polykin.math

roots_xcotx cached

roots_xcotx(
    a: float, N: int, xtol: float = 1e-06
) -> FloatVector

Determine the first N roots of the transcendental equation 1 - x_n*cot(x_n) = a.

PARAMETER	DESCRIPTION
a	Parameter. TYPE: float
N	Number of roots. TYPE: int
xtol	Tolerance. TYPE: float DEFAULT: 1e-06

RETURNS	DESCRIPTION
FloatVector	Roots of the equation.

Examples:

Determine the first 4 roots for a=1.

>>> from polykin.math import roots_xcotx
>>> roots_xcotx(2.0, 4)
array([ 2.0287575 ,  4.91318033,  7.97866578, 11.08553772])

Source code in src/polykin/math/special.py

@functools.cache
def roots_xcotx(a: float, N: int, xtol: float = 1e-6) -> FloatVector:
    r"""Determine the first `N` roots of the transcendental equation 
    `1 - x_n*cot(x_n) = a`.

    Parameters
    ----------
    a : float
        Parameter.
    N : int
        Number of roots.
    xtol : float
        Tolerance.

    Returns
    -------
    FloatVector
        Roots of the equation.

    Examples
    --------
    Determine the first 4 roots for a=1.
    >>> from polykin.math import roots_xcotx
    >>> roots_xcotx(2.0, 4)
    array([ 2.0287575 ,  4.91318033,  7.97866578, 11.08553772])
    """
    result = np.zeros(N)

    if a == 1:
        for i in range(N):
            result[i] = (i + 1/2)*pi
    elif a == inf:
        for i in range(N):
            result[i] = (i + 1)*pi
    elif a < 2:
        for i in range(N):
            sol = fzero_newton(f=lambda x: x/tan(x) + a - 1,
                               x0=(i + 1/2)*pi,
                               xtol=xtol,
                               ftol=1e-10)
            result[i] = sol.x
    else:
        for i in range(N):
            x0 = (i+1)*pi
            e = 0.0
            for _ in range(0, 30):
                e_new = np.arctan((x0 + e)/(1 - a))
                if abs(e_new - e) < xtol:
                    break
                e = e_new
            result[i] = x0 + e

    return result