[Arjen Markus] (10 april 2022) Having done some very simple experiments
with Ashok's [CFFI package] (which comes with an amazingly comprehensive
documentation) I decided to raise the stakes: what about using it for
solving partial differential equations? Do the heavy numerics on the [C]
or [Fortran] side and led Tcl do the control. The PDE of choice is a
simple two-dimensional version of the advection-diffusion equation:
======none
2 2
dC d C d C dC
-- = D ( --- + --- ) - v --
dt 2 2 dx
dx dy
======
on a square grid (n x n cells) and simple initial and boundary
conditions.
Of course, my favourite numerical language is Fortran, so let me first
introduce the Fortran part of this experiment:
======none
! advdiff.f90 --
! Module implementing a straightforeward solver for an advection-diffusion
! equation in two dimensions
!
! Note:
! I first used: ccentre = ccentre + veldt * (...) + diffdt * (...)
! but that gave odd results, with both gfortran and Intel Fortran oneAPI.
! This needs to be investigated
!
module advdiff
use iso_c_binding
implicit none
!
! Type holding the various parameters
! (alternatives: put them in separate arguments
! store them in module variables)
!
type, bind(C) :: param_def
real(kind=c_double) :: diffusion ! Diffusion coefficient (m2/s)
real(kind=c_double) :: flow_velocity ! Flow velocity (left to right; m/s)
real(kind=c_double) :: deltax ! Grid cell size (m)
real(kind=c_double) :: deltat ! Time step (s)
integer :: nreport ! Number of steps to set before returning
end type param_def
contains
subroutine update( n, n2, conc_in, conc_out, params ) bind(c, name = 'update')
!dec attributes dllexport :: update
integer, value :: n
integer, value :: n2 ! Dummy required for the Tcl side
real(kind=c_double), intent(in) :: conc_in(n,n)
real(kind=c_double), intent(out) :: conc_out(n,n)
type(param_def), intent(in) :: params
integer :: i
real(kind=c_double) :: diffdt, veldt
real(kind=c_double) :: dconc(n-2,n-2)
!
! We want a compact calculation
!
conc_out = conc_in
diffdt = params%diffusion / params%deltax**2 * params%deltat
veldt = params%flow_velocity / params%deltax * params%deltat
!
! Using associate we can formulate the difference equation in a way that avoids
! explicit loops and boundary conditions.
! Note:
! It is not necessarily very efficient ;)
!
! By first copying the initial values and then using the output array, we
! avoid intermediate copying.
!
associate( ccentre => conc_out(2:n-1,2:n-1), &
cleft => conc_out(1:n-2,2:n-1), &
cright => conc_out(3:n ,2:n-1), &
cup => conc_out(2:n-1,3:n ), &
cdown => conc_out(2:n-1,1:n-2) )
do i = 1,params%nreport
dconc = veldt * ( cleft - ccentre ) + &
diffdt * ( cleft + cright + cup + cdown - 4.0_c_double * ccentre )
ccentre = ccentre + dconc
enddo
end associate
end subroutine update
end module advdiff
======
Note: I use the current Fortran 2008 standard, so some of the
constructs may not be familiar to everyone. You can create a DLL or
shared object using `gfortran` with this command:
======none
gfortran -o advdiff.dll advdiff.f90 -shared
======
The routine `update` will update the incoming matrix `conc_in` and
put the result in the outgoing matrix `conc_out`. To avoid unnecessary
conversion steps, it does a number of steps before returning. This
number is contained in the derived type `params` as `nreport`.
So far the Fortran side of things. Let us move on to the Tcl side:
* Load the DLL or shared object via the `::cffi::Wrapper` command
* Define a convenient `cffi` structure for the derived type (C struct) to hold the numerical parameters
* Define the interface of the function
This is done in the first few lines of the Tcl program. The rest is
merely to:
* Set up the matrix
* Define a print procedure - `cffi` only knows about linear (non-nested) lists, so use that as the data structure
* Do the time loop
And so we have the following Tcl program:
======tcl
# model_advdiff.tcl --
# Demonstration of cffi used on an advection-diffusion problem
#set auto_path [concalist [file dirname [file dirname [file normalize [
file join [info script] ...]]]] {*}$auto_path]
package require cffi
::cffi::Wrapper create advdiff advdiff.dll
::cffi::Struct create PARAMS {diffusion double flow_velocity double deltax double deltat double nreport int}
advdiff function update void {n int nsq int conc_in double[nsq] conc_out {double[nsq] out} params {struct.PARAMS byref} }
set params [dict create diffusion 0.1 flow_velocity 0.1 deltax 0.1 deltat 0.02 nreport 10]
#
# Set up the model area: 12x12, the outer edges are the boundaries
# Boundary: concentration = 0
# Interior: concentration = 1
#
set n 12
set nsq [expr {$n**2}]
set row [lrepeat $n 0.0]
set conc_in [lrepeat $n $row]
for {set row 1} {$row < $n-1} {incr row} {
for {set col 1} {$col < $n-1} {incr col} {
lset conc_in $row $col 1.0
}
}
#
# cffi only support one-dimension arrays ...
#
set conc_in [concat {*}$conc_in]
# print_matrix --
# Print procedure
#
# Arguments:
# outfile File to print to
# n The list represents a matrix of n by n
# values List of n x n values, as a single long list
#
proc print_matrix {outfile n values} {
set i 0
for {set row 0} {$row < $n} {incr row} {
set line ""
for {set col 0} {$col < $n} {incr col} {
append line [format "%8.4f" [lindex $values $i]]
incr i
}
puts $outfile $line
}
}
# Run the calculation
set outfile [open "model_advdiff.out" w]
puts $outfile "Initial:"
print_matrix $outfile $n $conc_in
for {set time 0} {$time < 10} {incr time} {
puts $outfile "\nTime: [expr {$time+1}]"
update $n $nsq $conc_in conc_out $params
print_matrix $outfile $n $conc_out
set conc_in $conc_out
}
======
There are a few things to note:
* As `cffi` knows nothing about n-dimensional arrays, I had to include a dummy argument `nsq` to define a matrix.
* Care must be taken about the precise style of the arguments - pass by value or pass by reference etc. This is not always easy, especially with C pointers having very different roles without the syntax per se revealing that role. With Fortran similar issues appear.
* I have been careful to use the C-Fortran interfacing features as per the Fortran 2003 standard. This allowed me to avoid all interfacing via C.
* There is a quirk in my Fortran routine that I had to work around - see the comments. I do not entirely understand why this is required and that bugs me ;)
Just for fun, here is the output:
======none
Initial:
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000
0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000
0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000
0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000
0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000
0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000
0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000
0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000
0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000
0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Time: 1
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.1226 0.2296 0.3045 0.3462 0.3639 0.3662 0.3547 0.3217 0.2546 0.1454 0.0000
0.0000 0.2210 0.4137 0.5479 0.6223 0.6535 0.6576 0.6372 0.5784 0.4583 0.2619 0.0000
0.0000 0.2843 0.5316 0.7030 0.7973 0.8365 0.8416 0.8159 0.7414 0.5881 0.3364 0.0000
0.0000 0.3166 0.5915 0.7811 0.8848 0.9275 0.9329 0.9049 0.8230 0.6537 0.3742 0.0000
0.0000 0.3288 0.6139 0.8101 0.9169 0.9607 0.9663 0.9375 0.8531 0.6780 0.3883 0.0000
0.0000 0.3288 0.6139 0.8101 0.9169 0.9607 0.9663 0.9375 0.8531 0.6780 0.3883 0.0000
0.0000 0.3166 0.5915 0.7811 0.8848 0.9275 0.9329 0.9049 0.8230 0.6537 0.3742 0.0000
0.0000 0.2843 0.5316 0.7030 0.7973 0.8365 0.8416 0.8159 0.7414 0.5881 0.3364 0.0000
0.0000 0.2210 0.4137 0.5479 0.6223 0.6535 0.6576 0.6372 0.5784 0.4583 0.2619 0.0000
0.0000 0.1226 0.2296 0.3045 0.3462 0.3639 0.3662 0.3547 0.3217 0.2546 0.1454 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Time: 2
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0624 0.1226 0.1736 0.2109 0.2324 0.2377 0.2261 0.1965 0.1480 0.0813 0.0000
0.0000 0.1177 0.2313 0.3275 0.3977 0.4383 0.4482 0.4263 0.3705 0.2791 0.1533 0.0000
0.0000 0.1609 0.3160 0.4474 0.5433 0.5986 0.6121 0.5822 0.5061 0.3813 0.2095 0.0000
0.0000 0.1896 0.3724 0.5271 0.6400 0.7050 0.7208 0.6856 0.5960 0.4491 0.2468 0.0000
0.0000 0.2037 0.4001 0.5662 0.6874 0.7572 0.7740 0.7362 0.6401 0.4823 0.2651 0.0000
0.0000 0.2037 0.4001 0.5662 0.6874 0.7572 0.7740 0.7362 0.6401 0.4823 0.2651 0.0000
0.0000 0.1896 0.3724 0.5271 0.6400 0.7050 0.7208 0.6856 0.5960 0.4491 0.2468 0.0000
0.0000 0.1609 0.3160 0.4474 0.5433 0.5986 0.6121 0.5822 0.5061 0.3813 0.2095 0.0000
0.0000 0.1177 0.2313 0.3275 0.3977 0.4383 0.4482 0.4263 0.3705 0.2791 0.1533 0.0000
0.0000 0.0624 0.1226 0.1736 0.2109 0.2324 0.2377 0.2261 0.1965 0.1480 0.0813 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Time: 3
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0398 0.0793 0.1146 0.1423 0.1597 0.1650 0.1571 0.1358 0.1014 0.0553 0.0000
0.0000 0.0760 0.1515 0.2189 0.2718 0.3050 0.3151 0.3001 0.2594 0.1937 0.1056 0.0000
0.0000 0.1056 0.2105 0.3042 0.3776 0.4237 0.4378 0.4170 0.3604 0.2691 0.1467 0.0000
0.0000 0.1264 0.2519 0.3641 0.4520 0.5071 0.5239 0.4991 0.4313 0.3220 0.1756 0.0000
0.0000 0.1371 0.2732 0.3949 0.4901 0.5500 0.5682 0.5412 0.4678 0.3492 0.1904 0.0000
0.0000 0.1371 0.2732 0.3949 0.4901 0.5500 0.5682 0.5412 0.4678 0.3492 0.1904 0.0000
0.0000 0.1264 0.2519 0.3641 0.4520 0.5071 0.5239 0.4991 0.4313 0.3220 0.1756 0.0000
0.0000 0.1056 0.2105 0.3042 0.3776 0.4237 0.4378 0.4170 0.3604 0.2691 0.1467 0.0000
0.0000 0.0760 0.1515 0.2189 0.2718 0.3050 0.3151 0.3001 0.2594 0.1937 0.1056 0.0000
0.0000 0.0398 0.0793 0.1146 0.1423 0.1597 0.1650 0.1571 0.1358 0.1014 0.0553 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Time: 4
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0272 0.0544 0.0792 0.0992 0.1121 0.1166 0.1115 0.0965 0.0721 0.0393 0.0000
0.0000 0.0520 0.1043 0.1518 0.1901 0.2149 0.2234 0.2137 0.1850 0.1382 0.0753 0.0000
0.0000 0.0726 0.1455 0.2119 0.2653 0.3000 0.3119 0.2982 0.2582 0.1928 0.1051 0.0000
0.0000 0.0873 0.1749 0.2547 0.3188 0.3605 0.3748 0.3584 0.3103 0.2317 0.1263 0.0000
0.0000 0.0949 0.1902 0.2769 0.3466 0.3920 0.4075 0.3897 0.3374 0.2519 0.1373 0.0000
0.0000 0.0949 0.1902 0.2769 0.3466 0.3920 0.4075 0.3897 0.3374 0.2519 0.1373 0.0000
0.0000 0.0873 0.1749 0.2547 0.3188 0.3605 0.3748 0.3584 0.3103 0.2317 0.1263 0.0000
0.0000 0.0726 0.1455 0.2119 0.2653 0.3000 0.3119 0.2982 0.2582 0.1928 0.1051 0.0000
0.0000 0.0520 0.1043 0.1518 0.1901 0.2149 0.2234 0.2137 0.1850 0.1382 0.0753 0.0000
0.0000 0.0272 0.0544 0.0792 0.0992 0.1121 0.1166 0.1115 0.0965 0.0721 0.0393 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Time: 5
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0190 0.0381 0.0556 0.0699 0.0793 0.0828 0.0794 0.0689 0.0515 0.0281 0.0000
0.0000 0.0364 0.0730 0.1066 0.1340 0.1522 0.1587 0.1523 0.1321 0.0988 0.0539 0.0000
0.0000 0.0508 0.1020 0.1490 0.1872 0.2126 0.2218 0.2128 0.1846 0.1381 0.0753 0.0000
0.0000 0.0611 0.1227 0.1793 0.2253 0.2558 0.2669 0.2560 0.2221 0.1661 0.0906 0.0000
0.0000 0.0665 0.1335 0.1951 0.2451 0.2783 0.2903 0.2785 0.2417 0.1807 0.0986 0.0000
0.0000 0.0665 0.1335 0.1951 0.2451 0.2783 0.2903 0.2785 0.2417 0.1807 0.0986 0.0000
0.0000 0.0611 0.1227 0.1793 0.2253 0.2558 0.2669 0.2560 0.2221 0.1661 0.0906 0.0000
0.0000 0.0508 0.1020 0.1490 0.1872 0.2126 0.2218 0.2128 0.1846 0.1381 0.0753 0.0000
0.0000 0.0364 0.0730 0.1066 0.1340 0.1522 0.1587 0.1523 0.1321 0.0988 0.0539 0.0000
0.0000 0.0190 0.0381 0.0556 0.0699 0.0793 0.0828 0.0794 0.0689 0.0515 0.0281 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Time: 6
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0133 0.0268 0.0392 0.0494 0.0562 0.0588 0.0565 0.0491 0.0368 0.0201 0.0000
0.0000 0.0256 0.0515 0.0753 0.0948 0.1079 0.1128 0.1084 0.0942 0.0705 0.0385 0.0000
0.0000 0.0358 0.0719 0.1052 0.1325 0.1508 0.1576 0.1515 0.1317 0.0986 0.0538 0.0000
0.0000 0.0431 0.0866 0.1267 0.1595 0.1815 0.1897 0.1823 0.1585 0.1187 0.0648 0.0000
0.0000 0.0469 0.0942 0.1378 0.1735 0.1974 0.2064 0.1984 0.1724 0.1291 0.0705 0.0000
0.0000 0.0469 0.0942 0.1378 0.1735 0.1974 0.2064 0.1984 0.1724 0.1291 0.0705 0.0000
0.0000 0.0431 0.0866 0.1267 0.1595 0.1815 0.1897 0.1823 0.1585 0.1187 0.0648 0.0000
0.0000 0.0358 0.0719 0.1052 0.1325 0.1508 0.1576 0.1515 0.1317 0.0986 0.0538 0.0000
0.0000 0.0256 0.0515 0.0753 0.0948 0.1079 0.1128 0.1084 0.0942 0.0705 0.0385 0.0000
0.0000 0.0133 0.0268 0.0392 0.0494 0.0562 0.0588 0.0565 0.0491 0.0368 0.0201 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Time: 7
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0094 0.0190 0.0278 0.0350 0.0399 0.0418 0.0402 0.0350 0.0262 0.0143 0.0000
0.0000 0.0181 0.0364 0.0533 0.0672 0.0765 0.0801 0.0771 0.0671 0.0503 0.0275 0.0000
0.0000 0.0253 0.0509 0.0745 0.0939 0.1070 0.1120 0.1078 0.0938 0.0703 0.0384 0.0000
0.0000 0.0304 0.0612 0.0897 0.1130 0.1288 0.1348 0.1297 0.1128 0.0846 0.0462 0.0000
0.0000 0.0331 0.0666 0.0976 0.1230 0.1401 0.1467 0.1411 0.1228 0.0920 0.0503 0.0000
0.0000 0.0331 0.0666 0.0976 0.1230 0.1401 0.1467 0.1411 0.1228 0.0920 0.0503 0.0000
0.0000 0.0304 0.0612 0.0897 0.1130 0.1288 0.1348 0.1297 0.1128 0.0846 0.0462 0.0000
0.0000 0.0253 0.0509 0.0745 0.0939 0.1070 0.1120 0.1078 0.0938 0.0703 0.0384 0.0000
0.0000 0.0181 0.0364 0.0533 0.0672 0.0765 0.0801 0.0771 0.0671 0.0503 0.0275 0.0000
0.0000 0.0094 0.0190 0.0278 0.0350 0.0399 0.0418 0.0402 0.0350 0.0262 0.0143 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Time: 8
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0067 0.0134 0.0197 0.0248 0.0283 0.0297 0.0286 0.0249 0.0186 0.0102 0.0000
0.0000 0.0128 0.0258 0.0378 0.0476 0.0543 0.0569 0.0548 0.0477 0.0358 0.0195 0.0000
0.0000 0.0179 0.0360 0.0528 0.0666 0.0759 0.0795 0.0766 0.0667 0.0500 0.0273 0.0000
0.0000 0.0216 0.0434 0.0636 0.0802 0.0914 0.0957 0.0922 0.0803 0.0602 0.0329 0.0000
0.0000 0.0235 0.0472 0.0692 0.0872 0.0994 0.1042 0.1003 0.0873 0.0655 0.0358 0.0000
0.0000 0.0235 0.0472 0.0692 0.0872 0.0994 0.1042 0.1003 0.0873 0.0655 0.0358 0.0000
0.0000 0.0216 0.0434 0.0636 0.0802 0.0914 0.0957 0.0922 0.0803 0.0602 0.0329 0.0000
0.0000 0.0179 0.0360 0.0528 0.0666 0.0759 0.0795 0.0766 0.0667 0.0500 0.0273 0.0000
0.0000 0.0128 0.0258 0.0378 0.0476 0.0543 0.0569 0.0548 0.0477 0.0358 0.0195 0.0000
0.0000 0.0067 0.0134 0.0197 0.0248 0.0283 0.0297 0.0286 0.0249 0.0186 0.0102 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Time: 9
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0047 0.0095 0.0140 0.0176 0.0201 0.0211 0.0203 0.0177 0.0132 0.0072 0.0000
0.0000 0.0091 0.0183 0.0268 0.0338 0.0386 0.0404 0.0389 0.0339 0.0254 0.0139 0.0000
0.0000 0.0127 0.0256 0.0375 0.0473 0.0539 0.0565 0.0544 0.0474 0.0355 0.0194 0.0000
0.0000 0.0153 0.0308 0.0451 0.0569 0.0649 0.0680 0.0655 0.0570 0.0428 0.0234 0.0000
0.0000 0.0166 0.0335 0.0490 0.0619 0.0706 0.0740 0.0713 0.0621 0.0465 0.0254 0.0000
0.0000 0.0166 0.0335 0.0490 0.0619 0.0706 0.0740 0.0713 0.0621 0.0465 0.0254 0.0000
0.0000 0.0153 0.0308 0.0451 0.0569 0.0649 0.0680 0.0655 0.0570 0.0428 0.0234 0.0000
0.0000 0.0127 0.0256 0.0375 0.0473 0.0539 0.0565 0.0544 0.0474 0.0355 0.0194 0.0000
0.0000 0.0091 0.0183 0.0268 0.0338 0.0386 0.0404 0.0389 0.0339 0.0254 0.0139 0.0000
0.0000 0.0047 0.0095 0.0140 0.0176 0.0201 0.0211 0.0203 0.0177 0.0132 0.0072 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Time: 10
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0034 0.0068 0.0099 0.0125 0.0143 0.0150 0.0144 0.0125 0.0094 0.0051 0.0000
0.0000 0.0064 0.0130 0.0190 0.0240 0.0274 0.0287 0.0276 0.0241 0.0181 0.0099 0.0000
0.0000 0.0090 0.0181 0.0266 0.0335 0.0383 0.0401 0.0386 0.0337 0.0253 0.0138 0.0000
0.0000 0.0108 0.0218 0.0320 0.0404 0.0460 0.0483 0.0465 0.0405 0.0304 0.0166 0.0000
0.0000 0.0118 0.0237 0.0348 0.0439 0.0501 0.0525 0.0506 0.0441 0.0331 0.0181 0.0000
0.0000 0.0118 0.0237 0.0348 0.0439 0.0501 0.0525 0.0506 0.0441 0.0331 0.0181 0.0000
0.0000 0.0108 0.0218 0.0320 0.0404 0.0460 0.0483 0.0465 0.0405 0.0304 0.0166 0.0000
0.0000 0.0090 0.0181 0.0266 0.0335 0.0383 0.0401 0.0386 0.0337 0.0253 0.0138 0.0000
0.0000 0.0064 0.0130 0.0190 0.0240 0.0274 0.0287 0.0276 0.0241 0.0181 0.0099 0.0000
0.0000 0.0034 0.0068 0.0099 0.0125 0.0143 0.0150 0.0144 0.0125 0.0094 0.0051 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
======
<<categories>> Foreign interfaces | Example | FORTRAN | Numerical analysis