def test_LUdecomp():
testmat = Matrix([[0, 2, 5, 3], [3, 3, 7, 4], [8, 4, 0, 2], [-2, 6, 3, 4]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower()
assert U.is_upper()
assert (L * U).permuteBkwd(p) - testmat == zeros(4)
testmat = Matrix([[6, -2, 7, 4], [0, 3, 6, 7], [1, -2, 7, 4],
[-9, 2, 6, 3]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower()
assert U.is_upper()
assert (L * U).permuteBkwd(p) - testmat == zeros(4)
x, y, z = Symbol('x'), Symbol('y'), Symbol('z')
M = Matrix(((1, x, 1), (2, y, 0), (y, 0, z)))
L, U, p = M.LUdecomposition()
assert L.is_lower()
assert U.is_upper()
assert (L * U).permuteBkwd(p) - M == zeros(3)
# test FF LUdecomp
M = Matrix([[1, 3, 3], [3, 2, 6], [3, 2, 2]])
P, L, Dee, U = M.LUdecompositionFF()
assert P * M == L * Dee.inv() * U
M = Matrix([[1, 2, 3, 4], [3, -1, 2, 3], [3, 1, 3, -2], [6, -1, 0, 2]])
P, L, Dee, U = M.LUdecompositionFF()
assert P * M == L * Dee.inv() * U
def test_sparse_matrix():
return
def eye(n):
tmp = SMatrix(n, n, lambda i, j: 0)
for i in range(tmp.lines):
tmp[i, i] = 1
return tmp
def zeros(n):
return SMatrix(n, n, lambda i, j: 0)
# test_multiplication
a = SMatrix((
(1, 2),
(3, 1),
(0, 6),
))
b = SMatrix((
(1, 2),
(3, 0),
))
c = a * b
assert c[0, 0] == 7
assert c[0, 1] == 2
assert c[1, 0] == 6
assert c[1, 1] == 6
assert c[2, 0] == 18
assert c[2, 1] == 0
x = Symbol("x")
c = b * Symbol("x")
assert isinstance(c, SMatrix)
assert c[0, 0] == x
assert c[0, 1] == 2 * x
assert c[1, 0] == 3 * x
assert c[1, 1] == 0
c = 5 * b
assert isinstance(c, SMatrix)
assert c[0, 0] == 5
assert c[0, 1] == 2 * 5
assert c[1, 0] == 3 * 5
assert c[1, 1] == 0
#test_power
A = SMatrix([[2, 3], [4, 5]])
assert (A**5)[:] == [6140, 8097, 10796, 14237]
A = SMatrix([[2, 1, 3], [4, 2, 4], [6, 12, 1]])
assert (A**3)[:] == [290, 262, 251, 448, 440, 368, 702, 954, 433]
# test_creation
x = Symbol("x")
a = SMatrix([x, 0], [0, 0])
m = a
assert m.cols == m.lines
assert m.cols == 2
assert m[:] == [x, 0, 0, 0]
b = SMatrix(2, 2, [x, 0, 0, 0])
m = b
assert m.cols == m.lines
assert m.cols == 2
assert m[:] == [x, 0, 0, 0]
assert a == b
# test_determinant
x, y = Symbol('x'), Symbol('y')
assert SMatrix([[1]]).det() == 1
assert SMatrix(((-3, 2), (8, -5))).det() == -1
assert SMatrix(((x, 1), (y, 2 * y))).det() == 2 * x * y - y
assert SMatrix(((1, 1, 1), (1, 2, 3), (1, 3, 6))).det() == 1
assert SMatrix(((3, -2, 0, 5), (-2, 1, -2, 2), (0, -2, 5, 0),
(5, 0, 3, 4))).det() == -289
assert SMatrix(((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12),
(13, 14, 15, 16))).det() == 0
assert SMatrix(((3, 2, 0, 0, 0), (0, 3, 2, 0, 0), (0, 0, 3, 2, 0),
(0, 0, 0, 3, 2), (2, 0, 0, 0, 3))).det() == 275
assert SMatrix(((1, 0, 1, 2, 12), (2, 0, 1, 1, 4), (2, 1, 1, -1, 3),
(3, 2, -1, 1, 8), (1, 1, 1, 0, 6))).det() == -55
assert SMatrix(((-5, 2, 3, 4, 5), (1, -4, 3, 4, 5), (1, 2, -3, 4, 5),
(1, 2, 3, -2, 5), (1, 2, 3, 4, -1))).det() == 11664
assert SMatrix(((2, 7, -1, 3, 2), (0, 0, 1, 0, 1), (-2, 0, 7, 0, 2),
(-3, -2, 4, 5, 3), (1, 0, 0, 0, 1))).det() == 123
# test_submatrix
m0 = eye(4)
assert m0[0:3, 0:3] == eye(3)
assert m0[2:4, 0:2] == zeros(2)
m1 = SMatrix(3, 3, lambda i, j: i + j)
assert m1[0, :] == SMatrix(1, 3, (0, 1, 2))
assert m1[1:3, 1] == SMatrix(2, 1, (2, 3))
m2 = SMatrix([0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11], [12, 13, 14, 15])
assert m2[:, -1] == SMatrix(4, 1, [3, 7, 11, 15])
assert m2[-2:, :] == SMatrix([[8, 9, 10, 11], [12, 13, 14, 15]])
# test_submatrix_assignment
m = zeros(4)
m[2:4, 2:4] = eye(2)
assert m == SMatrix((0, 0, 0, 0), (0, 0, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1))
m[0:2, 0:2] = eye(2)
assert m == eye(4)
m[:, 0] = SMatrix(4, 1, (1, 2, 3, 4))
assert m == SMatrix((1, 0, 0, 0), (2, 1, 0, 0), (3, 0, 1, 0), (4, 0, 0, 1))
m[:, :] = zeros(4)
assert m == zeros(4)
m[:, :] = ((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16))
assert m == SMatrix(
((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)))
m[0:2, 0] = [0, 0]
assert m == SMatrix(
((0, 2, 3, 4), (0, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)))
# test_reshape
m0 = eye(3)
assert m0.reshape(1, 9) == SMatrix(1, 9, (1, 0, 0, 0, 1, 0, 0, 0, 1))
m1 = SMatrix(3, 4, lambda i, j: i + j)
assert m1.reshape(4, 3) == SMatrix((0, 1, 2), (3, 1, 2), (3, 4, 2),
(3, 4, 5))
assert m1.reshape(2, 6) == SMatrix((0, 1, 2, 3, 1, 2), (3, 4, 2, 3, 4, 5))
# test_applyfunc
m0 = eye(3)
assert m0.applyfunc(lambda x: 2 * x) == eye(3) * 2
assert m0.applyfunc(lambda x: 0) == zeros(3)
# test_LUdecomp
testmat = SMatrix([[0, 2, 5, 3], [3, 3, 7, 4], [8, 4, 0, 2], [-2, 6, 3,
4]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower()
assert U.is_upper()
assert (L * U).permuteBkwd(p) - testmat == zeros(4)
testmat = SMatrix([[6, -2, 7, 4], [0, 3, 6, 7], [1, -2, 7, 4],
[-9, 2, 6, 3]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower()
assert U.is_upper()
assert (L * U).permuteBkwd(p) - testmat == zeros(4)
x, y, z = Symbol('x'), Symbol('y'), Symbol('z')
M = Matrix(((1, x, 1), (2, y, 0), (y, 0, z)))
L, U, p = M.LUdecomposition()
assert L.is_lower()
assert U.is_upper()
assert (L * U).permuteBkwd(p) - M == zeros(3)
# test_LUsolve
A = SMatrix([[2, 3, 5], [3, 6, 2], [8, 3, 6]])
x = SMatrix(3, 1, [3, 7, 5])
b = A * x
soln = A.LUsolve(b)
assert soln == x
A = SMatrix([[0, -1, 2], [5, 10, 7], [8, 3, 4]])
x = SMatrix(3, 1, [-1, 2, 5])
b = A * x
soln = A.LUsolve(b)
assert soln == x
# test_inverse
A = eye(4)
assert A.inv() == eye(4)
assert A.inv("LU") == eye(4)
assert A.inv("ADJ") == eye(4)
A = SMatrix([[2, 3, 5], [3, 6, 2], [8, 3, 6]])
Ainv = A.inv()
assert A * Ainv == eye(3)
assert A.inv("LU") == Ainv
assert A.inv("ADJ") == Ainv
# test_cross
v1 = Matrix(1, 3, [1, 2, 3])
v2 = Matrix(1, 3, [3, 4, 5])
assert v1.cross(v2) == Matrix(1, 3, [-2, 4, -2])
assert v1.norm(v1) == 14
# test_cofactor
assert eye(3) == eye(3).cofactorMatrix()
test = SMatrix([[1, 3, 2], [2, 6, 3], [2, 3, 6]])
assert test.cofactorMatrix() == SMatrix([[27, -6, -6], [-12, 2, 3],
[-3, 1, 0]])
test = SMatrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
assert test.cofactorMatrix() == SMatrix([[-3, 6, -3], [6, -12, 6],
[-3, 6, -3]])
# test_jacobian
x = Symbol('x')
y = Symbol('y')
L = SMatrix(1, 2, [x**2 * y, 2 * y**2 + x * y])
syms = [x, y]
assert L.jacobian(syms) == Matrix([[2 * x * y, x**2], [y, 4 * y + x]])
L = SMatrix(1, 2, [x, x**2 * y**3])
assert L.jacobian(syms) == SMatrix([[1, 0],
[2 * x * y**3, x**2 * 3 * y**2]])
# test_QR
A = Matrix([[1, 2], [2, 3]])
Q, S = A.QRdecomposition()
R = Rational
assert Q == Matrix([[5**R(-1, 2), (R(2) / 5) * (R(1) / 5)**R(-1, 2)],
[2 * 5**R(-1, 2), (-R(1) / 5) * (R(1) / 5)**R(-1, 2)]])
assert S == Matrix([[5**R(1, 2), 8 * 5**R(-1, 2)],
[0, (R(1) / 5)**R(1, 2)]])
assert Q * S == A
assert Q.T * Q == eye(2)
# test nullspace
# first test reduced row-ech form
R = Rational
M = Matrix([[5, 7, 2, 1], [1, 6, 2, -1]])
out, tmp = M.rref()
assert out == Matrix([[1, 0, -R(2) / 23, R(13) / 23],
[0, 1, R(8) / 23, R(-6) / 23]])
M = Matrix([[1, 3, 0, 2, 6, 3, 1], [-2, -6, 0, -2, -8, 3, 1],
[3, 9, 0, 0, 6, 6, 2], [-1, -3, 0, 1, 0, 9, 3]])
out, tmp = M.rref()
assert out == Matrix([[1, 3, 0, 0, 2, 0, 0], [0, 0, 0, 1, 2, 0, 0],
[0, 0, 0, 0, 0, 1, R(1) / 3], [0, 0, 0, 0, 0, 0, 0]])
# now check the vectors
basis = M.nullspace()
assert basis[0] == Matrix([[-3, 1, 0, 0, 0, 0, 0]])
assert basis[1] == Matrix([[0, 0, 1, 0, 0, 0, 0]])
assert basis[2] == Matrix([[-2, 0, 0, -2, 1, 0, 0]])
assert basis[3] == Matrix([[0, 0, 0, 0, 0, R(-1) / 3, 1]])
# test eigen
x = Symbol('x')
y = Symbol('y')
eye3 = eye(3)
assert eye3.charpoly(x) == (1 - x)**3
assert eye3.charpoly(y) == (1 - y)**3
# test values
M = Matrix([(0, 1, -1), (1, 1, 0), (-1, 0, 1)])
vals = M.eigenvals()
vals.sort()
assert vals == [-1, 1, 2]
R = Rational
M = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
assert M.eigenvects() == [[
1, 3,
[
Matrix(1, 3, [1, 0, 0]),
Matrix(1, 3, [0, 1, 0]),
Matrix(1, 3, [0, 0, 1])
]
]]
M = Matrix([[5, 0, 2], [3, 2, 0], [0, 0, 1]])
assert M.eigenvects() == [[1, 1,
[Matrix(1, 3,
[R(-1) / 2, R(3) / 2, 1])]],
[2, 1, [Matrix(1, 3, [0, 1, 0])]],
[5, 1, [Matrix(1, 3, [1, 1, 0])]]]
assert M.zeros((3, 5)) == SMatrix(3, 5, {})
from sympy import Matrix
from sympy import init_printing
L = []
print("LU Decomposition of n*n matrix.")
n = int(input("Enter value of n :\n"))
print("Enter the matrix elements :")
for i in range(n):
l = []
for j in range(n):
x = int(input())
l.append(x)
L.append(l)
A = Matrix(L)
L, U, _ = A.LUdecomposition()
print("L = ", L, "\n")
print("U = ", U)
import numpy as np
from sympy import Matrix
A = np.array([[2, 3, 1, 0], [4, -1, 9, 0], [3, 8, -2, 0], [1, -2, 4, 0]])
# 将numpy.array转为sympy.Matrix
MA = Matrix(A)
print("------------------行简化阶梯型-------------------")
print(MA.rref()) # 行简化阶梯型
print("------------------对角化矩阵-------------------")
M = Matrix([[3,-2,4,-2],[5,3,-3,-2],[5,-2,2,-2],[5,-2,-3,3]])
P, D = M.diagonalize() # 对角化矩阵
print(P)
print("对角矩阵 D：")
print(D)
print("------------------LU 分解-------------------")
print(MA.LUdecomposition()) # LU 分解
print("------------------QR 分解-------------------")
print(MA.QRdecomposition()) # QR 分解
# (把矩阵分解成一个正交矩阵与一个上三角矩阵的积 ,Q 是正交矩阵（意味着 QTQ = I）而 R 是上三角矩阵,此外，原矩阵A不必为正方矩阵； 如果矩阵A大小为n*m，则矩阵Q大小为n*m，矩阵R大小为m*m。)
