def _ell(A, m):
"""
A helper function for expm_2009.
Parameters
----------
A : linear operator
A linear operator whose norm of power we care about.
m : int
The power of the linear operator
Returns
-------
value : int
A value related to a bound.
"""
if len(A.shape) != 2 or A.shape[0] != A.shape[1]:
raise ValueError('expected A to be like a square matrix')
# The c_i are explained in (2.2) and (2.6) of the 2005 expm paper.
# They are coefficients of terms of a generating function series expansion.
choose_2m_m = comb(2 * m, m, exact=True)
abs_c_recip = float(choose_2m_m * factorial(2 * m + 1))
# This is explained after Eq. (1.2) of the 2009 expm paper.
# It is the "unit roundoff" of IEEE double precision arithmetic.
u = 2**-53
# Compute the one-norm of matrix power p of abs(A).
A_abs_onenorm = matfuncs._onenorm_matrix_power_nnm(abs(A), 2 * m + 1)
# Treat zero norm as a special case.
if not A_abs_onenorm:
return 0
alpha = A_abs_onenorm / (_onenorm(A) * abs_c_recip)
log2_alpha_div_u = np.log2(alpha / u)
value = int(np.ceil(log2_alpha_div_u / (2 * m)))
return max(value, 0)
def d10_tight(self):
if self._d10_exact is None:
self._d10_exact = _onenorm(self.A10)**(1 / 10.)
return self._d10_exact
def d6_tight(self):
if self._d6_exact is None:
self._d6_exact = _onenorm(self.A6)**(1 / 6.)
return self._d6_exact
def d8_tight(self):
if self._d8_exact is None:
self._d8_exact = _onenorm(self.A8)**(1 / 8.)
return self._d8_exact
def d4_tight(self):
if self._d4_exact is None:
self._d4_exact = _onenorm(self.A4)**(1 / 4.)
return self._d4_exact