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 diofant.matrices.densesolve import LDL
>>> from diofant 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 LU(matlist, K, reverse=0):
"""
It computes the LU decomposition of a matrix and returns L and U
matrices.
Examples
========
>>> from diofant.matrices.densesolve import LU
>>> from diofant import QQ
>>> a = [
... [QQ(1), QQ(2), QQ(3)],
... [QQ(2), QQ(-4), QQ(6)],
... [QQ(3), QQ(-9), QQ(-3)]]
>>> LU(a, QQ)
([[1, 0, 0], [2, 1, 0], [3, 15/8, 1]], [[1, 2, 3], [0, -8, 0], [0, 0, -12]])
See Also
========
upper_triangle
lower_triangle
"""
nrow = len(matlist)
new_matlist1, new_matlist2 = eye(nrow, K), copy.deepcopy(matlist)
for i in range(nrow):
for j in range(i + 1, nrow):
if (new_matlist2[j][i] != 0):
new_matlist1[j][i] = new_matlist2[j][i] / new_matlist2[i][i]
rowadd(new_matlist2, j, i,
-new_matlist2[j][i] / new_matlist2[i][i], K)
return new_matlist1, new_matlist2
def cholesky(matlist, K):
"""
Performs the cholesky decomposition of a Hermitian matrix and
returns L and it's conjugate transpose.
Examples
========
>>> from diofant.matrices.densesolve import cholesky
>>> from diofant 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)
def test_mulmatmat():
a = [[ZZ(3), ZZ(4)], [ZZ(5), ZZ(6)]]
b = [[ZZ(1), ZZ(2)], [ZZ(7), ZZ(8)]]
c = eye(2, ZZ)
d = [[ZZ(6)], [ZZ(7)]]
assert mulmatmat(a, b, ZZ) == [[ZZ(31), ZZ(38)], [ZZ(47), ZZ(58)]]
assert mulmatmat(b, d, ZZ) == [[ZZ(20)], [ZZ(98)]]
def test_transpose():
a = [[ZZ(3), ZZ(7), ZZ(4)], [ZZ(2), ZZ(4), ZZ(5)], [ZZ(6), ZZ(2), ZZ(3)]]
b = eye(4, ZZ)
assert transpose(a, ZZ) == ([[ZZ(3), ZZ(2), ZZ(6)], [ZZ(7),
ZZ(4),
ZZ(2)],
[ZZ(4), ZZ(5), ZZ(3)]])
assert transpose(b, ZZ) == b
def test_mulmatscaler():
a = eye(3, ZZ)
b = [[ZZ(3), ZZ(7), ZZ(4)], [ZZ(2), ZZ(4), ZZ(5)], [ZZ(6), ZZ(2), ZZ(3)]]
assert mulmatscaler(a, ZZ(4), ZZ) == [[ZZ(4), ZZ(0), ZZ(0)],
[ZZ(0), ZZ(4), ZZ(0)],
[ZZ(0), ZZ(0), ZZ(4)]]
assert mulmatscaler(b, ZZ(1), ZZ) == [[ZZ(3), ZZ(7), ZZ(4)],
[ZZ(2), ZZ(4), ZZ(5)],
[ZZ(6), ZZ(2), ZZ(3)]]
def test_trace():
a = [[ZZ(3), ZZ(7), ZZ(4)], [ZZ(2), ZZ(4), ZZ(5)], [ZZ(6), ZZ(2), ZZ(3)]]
b = eye(2, ZZ)
assert trace(a, ZZ) == ZZ(10)
assert trace(b, ZZ) == ZZ(2)