"""
Here we perform some symbolic computations required for the N-D
interpolation routines in `interpnd.pyx`.

"""
from sympy import symbols, binomial, Matrix  # type: ignore[import]


def _estimate_gradients_2d_global():

    #
    # Compute
    #
    #

    f1, f2, df1, df2, x = symbols(['f1', 'f2', 'df1', 'df2', 'x'])
    c = [f1, (df1 + 3*f1)/3, (df2 + 3*f2)/3, f2]

    w = 0
    for k in range(4):
        w += binomial(3, k) * c[k] * x**k*(1-x)**(3-k)

    wpp = w.diff(x, 2).expand()
    intwpp2 = (wpp**2).integrate((x, 0, 1)).expand()

    A = Matrix([[intwpp2.coeff(df1**2), intwpp2.coeff(df1*df2)/2],
                [intwpp2.coeff(df1*df2)/2, intwpp2.coeff(df2**2)]])

    B = Matrix([[intwpp2.coeff(df1).subs(df2, 0)],
                [intwpp2.coeff(df2).subs(df1, 0)]]) / 2

    print("A")
    print(A)
    print("B")
    print(B)
    print("solution")
    print(A.inv() * B)