BUG: Fix mistranslated expression in amos.h

[WarrenWeckesser]

Closes gh-46.

Thanks to github user hchau630 for identifying the problem and suggesting the fix.

[WarrenWeckesser]

I didn't add new unit tests. At the moment, I'm not sure what the recommended way to do that is here in xsf. Edit: tests added.

In the mean time, here's a C++ program that accepts the input parameters of cyl_bessel_k() on the command line and prints the result. It allows a quick comparison to the result computed with mpmath.

cyl_bessel_k_demo.cpp

// Requires at least C++17.

#include <complex>
#include <iomanip>
#include <iostream>
#include <xsf/bessel.h>

int main(int argc, char *argv[])
{
    using namespace std;

    double nu = 0.0;
    double re = 0.0;
    double im = 0.0;

    if (argc > 1) {
        nu = stod(argv[1]);
        if (argc > 2) {
            re = stod(argv[2]);
            if (argc > 3) {
                im = stod(argv[3]);
            }
        }
    }
    complex<double> z(re, im);
    complex<double> w = xsf::cyl_bessel_k(nu, z);

    cout << setprecision(15);
    cout << w.real() << " " << w.imag() << endl;
    return 0;
}

For example, with the changes in this PR:

% ./cyl_bessel_k_demo 0 680 680 
-4.55373003294427e-298 -1.87872701010989e-297
% python3 -c "from mpmath import mp; mp.dps = 100; z = 680+680j; w = complex(mp.besselk(0, z)); print(f'{w.real:.15e} {w.imag:.15e}')"
-4.553730032944803e-298 -1.878727010109855e-297
% 
% ./cyl_bessel_k_demo 0 680 1000
1.90168487199878e-298 -1.71341234147958e-297
% python3 -c "from mpmath import mp; mp.dps = 100; z = 680+1000j; w = complex(mp.besselk(0, z)); print(f'{w.real:.15e} {w.imag:.15e}')"
1.901684871999608e-298 -1.713412341479591e-297

[WarrenWeckesser]

After poking around some more, I see there are tests that don't rely on those parquet files. For example, @jorenham's pull request added test_lqmn.cpp. So I can put together a simple test for this change, and we can iterate on the preferred test idioms to use.

[WarrenWeckesser]

I added unit tests. Since these are new tests that are not related to old scipy tests, I put the test file in a new subdirectory, xsf_tests (instead of scipy_special_tests).

[WarrenWeckesser]

Windows failures are not new: #70.
The GPU job will be canceled in about 24 hours. ☹️