Problem with calling C++ from Fortran

db

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Sep 9, 2023, 4:54:53 AM9/9/23

to

I want to call a C++ function from Fortran. The function is
called jband, in the file jband.cpp. The function is of type
long double, and so is its argument tbar.
I wrote this:

module STUFF
integer, parameter :: dbl=selected_real_kind(14)
integer, parameter :: qud=selected_real_kind(30)
end module STUFF
program JBAND_TEST
use STUFF; implicit none
real(dbl) :: curr, FORTJBAND
real(qud) :: tbar
do
read *, tbar
if (tbar < 0) exit
curr = FORTJBAND (tbar)
print '(" curr =", f10.4)', curr
enddo
end program JBAND_TEST

function FORTJBAND (tbar)
use STUFF; implicit none
interface
function JBAND (tbar) bind(c)
import JBAND, c_long_double
import tbar, c_long_double
end function JBAND
end interface
real(dbl) :: FORTJBAND
real(qud) :: tbar
FORTJBAND = JBAND (tbar)
end function FORTJBAND

I run this using this script

g++ -o jband.o jband.cpp
gfortran -o jbandtest.o jbandtest.f90
gfortran jband.o jbandtest.o
./a.out << eoi
1.0
-1
eoi
rm a.out jband.o jbandtest.o

and I get this output

/usr/bin/ld:
/usr/lib/gcc/x86_64-linux-gnu/11/../../../x86_64-linux-gnu/Scrt1.o: in
function `_start':
(.text+0x1b): undefined reference to `main'
collect2: error: ld returned 1 exit status
jbandtest.f90:21:31:

21 | import JBAND, c_long_double
| 1
Error: Cannot IMPORT 'c_long_double' from host scoping unit at (1) -
does not exist.
jbandtest.f90:22:30:

22 | import tbar, c_long_double
| 1
Error: Cannot IMPORT 'c_long_double' from host scoping unit at (1) -
does not exist.
jbandtest.f90:27:14:

27 | FORTJBAND = JBAND (tbar)
| 1
Error: Type mismatch in argument 'tbar' at (1); passed REAL(16) to REAL(4)
[...]

So there is a communication problem with agreeing on the C++
type long double.
I have tried making everything in jband plain double, making
the two statements

import JBAND, c_double
import tbar, c_double

but now it says that C-double doesn't exist.

Any ideas?

--
Dieter Britz

Mut...@dastardlyhq.com

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Sep 9, 2023, 5:02:28 AM9/9/23

to

On Sat, 9 Sep 2023 10:54:33 +0200
db <dieterh...@gmail.com> wrote:
>but now it says that C-double doesn't exist.
>
>Any ideas?

C++ mangles function names so that the name you give the function isn't the
name the dynamic loader sees in the binary. You need to put this around whatever
function you want to call from Fortran so it has C style linkage which means
the name remains the same:

extern "C" {

void calledFromFortran()
{
:
}

}

db

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Sep 9, 2023, 6:47:58 AM9/9/23

to

Like this:

const long double iband(const long double tbar)
extern "C" {

void calledFromFortran()
{
:
}

}

? Why the extra } ?
--
Dieter Britz

Mut...@dastardlyhq.com

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Sep 9, 2023, 6:52:09 AM9/9/23

to

On Sat, 9 Sep 2023 12:47:38 +0200
db <dieterh...@gmail.com> wrote:
>On 09.09.2023 11.02, Mut...@dastardlyhq.com wrote:
>> On Sat, 9 Sep 2023 10:54:33 +0200
>> db <dieterh...@gmail.com> wrote:
>>> but now it says that C-double doesn't exist.
>>>
>>> Any ideas?
>>
>> C++ mangles function names so that the name you give the function isn't the
>> name the dynamic loader sees in the binary. You need to put this around
>whatever
>> function you want to call from Fortran so it has C style linkage which means
>> the name remains the same:
>>
>> extern "C" {
>>
>> void calledFromFortran()
>> {
>> :
>> }
>>
>> }
>>
>
>Like this:

No. calledFromFortran was an example function.

extern "C" {

const long double iband(const long double tbar)

{
:
}

}

>? Why the extra } ?

What extra } ? The brackets balance.

db

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Sep 9, 2023, 6:53:00 AM9/9/23

to

Now I get a new error, due to this "extern"

~/ownlib90/tests> ./jbandtest
jband.cpp:88:1: error: expected initializer before 'extern'
88 | extern "C" {
| ^~~~~~
jband.cpp:96:1: error: expected unqualified-id before '{' token
96 | {
| ^

jbandtest.f90:21:31:

21 | import JBAND, c_long_double
| 1
Error: Cannot IMPORT 'c_long_double' from host scoping unit at (1) -
does not exist.

[...]

--
Dieter Britz

Mut...@dastardlyhq.com

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Sep 9, 2023, 7:00:30 AM9/9/23

to

If your knowledge of C++ syntax is so basic you don't know which bits go
where I'd suggest getting someone else to write the C++ code and you stick
to fortran.

Paavo Helde

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Sep 9, 2023, 9:56:29 AM9/9/23

to

Assuming you are indeed compiling this as C++ as intended, the problem
must be with the preceding content.

> jbandtest.f90:21:31:
>
> 21 | import JBAND, c_long_double
> | 1
> Error: Cannot IMPORT 'c_long_double' from host scoping unit at (1) -
> does not exist.
> [...]

This is a fortran error message. It's not very likely you will get any
help about this here, most people here do not know anything about fortran.

Alf P. Steinbach

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Sep 9, 2023, 10:30:53 AM9/9/23

to

On 2023-09-09 12:47 PM, db wrote:
> On 09.09.2023 11.02, Mut...@dastardlyhq.com wrote:
>> On Sat, 9 Sep 2023 10:54:33 +0200
>> db <dieterh...@gmail.com> wrote:
>>> but now it says that C-double doesn't exist.
>>>
>>> Any ideas?
>>
>> C++ mangles function names so that the name you give the function
>> isn't the
>> name the dynamic loader sees in the binary. You need to put this
>> around whatever
>> function you want to call from Fortran so it has C style linkage which
>> means
>> the name remains the same:
>>
>> extern "C" {
>>
>> void calledFromFortran()
>> {
>> :
>> }
>>
>> }
>>
>
> Like this:
>
> const long double iband(const long double tbar)

Why did you add that?

You could at least try to do /exactly/ the example given by a
knowledgeable person.

db

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Sep 10, 2023, 8:33:45 AM9/10/23

to

An unhelpful comment..
The C++ function was written by an experienced C++
programmer, and my problem is calling it from Fortran.
OK, I missed the extra opening {. The problem lies
in the communication between Fortran and C++.
--
Dieter Britz

db

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Sep 10, 2023, 8:36:36 AM9/10/23

to

I have tried that but so far no one was able to help there,
so I thought I'd try at the other end. No go here either.
Ar well, I might have to myself translate the C++ code into
Fortran. I have done that with another routine and it was
quite laborious.
--
Dieter Britz

db

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Sep 10, 2023, 8:48:24 AM9/10/23

to

On 09.09.2023 11.02, Mut...@dastardlyhq.com wrote:

Where should this go in the code? I added it at the end as is,
and now I get this:

~/ownlib90/tests> ./jbandtest
jband.cpp:88:1: error: expected initializer before 'extern'
88 | extern "C" {
| ^~~~~~
jband.cpp:96:1: error: expected unqualified-id before '{' token
96 | {
| ^

jband.cpp: In function 'void calledFromFortran()':
jband.cpp:350:9: error: expected primary-expression before ':' token
350 | :
...

--
Dieter Britz

Ben Bacarisse

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Sep 10, 2023, 9:22:28 AM9/10/23

to

You are giving people very little to go on. There's an unknown command
(./jbandtest) the gives some errors where only one line of code can be
seen.

Can you produce a cut-down version to the program that shows the same
error -- mimimal example of the C++ and of the Fortran that should
calls it but which fails? That would mean you could post the code.

If that is not possible, can you link to the source so those who know
both C++ and Fortran can take a proper look?

--
Ben.

Bo Persson

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Sep 10, 2023, 10:08:24 AM9/10/23

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I wild guess here is that the line *before* this is where something is
missing. The message is just that 'extern' was not expected here, but
something else.

Michael S

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Sep 10, 2023, 11:38:40 AM9/10/23

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That's command is wrong. Should be:
g++ -c -o jband.o jband.cpp

Paavo Helde

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Sep 10, 2023, 11:49:17 AM9/10/23

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10.09.2023 15:33 db kirjutas:

> The C++ function was written by an experienced C++
> programmer, and my problem is calling it from Fortran.
> OK, I missed the extra opening {. The problem lies
> in the communication between Fortran and C++.

From C++ side it's necessary to declare and compile the function extern
"C" and it must be exported from the resulting shared library (by
__declspec(dllexport) or __attribute__ ((visibility ("default"))),
depending on the platform).

For sure an experienced C++ programmer is able to do that easily. After
that, the ball is on the Fortran side.

Ben Bacarisse

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Sep 10, 2023, 12:27:58 PM9/10/23

to

I see you have indeed posted half of it. If you post the C++ source,
someone might be able to help with both parts. I can tell you a bit
about what's wrong with the Fortran you posted, but I need to see the
other half.

--
Ben.

Michael S

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Sep 10, 2023, 12:51:56 PM9/10/23

to

On Saturday, September 9, 2023 at 11:54:53 AM UTC+3, db wrote:

I manged to pass link. 99% of the problems appear to have nothing to do
with C++ language. It's partly bad Fortran, partly wrong size of floating-point
types and partly forgotten -c flag in compilation commands.

That's my Fortran (f90) source code:

module STUFF
integer, parameter :: dbl=selected_real_kind(14)
integer, parameter :: qud=selected_real_kind(30)
end module STUFF

module abc
use, intrinsic :: iso_c_binding

interface
function JBAND (tbar) bind(c)

import c_long_double
real(c_long_double) :: JBAND
real(c_long_double) :: tbar

end function JBAND
end interface

end module abc

function FORTJBAND (tbar)
use STUFF;

use abc;
implicit none
real(dbl) :: FORTJBAND
real(qud) :: tbar
real(10) :: tbar10
real(10) :: res10
tbar10 = tbar
res10 = JBAND (tbar10)
FORTJBAND = res10
end function FORTJBAND

program JBAND_TEST
use STUFF;
implicit none
real(dbl) :: curr, FORTJBAND
real(qud) :: tbar
do
read *, tbar
if (tbar < 0) exit
curr = FORTJBAND (tbar)
print '(" curr =", f10.4)', curr
enddo
end program JBAND_TEST

That's my C++ source code (1 line)
extern "C" long double jband(long double x) { return x; }

That's my compilation commands:

g++ -c -o jband.o jband.cpp

gfortran -c -o jbandtest.o jbandtest.f90
gfortran jband.o jbandtest.o

Now I have no idea if it works like intended or not

James Kuyper

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Sep 11, 2023, 12:14:46 AM9/11/23

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While "C" and "C++" are the only language linkages for which support is
mandated by the standard, the description of language linkage is clearly
intended to allow support for a wider range of language linkages. Could
anyone tell me which implementatins, if any, support language linkages
other than "C" and C++"? If so, what languages do they support? How well
does the feature work? Is there any significant amount of use relying on
that feature?
In particular, and relevant to this thread, does any implementation
support "Fortran" language linkage?

Mut...@dastardlyhq.com

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Sep 11, 2023, 4:23:41 AM9/11/23

to

On Sun, 10 Sep 2023 14:33:26 +0200

If you're having trouble with this kind of basic stuff I think you're going
to have a lot more problems down the line with this.

If you're in contact with the guy who wrote the C++ code I would suggest
talking to him about it.

Michael S

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Sep 11, 2023, 5:49:40 AM9/11/23

to

Tested it. It was incorrect, but by lucky chance worked on x86-64 Windows.
No such luck on x86-64 Linux.
Here is the correct syntax that works both on Windows and on Linux:

module abc
use, intrinsic :: iso_c_binding
interface

real(c_long_double) function JBAND (tbar) bind(c)
import c_long_double
real(c_long_double), value :: tbar

end function JBAND
end interface
end module abc

I actually understand why the previous version worked on Windows
despite being incorrect, but I don't believe that explanation is of
interest for OP.

db

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Sep 11, 2023, 8:00:26 AM9/11/23

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I put the C++ function here: www.dieterbritz.dk/jband.cpp
After getting errors about the extern lines, whch I had
added at the end, I shifted it to near the start:

#include <iostream>
#include <fstream>
#include <iomanip>
#include "math.h"

using namespace std;

extern "C" {

void calledFromFortran()
{
:
}

}

...

I still get errors pertaining to these lines. But after adding
the -c to the two compilation lines in the script, these are
now the only errors. I think I am getting closer. The errors
are now

jband.cpp: In function 'void calledFromFortran()':

jband.cpp:12:9: error: expected primary-expression before ':' token
12 | :
| ^
jband.cpp: At global scope:
jband.cpp:96:1: error: expected initializer before 'extern'
96 | extern "C" {
| ^~~~~~
jband.cpp:104:1: error: expected unqualified-id before '{' token
104 | {
| ^
jbandtest.f90:28:14:

28 | FORTJBAND = JBAND (tbar)
| 1
Error: Type mismatch in argument 'tbar' at (1); passed REAL(16) to REAL(10)

The last error will go away together with the others.

--
Dieter Britz

https://psychedelicminds.shop/

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Sep 11, 2023, 8:14:34 AM9/11/23

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Michael S

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Sep 11, 2023, 8:21:27 AM9/11/23

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You have got a special talent for messing up things. :(
Please, put the code as written by your C++ specialist on your web site.
As is, without modifications!
Then we can show you where to insert ' extern "C" '.

> jbandtest.f90:28:14:
>
> 28 | FORTJBAND = JBAND (tbar)
> | 1
> Error: Type mismatch in argument 'tbar' at (1); passed REAL(16) to REAL(10)
>

FORTJBAND = real(JBAND(Real(tbar, c_long_double)), dbl)

Mut...@dastardlyhq.com

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Sep 11, 2023, 9:33:34 AM9/11/23

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On Mon, 11 Sep 2023 14:00:06 +0200
db <dieterh...@gmail.com> wrote:
>jband.cpp: In function 'void calledFromFortran()':

I told you before that calledFromFortran() was simply an example function
name I chose. I could have called it deathToTurnips(). You need to use the
prototype of the function you're going to call.

>jband.cpp:12:9: error: expected primary-expression before ':' token
> 12 | :

FFS, that was just a space to indicate code goes there, its not C++ syntax!

You need to talk to your C++ guy, your knowledge of C/C++ is clearly
non existent so its pointless you trying to do something like this which can
have many gotchas beyond simple compilation.

Ben Bacarisse

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Sep 11, 2023, 7:26:49 PM9/11/23

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Remove them all. You have two options. One: correct the C++. To do
this put this, and only this,

extern "C" {
const long double iband(const long double tbar);
}

just after the #include lines. This is perfect (the first const is a
meaningless word here and the compiler may warn you about that) but it
will do.

Next replace

#include "math.h"

with

#include <math.h>

I'd also remove the two unnecessary #include lines (for iomanip and
fstream). All of these are "tidying up". Only the extern "C" part I
gave corrects an actual mistake.

The second option is the turn the code in C which links directly with
Fortran. The only C++ used by the code is one "cout << " line (not a
good way to report an error, but that's not important right now) and a
setting for pi that is not a compile-time constant (4*atanl(1)) written
in a more fussy way.

To convert to C, copy the file to jband.c (rather than .cpp) and replce
the declaration for pi. Given the code is littered with explicit 21
digits decimals (a little more than the precision of C's usualy long
double type) I would just write

static const long double pi = 3.14159265358979323846L;

Now replace the cout line with

fputs("\nError in stredges::iband(): tbar must be > 0", stderr);

This writes the message to the error stream and put only these two
include lines at the top:

#include <math.h>
#include <stdio.h>

Now the file is C code.

Frankly, this is the way I'd go, since linking C is just that bit
simpler than linking C++. But if the code is going to become fully
fledged (and complex) C++, obviously use my first option.

> I still get errors pertaining to these lines. But after adding
> the -c to the two compilation lines in the script, these are
> now the only errors. I think I am getting closer. The errors
> are now

Yes, you must compile it like this:

g++ -c jband.cpp (for C++ code)
gcc -c jband.c (for C code)

You don't need any -o but you could add in lots of warnings like

gcc -Wall -Wextra -c jband.c

> jband.cpp: In function 'void calledFromFortran()':
> jband.cpp:12:9: error: expected primary-expression before ':' token
> 12 | :
> | ^
> jband.cpp: At global scope:
> jband.cpp:96:1: error: expected initializer before 'extern'
> 96 | extern "C" {
> | ^~~~~~
> jband.cpp:104:1: error: expected unqualified-id before '{' token
> 104 | {
> | ^
> jbandtest.f90:28:14:
>
> 28 | FORTJBAND = JBAND (tbar)
> | 1
> Error: Type mismatch in argument 'tbar' at (1); passed REAL(16) to REAL(10)
>
> The last error will go away together with the others.

Michael S has given you one way to go for that. My comments were going
to be more general. You have several ways of specifying the types, and
I could not tell if you really wanted all of them. In general, I would
define the Fortran types from the C ones (since those are the types you
need for the call and return) and leave it at that.

But the big one is that the Fortran code seems to think the function is
called jband, when in fact it's called iband.

I went for this:

module STUFF
use iso_c_binding
integer, parameter :: qud=real(c_long_double)

end module STUFF
program JBAND_TEST
use STUFF; implicit none

real(qud) :: tbar, curr, FORTJBAND

do
read *, tbar
if (tbar < 0) exit
curr = FORTJBAND (tbar)
print '(" curr =", f10.4)', curr
enddo
end program JBAND_TEST

function FORTJBAND (tbar)
use STUFF; implicit none
interface

function IBAND (tbar) bind(c)
use STUFF
real(qud), VALUE :: tbar
real(qud) :: IBAND
end function iband
end interface
real(qud) :: FORTJBAND, tbar
FORTJBAND = IBAND(tbar)
end function FORTJBAND

Note that you can compile the Fortran along with the C at the same time
if you like:

$ gfortran -o jtest jbandtest.f90 jband.c -lm
$ ./jtest
1
curr = 1.5224
1e-10
curr =56419.9584
-1
$

($ is my Linux command-line prompt). I have no idea if these are
correct results and I can't be sure what you need to run on your system,
but the code links and appears to work.

If you go with the C++ option, the linking needs the standard C++
library:

$ gfortran -o jtest jbandtest.f90 jband.cpp -lstdc++ -lm
$ ./jtest
1
curr = 1.5224
1e-10
curr =56419.9584
-1
$

BTW, you said the C++ was written by a C++ expert. I don't think it
was!

--
Ben.

db

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Sep 12, 2023, 4:48:55 AM9/12/23

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The code, apart from the extra lines suggested by someone here,
but without the information on where to put them, is runnable
C++ code written by another person who knows C++ very well. I
myself use Fortran and you are right, I know next to nothing
about C++. I did use the extra code exactly as is, but possibly
put it in the wrong place. I tried two, without success.
I am giving up, and will translate the C++ function into Fortran,
where I know what I'm doing.
--
Dieter Britz

Michael S

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Sep 12, 2023, 5:01:03 AM9/12/23

to

I think, you are taking chances here. Even if it happens to work
on x86-64 Linux, it can fail on some other target where calling
conventions for long double return value in 'C' differ from those
of Fortran. The method shown in one of my previous posts
appear more robust:

interface
real(c_long_double) function IBAND (tbar) bind(c)

import c_long_double
real(c_long_double), value :: tbar

end function IBAND

With no relationships to C, C++ or Fortran, I don't like the way they evaluate
their polynomials from numerical perspective . But I assume that algorithm
was copied 'as is' from 40 y.o. book so it's not a fault of C++ programmer.

Despite my disliking, their method is good enough to achieve their modest
specified goal for relative accuracy: of 14-15 significant digits.
With smarter method the same or better accuracy can be achieved using
standard 'double" arithmetic (IEEE-754 binary64) thus making the routine
portable to platforms that have no support for extended precision math.

>
> --
> Ben.

Ben Bacarisse

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Sep 12, 2023, 9:14:24 AM9/12/23

to

db <dieterh...@gmail.com> writes:

>> BTW, you said the C++ was written by a C++ expert. I don't think it
>> was!
>>
> The code, apart from the extra lines suggested by someone here,
> but without the information on where to put them, is runnable
> C++ code written by another person who knows C++ very well.

I was just saying that the code does not look like it was written by
someone who knows C++ very well. All those unnecessary duplicated
sizes. A const qualified function return type. Including "math.h"
rather that either <math.h> or <cmath>.

> I myself use Fortran and you are right, I know next to nothing about
> C++.

Did I say that? Sorry.

> I did use the extra code exactly as is, but possibly
> put it in the wrong place. I tried two, without success.
> I am giving up, and will translate the C++ function into Fortran,
> where I know what I'm doing.

OK. Odd to give up just as two people have given you working code. Oh
well...

Mind you, the C++ (or C) just picks an array and an initial value (an
'x' in a polynomial evaluation) and calculates a simple sum. It will be
trivial in Fortran. The hard work is in the coefficients and those can
just be coped.

--
Ben.

Ben Bacarisse

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Sep 12, 2023, 9:23:09 AM9/12/23

to

I think I see -- keep the C-related types in the smallest interface? I
didn't study your example, but the OP renamed the C function and gave
the resulting Fortran function a native Fortran type. Is that what you
advise?

The OP should have taken your Fortran and applied my C++ edits. Maybe
they took my Fortran and it did not work on Windows. That would be a
shame.

>> BTW, you said the C++ was written by a C++ expert. I don't think it
>> was!
>
> With no relationships to C, C++ or Fortran, I don't like the way they
> evaluate their polynomials from numerical perspective . But I assume
> that algorithm was copied 'as is' from 40 y.o. book so it's not a
> fault of C++ programmer.

I was just remarking on the C++ itself which has some 'non-expert'
attributes.

> Despite my disliking, their method is good enough to achieve their modest
> specified goal for relative accuracy: of 14-15 significant digits.
> With smarter method the same or better accuracy can be achieved using
> standard 'double" arithmetic (IEEE-754 binary64) thus making the routine
> portable to platforms that have no support for extended precision
> math.

Interesting. I don't know the function being evaluated at all.

--
Ben.

Michael S

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Sep 12, 2023, 10:15:18 AM9/12/23

to

It seems to me, you are giving up too early.
If I guessed you original code correctly then all you have to do
on C++ side is to replace that line:

const long double iband(const long double tbar)

by that line:
extern "C" long double iband(const long double tbar)

> and will translate the C++ function into Fortran,
> where I know what I'm doing.

Understanding what you are doing is valuable, yes.

> --
> Dieter Britz

Michael S

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Sep 12, 2023, 10:33:09 AM9/12/23

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I don't know it either. But I had drawn the graphs and looked at
coefficients in two ranges where precision is worst: [0.0081: 0.0158] and
[0.0158:0.0631].
A simple affine transform of the polynomial could either improve precision
of result by 4 orders of magnitude or reduce needs for precision of arithmetic
by 3 orders of magnitude while still somewhat improving the precision of result.
I'd guess, more advanced techniques can improve it further in both dimensions,
but low hanging fruits first.

> --
> Ben.

db

unread,

Sep 12, 2023, 12:36:32 PM9/12/23

to

The person who gave me the C++ code is an older experienced C++
programmer. Maybe he uses out of data C++ features but the
function works as is (without the extra lines given to me here,
which were meant to help with calling it from Fortran). He also
gave me a little C++ program to call it, and it produced
output that I was able to verify.

The problem is electrochemical, the current vs time at a flat band
electrode flush with an insulating plane, for which there is no
analytical solution, and the function gives highly accurate values
using piecewise polynomial fits. Actually, he is over the top with
the accuracy because in practice, comparing with experiments, 3-4
figures are all that we need but he likes to do the best that can
be done.

--
Dieter Britz

Michael S

unread,

Sep 19, 2023, 5:39:46 AM9/19/23

to

If all you need is 3-4 significant figures then you can use much simpler
calculation. Like that (pay attention, it's C rather than C++).

#include <math.h>
#include <stdio.h>

static double polyval(const double coef[], int order, double x);

double iband(double tbar)
{
if (tbar <= 0) {
printf("\nError in stredges::iband(): tbar must be > 0");
return 0;
}

// Coefficients of the polynomial approximation for tbar < 0.76
static const double c1[] = {
-2.0610398523438721606087449e-01,
+3.8436669168692465117185331e-01,
-2.0618169531912929413418346e-01,
-2.6474292774634270832496992e-02,
+3.1131226184858380642960328e-03,
+5.6415965971931841142037414e-01,
};
const int c1_order = sizeof(c1)/sizeof(c1[0])-1;

if (tbar < 0.76)
return polyval(c1, c1_order, tbar)/sqrt(tbar) + 1;

// Coefficients of the polynomial approximation for tbar >= 0.76
static const double c2[] = {
+1.3247948853085559402034700e+01,
-2.8936433794424133211569633e+01,
+2.2857148655763359956491203e+01,
-5.3743813763778698859386414e+00,
-3.5789894047006239277645653e+00,
+6.2827953597648754628587263e+00,
};
const int c2_order = sizeof(c2)/sizeof(c2[0])-1;
static const double c2_b2 = 3.00445175355660613529;
static const double c2_b1 = 1.93;

double log_t = log(tbar);
return polyval(c2, c2_order, 1/(log_t+c2_b1))/(log_t+c2_b2);
}

static double polyval(const double coef[], int order, double x)
{
double sum = coef[0];
for (int i = 0; i < order; ++i)
sum = sum * x + coef[i+1];
return sum;
}