def _LDLdecomposition(self, hermitian=True):
"""Helper function of LDLdecomposition.
Without the error checks.
To be used privately.
Returns L and D such that L*D*L.H == self if hermitian flag is True,
or L*D*L.T == self if hermitian is False.
"""
# https://en.wikipedia.org/wiki/Cholesky_decomposition#LDL_decomposition_2
D = zeros(self.rows, self.rows)
L = eye(self.rows)
if hermitian:
for i in range(self.rows):
for j in range(i):
L[i, j] = (1 / D[j, j]) * expand_mul(self[i, j] - sum(
L[i, k] * L[j, k].conjugate() * D[k, k]
for k in range(j)))
D[i,
i] = expand_mul(self[i, i] -
sum(L[i, k] * L[i, k].conjugate() * D[k, k]
for k in range(i)))
if D[i, i].is_positive is False:
raise NonPositiveDefiniteMatrixError(
"Matrix must be positive-definite")
else:
for i in range(self.rows):
for j in range(i):
L[i, j] = (1 / D[j, j]) * (self[i, j] -
sum(L[i, k] * L[j, k] * D[k, k]
for k in range(j)))
D[i,
i] = self[i, i] - sum(L[i, k]**2 * D[k, k] for k in range(i))
return self._new(L), self._new(D)
def _cholesky(self, hermitian=True):
"""Helper function of cholesky.
Without the error checks.
To be used privately.
Implements the Cholesky-Banachiewicz algorithm.
Returns L such that L*L.H == self if hermitian flag is True,
or L*L.T == self if hermitian is False.
"""
L = zeros(self.rows, self.rows)
if hermitian:
for i in range(self.rows):
for j in range(i):
L[i, j] = (1 / L[j, j]) * expand_mul(self[i, j] - sum(
L[i, k] * L[j, k].conjugate() for k in range(j)))
Lii2 = expand_mul(self[i, i] -
sum(L[i, k] * L[i, k].conjugate()
for k in range(i)))
if Lii2.is_positive is False:
raise NonPositiveDefiniteMatrixError(
"Matrix must be positive-definite")
L[i, i] = sqrt(Lii2)
else:
for i in range(self.rows):
for j in range(i):
L[i, j] = (1 / L[j, j]) * (self[i, j] -
sum(L[i, k] * L[j, k]
for k in range(j)))
L[i, i] = sqrt(self[i, i] - sum(L[i, k]**2 for k in range(i)))
return self._new(L)