_
<%= decl %>:: abs_err_
Logical(kind=kind(err)):: err_
Integer(kind=kind(n_eval)):: n_eval_
rtol_ = sqrt(epsilon(rtol_))
if(present(rtol)) rtol_ = rtol
atol_ = sqrt(epsilon(atol_))
if(present(atol)) atol_ = atol
ASSERT(rtol_ >= 0 .or. atol_ >= 0)
ret = romberg_impl<%= suffix %>(f, a, b <%= args %>, rtol_, atol_, abs_err_, err_, n_eval_)
if(present(abs_err)) abs_err = abs_err_
if(present(err)) err = err_
if(present(n_eval)) n_eval = n_eval_
end function romberg<%= suffix %>
function romberg_impl<%= suffix %>(f, a, b <%= args %>, rtol, atol, abs_err, err, n_eval) result(ret)
interface
function f(x <%= args %>) result(ret)
use, intrinsic:: iso_fortran_env, only: real<%= k %>
<% if t == :Dual %>
use, non_intrinsic:: ad_lib
<% end %>
<%= decl %>:: ret
Real(kind=real<%= k %>), intent(in):: x
<%= decl_args %>, intent(in):: args(:)
end function f
end interface
! Maximum number of function evaluation is 2^(iter_max) + 1
Integer(kind=int64), parameter:: iter_max = <%= iter_max %>
Integer(kind=int64), parameter:: i_zero = 0
<%= decl %>:: ret
Real(kind=real<%= k %>), intent(in):: a, b
<%= decl_args %>, intent(in):: args(:)
Real(kind=real<%= k %>), intent(in):: rtol, atol
<%= decl %>, intent(out):: abs_err
Logical, intent(out):: err
Integer(kind=int64), intent(out):: n_eval
Real(kind=kind(a)):: h, h_new
<%= decl %>:: integ_pre
! integs:
! i\j 1 2 3 4
! 1 ⏢
! ↓
! 2 o ← ⏢
! ↓ ↓
! 3 o ← o ← ⏢
! ↓ ↓ ↓
! 4 o ← o ← o ← ⏢
<%= decl %>:: integs(iter_max)
Integer(kind=kind(n_eval)):: n, i, j
h = b - a
integs(1) = h*(f(a <%= args %>) + f(b <%= args %>))/2
ret = integs(1)
n = 1
err = .true.
do i = 2, iter_max
integ_pre = ret
h_new = h/2
integs(i) = (integs(i - 1) + h*sum_f<%= suffix %>(f, a + h_new, h, i_zero, n - 1 <%= args %>))/2
do j = i - 1, 1, -1
integs(j) = sub_err<%= _suffix(t, k, o, n, nil) %>(integs(j + 1), integs(j), i - j)
end do
ret = integs(1)
abs_err = abs(ret - integ_pre)
if(abs_err <= atol .or. abs_err <= rtol*ret)then
err = .false.
exit
end if
n = 2*n
h = h_new
end do
n_eval = 2*n + 1
end function romberg_impl<%= suffix %>
recursive function sum_f<%= suffix %>(f, x1, h, i1, i2 <%= args %>) result(ret)
interface
function f(x <%= args %>) result(ret)
use, intrinsic:: iso_fortran_env, only: real<%= k %>
<% if t == :Dual %>
use, non_intrinsic:: ad_lib
<% end %>
<%= decl %>:: ret
Real(kind=real<%= k %>), intent(in):: x
<%= decl_args %>, intent(in):: args(:)
end function f
end interface
Integer, parameter:: block_size = 2**5
<%= decl %>:: ret
Real(kind=real<%= k %>), intent(in):: x1, h
<%= decl_args %>, intent(in):: args(:)
Integer(kind=int64), intent(in):: i1, i2
Integer(kind=kind(i1)):: i, i_mid
if(i1 + block_size > i2)then
<% if t == :Real %>
ret = 0
<% else %>
! Dual is initialized to 0
<% end %>
do i = i1, i2
ret = ret + f(x1 + h*i <%= args %>)
end do
else
i_mid = (i1 + i2)/2
ret = sum_f<%= suffix %>(f, x1, h, i1, i_mid <%= args %>) + sum_f<%= suffix %>(f, x1, h, i_mid + 1, i2 <%= args %>)
end if
end function sum_f<%= suffix %>
<% if args.nil? %>
function sub_err<%= suffix %>(integ_right, integ_above, k) result(ret)
Real(kind=real<%= k %>), parameter:: four = 4
<%= decl %>:: ret
<%= decl %>, intent(in):: integ_right, integ_above
Integer(kind=int64), intent(in):: k
Real(kind=kind(four)):: four_pow_k
DEBUG_ASSERT(k > 0)
four_pow_k = four**k
ret = (four_pow_k*integ_right - integ_above)/(four_pow_k - 1)
end function sub_err<%= suffix %>
<% end %>
<% } %>
end module quadrature_lib