gftool.hermpade.hermite2

gftool.hermpade.hermite2(an, p_deg: int, q_deg: int, r_deg: int) → Tuple[Polynomial, Polynomial, Polynomial]

Return the polynomials p, q, r for the quadratic Hermite-Padé approximant.

The polynomials fulfill the equation

\[p(x) + q(x) f(x) + r(x) f^2(x) = 𝒪(x^{N_p + N_q + N_r + 2})\]

where \(f(x)\) is the function with Taylor coefficients an, and \(N_x\) are the degrees of the polynomials. The approximant has two branches

\[F^±(z) = [-q(z) ± \sqrt{q^2(z) - 4p(z)r(z)}] / 2r(z)\]

Parameters:

an(L,) array_like

Taylor series coefficients representing polynomial of order L-1.

p_deg, q_deg, r_degint

The order of the polynomials of the quadratic Hermite-Padé approximant. The sum must be at most p_deg + q_deg + r_deg + 2 <= L.

Returns:

p, q, rPolynom

The polynomials p, q, and r building the quadratic Hermite-Padé approximant.

See also

Hermite2

High-level interface, guessing the correct branch.

Examples

The quadratic Hermite-Padé approximant can reproduce the square root f(z) = (1 + z)**(1/2):

>>> from scipy.special import binom
>>> an = binom(1/2, np.arange(5+5+5+2))  # Taylor of (1+x)**(1/2)
>>> x = np.linspace(-3, 3, num=500)
>>> fx = np.emath.power(1+x, 1/2)

>>> p, q, r = gt.hermpade.hermite2(an, 5, 5, 5)
>>> px, qx, rx = p(x), q(x), r(x)
>>> pos_branch = (-qx + np.emath.sqrt(qx**2 - 4*px*rx)) / (2*rx)

>>> import matplotlib.pyplot as plt
>>> __ = plt.plot(x, fx.real, label='exact', color='black')
>>> __ = plt.plot(x, fx.imag, '--', color='black')
>>> __ = plt.plot(x, pos_branch.real, '--', label='Herm2', color='C1')
>>> __ = plt.plot(x, pos_branch.imag, ':', color='C1')
>>> plt.show()

(png, pdf)