def run_eigsy(A, verbose=False):
if verbose:
print("original matrix:\n", str(A))
D, Q = mp.eigsy(A)
B = Q * mp.diag(D) * Q.transpose()
C = A - B
E = Q * Q.transpose() - mp.eye(A.rows)
if verbose:
print("eigenvalues:\n", D)
print("eigenvectors:\n", Q)
NC = mp.mnorm(C)
NE = mp.mnorm(E)
if verbose:
print("difference:", NC, "\n", C, "\n")
print("difference:", NE, "\n", E, "\n")
eps = mp.exp(0.8 * mp.log(mp.eps))
assert NC < eps
assert NE < eps
return NC
def run_eigsy(A, verbose = False):
if verbose:
print("original matrix:\n", str(A))
D, Q = mp.eigsy(A)
B = Q * mp.diag(D) * Q.transpose()
C = A - B
E = Q * Q.transpose() - mp.eye(A.rows)
if verbose:
print("eigenvalues:\n", D)
print("eigenvectors:\n", Q)
NC = mp.mnorm(C)
NE = mp.mnorm(E)
if verbose:
print("difference:", NC, "\n", C, "\n")
print("difference:", NE, "\n", E, "\n")
eps = mp.exp( 0.8 * mp.log(mp.eps))
assert NC < eps
assert NE < eps
return NC
def _compute_roots(w, x, use_mp):
# Cf.:
# Knockaert, L. (2008). A simple and accurate algorithm for barycentric
# rational interpolation. IEEE Signal processing letters, 15, 154-157.
if use_mp:
from mpmath import mp
if use_mp is True:
use_mp = 100
mp.dps = use_mp
ak = mp.matrix(w)
ak /= sum(ak)
bk = mp.matrix(x)
M = mp.diag(bk)
for i in range(M.rows):
for j in range(M.cols):
M[i, j] -= ak[i] * bk[j]
lam = mp.eig(M, left=False, right=False)
lam = np.array(lam, dtype=complex)
else:
# the same procedure in standard double precision
ak = w / w.sum()
M = np.diag(x) - np.outer(ak, x)
lam = scipy.linalg.eigvals(M)
# remove one simple root
lam = np.delete(lam, np.argmin(abs(lam)))
return np.real_if_close(lam)
def _compute_roots(w, x, use_mp):
# Cf.:
# Knockaert, L. (2008). A simple and accurate algorithm for barycentric
# rational interpolation. IEEE Signal processing letters, 15, 154-157.
#
# This version requires solving only a standard eigenvalue problem, but
# has troubles when the polynomial has leading 0 coefficients.
if _is_mp_array(w) or _is_mp_array(x):
use_mp = True
if use_mp:
assert mpmath, 'mpmath package is not installed'
ak = mp.matrix(w)
ak /= sum(ak)
bk = mp.matrix(x)
M = mp.diag(bk)
for i in range(M.rows):
for j in range(M.cols):
M[i, j] -= ak[i] * bk[j]
lam = np.array(mp.eig(M, left=False, right=False))
# remove one simple root
lam = np.delete(lam, np.argmin(abs(lam)))
return lam
else:
# the same procedure in standard double precision
ak = w / w.sum()
M = np.diag(x) - np.outer(ak, x)
lam = scipy.linalg.eigvals(M)
# remove one simple root
lam = np.delete(lam, np.argmin(abs(lam)))
return np.real_if_close(lam)
def takagi_fact(a):
A = mp.matrix(a)
sing_vs, Q = symmetric_svd(A)
phase_mat = mp.diag(
[mp.exp(-1j * mp.arg(sing_v) / 2.0) for sing_v in sing_vs])
vs = [mp.fabs(sing_v) for sing_v in sing_vs]
Qp = Q * phase_mat
return vs, Qp
def apply(self, m, evaluation):
'SingularValueDecomposition[m_]'
if not any(leaf.is_inexact() for row in m.leaves for leaf in row.leaves):
# symbolic argument (not implemented)
evaluation.message('SingularValueDecomposition', 'nosymb')
matrix = to_mpmath_matrix(m)
U, S, V = mp.svd(matrix)
S = mp.diag(S)
U_list = Expression('List', *U.tolist())
S_list = Expression('List', *S.tolist())
V_list = Expression('List', *V.tolist())
return Expression('List', *[U_list, S_list, V_list])
def apply(self, m, evaluation):
'SingularValueDecomposition[m_?MatrixQ]'
if not any(leaf.is_inexact() for row in m.leaves for leaf in row.leaves):
# symbolic argument (not implemented)
evaluation.message('SingularValueDecomposition', 'nosymb')
matrix = to_mpmath_matrix(m)
U, S, V = mp.svd(matrix)
S = mp.diag(S)
U_list = Expression('List', *U.tolist())
S_list = Expression('List', *S.tolist())
V_list = Expression('List', *V.tolist())
return Expression('List', *[U_list, S_list, V_list])
def apply(self, m, evaluation):
"SingularValueDecomposition[m_]"
matrix = to_mpmath_matrix(m)
if matrix is None:
return evaluation.message("SingularValueDecomposition", "matrix",
m, 1)
if not any(leaf.is_inexact() for row in m.leaves
for leaf in row.leaves):
# symbolic argument (not implemented)
evaluation.message("SingularValueDecomposition", "nosymb")
U, S, V = mp.svd(matrix)
S = mp.diag(S)
U_list = Expression("List", *U.tolist())
S_list = Expression("List", *S.tolist())
V_list = Expression("List", *V.tolist())
return Expression("List", *[U_list, S_list, V_list])
def conjugator_into_SL2R(SL2C_matrices):
"""
Returns a matrix C in SL(2, C) so that C^-1 * M * C is
(essentially) in SL(2, R) for all the input matrices M.
"""
ans, sig, form = preserves_hermitian_form(SL2C_matrices)
if ans is None:
raise ValueError('No invariant hermitian form found')
if sig == 'definite':
raise ValueError('Conjugate into SU(2), not SL(2, R)')
if sig == 'both':
raise ValueError('This degnerate case not implemented')
assert sig == 'indefinite'
J = sage_matrix_to_mpmath(form)
eigs, U = mp.eighe(J)
C = U * mp.diag([1 / mp.sqrt(abs(e)) for e in eigs])
sq_two = mp.sqrt(2)
sq_two_i = mp.mpc(imag=sq_two)
S = mp.matrix([[1 / sq_two_i, 1 / sq_two_i], [-1 / sq_two, 1 / sq_two]])
C = C * S.H
C = (1 / mp.sqrt(mp.det(C))) * C
return mpmath_matrix_to_sage(C)
def nearly_diagonal(A):
assert A.rows == A.cols == 2
a = A[0, 0]
return mp.norm(A - mp.diag([a, a])) < 1000 * mp.eps
