misc.py 文件源码

def compute_dobrodeev(n, I0, I2, I22, I4, pm_type, i, j, k):
    '''Compute some helper quantities used in

    L.N. Dobrodeev,
    Cubature rules with equal coefficients for integrating functions with
    respect to symmetric domains,
    USSR Computational Mathematics and Mathematical Physics,
    Volume 18, Issue 4, 1978, Pages 27-34,
    <https://doi.org/10.1016/0041-5553(78)90064-2>.
    '''
    t = 1 if pm_type == 'I' else -1

    L = binomial(n, i) * 2**i
    M = fact(n) // (fact(j) * fact(k) * fact(n-j-k)) * 2**(j+k)
    N = L + M
    F = I22/I0 - I2**2/I0**2 + (I4/I0 - I22/I0) / n
    R = -(j+k-i) / i * I2**2/I0**2 + (j+k-1)/n * I4/I0 - (n-1)/n * I22/I0
    H = 1/i * (
        (j+k-i) * I2**2/I0**2 + (j+k)/n * ((i-1) * I4/I0 - (n-1)*I22/I0)
        )
    Q = L/M*R + H - t * 2*I2/I0 * (j+k-i)/i * sqrt(L/M*F)

    G = 1/N
    a = sqrt(n/i * (I2/I0 + t * sqrt(M/L*F)))
    b = sqrt(n/(j+k) * (I2/I0 - t * sqrt(L/M*F) + t * sqrt(k/j*Q)))
    c = sqrt(n/(j+k) * (I2/I0 - t * sqrt(L/M*F) - t * sqrt(j/k*Q)))
    return G, a, b, c