def LU(matlist, K, reverse=0):
"""
It computes the LU decomposition of a matrix and returns L and U
matrices.
Examples
========
>>> from sympy.matrices.densesolve import LU
>>> from sympy 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 rref(matlist, K):
"""
Returns the reduced row echelon form of a Matrix.
Examples
========
>>> from sympy.matrices.densesolve import rref
>>> from sympy import QQ
>>> a = [
... [QQ(1), QQ(2), QQ(1)],
... [QQ(-2), QQ(-3), QQ(1)],
... [QQ(3), QQ(5), QQ(0)]]
>>> rref(a, QQ)
[[1, 0, -5], [0, 1, 3], [0, 0, 0]]
See Also
========
row_echelon
"""
result_matlist = copy.deepcopy(matlist)
result_matlist = row_echelon(result_matlist, K)
nrow = len(result_matlist)
for i in range(nrow):
if result_matlist[i][i] == 1:
for j in range(i):
rowadd(result_matlist, j, i, -result_matlist[j][i], K)
return result_matlist
def row_echelon(matlist, K):
"""
Returns the row echelon form of a matrix with diagonal elements
reduced to 1.
Examples
========
>>> from sympy.matrices.densesolve import row_echelon
>>> from sympy import QQ
>>> a = [
... [QQ(3), QQ(7), QQ(4)],
... [QQ(2), QQ(4), QQ(5)],
... [QQ(6), QQ(2), QQ(3)]]
>>> row_echelon(a, QQ)
[[1, 7/3, 4/3], [0, 1, -7/2], [0, 0, 1]]
See Also
========
rref
"""
result_matlist = copy.deepcopy(matlist)
nrow = len(result_matlist)
for i in range(nrow):
if result_matlist[i][i] != 1 and result_matlist[i][i] != 0:
rowmul(result_matlist, i, 1 / result_matlist[i][i], K)
for j in range(i + 1, nrow):
if result_matlist[j][i] != 0:
rowadd(result_matlist, j, i, -result_matlist[j][i], K)
return result_matlist
def LU(matlist, K, reverse = 0):
"""
It computes the LU decomposition of a matrix and returns L and U
matrices.
Examples
========
>>> from sympy.matrices.densesolve import LU
>>> from sympy 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 rref(matlist, K):
"""
Returns the reduced row echelon form of a Matrix.
Examples
========
>>> from sympy.matrices.densesolve import rref
>>> from sympy import QQ
>>> a = [
... [QQ(1), QQ(2), QQ(1)],
... [QQ(-2), QQ(-3), QQ(1)],
... [QQ(3), QQ(5), QQ(0)]]
>>> rref(a, QQ)
[[1, 0, -5], [0, 1, 3], [0, 0, 0]]
See Also
========
row_echelon
"""
result_matlist = copy.deepcopy(matlist)
result_matlist = row_echelon(result_matlist, K)
nrow = len(result_matlist)
for i in range(nrow):
if result_matlist[i][i] == 1:
for j in range(i):
rowadd(result_matlist, j, i, -result_matlist[j][i], K)
return result_matlist
def row_echelon(matlist, K):
"""
Returns the row echelon form of a matrix with diagonal elements
reduced to 1.
Examples
========
>>> from sympy.matrices.densesolve import row_echelon
>>> from sympy import QQ
>>> a = [
... [QQ(3), QQ(7), QQ(4)],
... [QQ(2), QQ(4), QQ(5)],
... [QQ(6), QQ(2), QQ(3)]]
>>> row_echelon(a, QQ)
[[1, 7/3, 4/3], [0, 1, -7/2], [0, 0, 1]]
See Also
========
rref
"""
result_matlist = copy.deepcopy(matlist)
nrow = len(result_matlist)
for i in range(nrow):
if (result_matlist[i][i] != 1 and result_matlist[i][i] != 0):
rowmul(result_matlist, i, 1/result_matlist[i][i], K)
for j in range(i + 1, nrow):
if (result_matlist[j][i] != 0):
rowadd(result_matlist, j, i, -result_matlist[j][i], K)
return result_matlist