def LDL(matlist, K):
"""
Performs the LDL decomposition of a hermitian matrix and returns L, D and
transpose of L. Only applicable to rational entries.
Examples
========
>>> from sympy.matrices.densesolve import LDL
>>> from sympy import QQ
>>> a = [
... [QQ(4), QQ(12), QQ(-16)],
... [QQ(12), QQ(37), QQ(-43)],
... [QQ(-16), QQ(-43), QQ(98)]]
>>> LDL(a, QQ)
([[1, 0, 0], [3, 1, 0], [-4, 5, 1]], [[4, 0, 0], [0, 1, 0], [0, 0, 9]], [[1, 3, -4], [0, 1, 5], [0, 0, 1]])
"""
new_matlist = copy.deepcopy(matlist)
nrow = len(new_matlist)
L, D = eye(nrow, K), eye(nrow, K)
for i in range(nrow):
for j in range(i + 1):
a = K.zero
for k in range(j):
a += L[i][k] * L[j][k] * D[k][k]
if i == j:
D[j][j] = new_matlist[j][j] - a
else:
L[i][j] = (new_matlist[i][j] - a) / D[j][j]
return L, D, conjugate_transpose(L, K)
def LDL(matlist, K):
"""
Performs the LDL decomposition of a hermitian matrix and returns L, D and
transpose of L. Only applicable to rational entries.
Examples
========
>>> from sympy.matrices.densesolve import LDL
>>> from sympy import QQ
>>> a = [
... [QQ(4), QQ(12), QQ(-16)],
... [QQ(12), QQ(37), QQ(-43)],
... [QQ(-16), QQ(-43), QQ(98)]]
>>> LDL(a, QQ)
([[1, 0, 0], [3, 1, 0], [-4, 5, 1]], [[4, 0, 0], [0, 1, 0], [0, 0, 9]], [[1, 3, -4], [0, 1, 5], [0, 0, 1]])
"""
new_matlist = copy.deepcopy(matlist)
nrow = len(new_matlist)
L, D = eye(nrow, K), eye(nrow, K)
for i in range(nrow):
for j in range(i + 1):
a = K.zero
for k in range(j):
a += L[i][k]*L[j][k]*D[k][k]
if i == j:
D[j][j] = new_matlist[j][j] - a
else:
L[i][j] = (new_matlist[i][j] - a)/D[j][j]
return L, D, conjugate_transpose(L, K)
def cholesky(matlist, K):
"""
Performs the cholesky decomposition of a Hermitian matrix and
returns L and it's conjugate transpose.
Examples
========
>>> from sympy.matrices.densesolve import cholesky
>>> from sympy import QQ
>>> cholesky([[QQ(25), QQ(15), QQ(-5)], [QQ(15), QQ(18), QQ(0)], [QQ(-5), QQ(0), QQ(11)]], QQ)
([[5, 0, 0], [3, 3, 0], [-1, 1, 3]], [[5, 3, -1], [0, 3, 1], [0, 0, 3]])
See Also
========
cholesky_solve
"""
new_matlist = copy.deepcopy(matlist)
nrow = len(new_matlist)
L = eye(nrow, K)
for i in range(nrow):
for j in range(i + 1):
a = K.zero
for k in range(j):
a += L[i][k] * L[j][k]
if i == j:
L[i][j] = isqrt(new_matlist[i][j] - a)
else:
L[i][j] = (new_matlist[i][j] - a) / L[j][j]
return L, conjugate_transpose(L, K)
def cholesky(matlist, K):
"""
Performs the cholesky decomposition of a Hermitian matrix and
returns L and it's conjugate transpose.
Examples
========
>>> from sympy.matrices.densesolve import cholesky
>>> from sympy import QQ
>>> cholesky([[QQ(25), QQ(15), QQ(-5)], [QQ(15), QQ(18), QQ(0)], [QQ(-5), QQ(0), QQ(11)]], QQ)
([[5, 0, 0], [3, 3, 0], [-1, 1, 3]], [[5, 3, -1], [0, 3, 1], [0, 0, 3]])
See Also
========
cholesky_solve
"""
new_matlist = copy.deepcopy(matlist)
nrow = len(new_matlist)
L = eye(nrow, K)
for i in range(nrow):
for j in range(i + 1):
a = K.zero
for k in range(j):
a += L[i][k]*L[j][k]
if i == j:
L[i][j] = int(sqrt(new_matlist[i][j] - a))
else:
L[i][j] = (new_matlist[i][j] - a)/L[j][j]
return L, conjugate_transpose(L, K)
