gftool.hermpade.pade

gftool.hermpade.pade(an, num_deg: int, den_deg: int, fast=False) → RatPol

Return the [num_deg/den_deg] Padé approximant to the polynomial an.

Parameters:

an(L,) array_like

Taylor series coefficients representing polynomial of order L-1.

num_deg, den_degint

The order of the return approximating numerator/denominator polynomial. The sum must be at most L: L >= num_deg + den_deg + 1.

fastbool, optional

If fast, use faster solve_toeplitz algorithm. Else use QR and calculate null-vector (default: False).

Returns:

RatPol

The rational polynomial with numerator RatPol.numer, and denominator RatPol.denom.

Examples

Let’s approximate the cubic root f(z) = (1 + z)**(1/3) by the [8/8] Padé approximant:

>>> from scipy.special import binom
>>> an = binom(1/3, np.arange(8+8+1))  # Taylor of (1+x)**(1/3)
>>> x = np.linspace(-1, 3, num=500)
>>> fx = np.emath.power(1+x, 1/3)

>>> pade = gt.hermpade.pade(an, num_deg=8, den_deg=8)

>>> import matplotlib.pyplot as plt
>>> __ = plt.plot(x, fx, label='exact', color='black')
>>> __ = plt.plot(x, np.polynomial.Polynomial(an)(x), '--', label='Taylor')
>>> __ = plt.plot(x, pade.eval(x), ':', label='Pade')
>>> __ = plt.ylim(ymin=0, ymax=2)
>>> __ = plt.legend(loc='upper left')
>>> plt.show()

(png, pdf)

The Padé approximation is able to approximate the function even for larger x.

Using fast=True, the Toeplitz structure is used to evaluate the pade faster using Levinson recursion. This might, however, be less accurate in some cases.

>>> padef = gt.hermpade.pade(an, num_deg=8, den_deg=8, fast=True)
>>> __ = plt.plot(x, abs(np.polynomial.Polynomial(an)(x) - fx), label='Taylor')
>>> __ = plt.plot(x, abs(pade.eval(x) - fx), label='QR')
>>> __ = plt.plot(x, abs(padef.eval(x) - fx), label='Levinson')
>>> __ = plt.legend()
>>> plt.yscale('log')
>>> plt.show()

(png, pdf)