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