Mutually orthogonality of the wave functions - Armadillo - c++

Question

Not sure if this is the best place to ask this.
So I'm trying to study the orthogonality of the wave function solutions by calculating the Integral of the product of two solutions of different orders m and n. Now I get to the part where I have to do the product of 2 Hermite matrices of different dimensions, which I can't mathematically perform, one being 3x20 and the other one 4x20. Is there a way around this?
arma::mat Orthonormality::gaussHermiteG(int n, int m, arma::mat Z)
{
Miscellaneous misc;
Calcul *caln = new Calcul(n,Z);
Calcul *calm = new Calcul(m,Z);
double f1;
arma::mat Hnm;
arma::mat res;
f1 = (1 / std::sqrt(std::exp(n * std::log(2)) * misc.factorial(n))) * (1 / std::sqrt(std::exp(m * std::log(2)) * misc.factorial(m))) * std::sqrt(1 / M_PI);
Hnm = caln->calculPolynomeHermite() % calm->calculPolynomeHermite();
res = f1 * Hnm;
return res;
}
Here's my function for getting the quadrature. It this the way to prove the orthogonality, or am I doing it wrong?
long double Orthonormality::quadrature(int n, int m)
{
arma::mat gx;
arma::mat gauss_point = {{
-2.453407083009012499038365306336166239661e-1,
2.453407083009012499038365306336166239661e-1,
-7.374737285453943587056051442521042290772e-1,
7.374737285453943587056051442521042290772e-1,
1.234076215395323007885818346959410229585,
-1.234076215395323007885818346959410229584,
-1.738537712116586206780865662136406442958,
1.738537712116586206780865662136406442953,
2.254974002089275523082333344734565128082,
-2.254974002089275523082333344734565128065,
-2.788806058428130480525033756403185410695,
2.788806058428130480525033756403185410655,
3.347854567383216326914924522996463698566,
-3.347854567383216326914924522996463698495,
-3.94476404011562521037562880052441180715,
3.944764040115625210375628800524411807067,
4.603682449550744273077675248978347585171,
-4.603682449550744273077675248978347585109,
5.387480890011232862016900410681120753981,
-5.387480890011232862016900410681120754003,
}
};
arma::mat gauss_point_weight = {{
4.622436696006100896503286398612081142142e-1,
4.622436696006100896503286398612081142142e-1,
2.866755053628341297196597062280879168236e-1,
2.866755053628341297196597062280879168236e-1,
1.090172060200233200137550335354255770852e-1,
1.090172060200233200137550335354255770846e-1,
2.481052088746361088216495255894039439922e-2,
2.481052088746361088216495255894039440028e-2,
3.24377334223786183218324713235370544232e-3,
3.243773342237861832183247132353705443042e-3,
2.283386360163539672571459179634955394906e-4,
2.283386360163539672571459179634955393512e-4,
7.802556478532063694145991999647569104495e-6,
7.802556478532063694145991999647569095955e-6,
1.086069370769281693999524563447163430255e-7,
1.086069370769281693999524563447163432688e-7,
4.399340992273180553628851455467928211995e-10,
4.399340992273180553628851455467928212879e-10,
2.229393645534151292522500616029095785758e-13,
2.22939364553415129252250061602909578525e-13,
}
};
gx = Orthonormality::gaussHermiteG(n, m, gauss_point);
arma::mat res;
res = gx * gauss_point_weight.t();
long double resDouble = res(0, 0);
return resDouble;
}
Here's the Hermite Polynomial function and its output for the 3 an 4 modes:
mat Calcul::calculPolynomeHermite(int n_max, mat z)
{
mat H(n_max, z.n_elem);
if (n_max == 0)
{
H = z.ones(size(z));
}
else
{
if (n_max == 1)
{
return z.for_each([](arma::mat::elem_type& val)
{
val = 2 * val;
});
}
else {
for(int i = 0; i < z.n_elem; ++i)
{
H(0, i) = 1;
}
rowvec h2 = rowvec(z.n_elem);
h2 = 2 * z;
H.row(1) = h2;
for(int i = 2; i < n_max; i++)
{
rowvec hn = rowvec(z.n_elem);
hn = h2 % H.row(i - 1) - (2 * i) * H.row(i - 2);
H.row(i) = hn;
}
}
}
return H;
}
output :
H(3,z):
1.0000 1.0000 1.0000 1.0000 1.0000
-4.0000 -2.0000 0 2.0000 4.0000
12.0000 0 -4.0000 0 12.0000
H(4,z):
1.0000 1.0000 1.0000 1.0000 1.0000
-4.0000 -2.0000 0 2.0000 4.0000
12.0000 0 -4.0000 0 12.0000
-24.0000 12.0000 0 -12.0000 24.0000