def test_sparse_matrix():
return
def eye(n):
tmp = SMatrix(n,n,lambda i,j:0)
for i in range(tmp.rows):
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.rows
assert m.cols == 2
assert m[:] == [x,0,0,0]
b = SMatrix(2,2, [x, 0, 0, 0])
m = b
assert m.cols == m.rows
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, {})
def zeros(n):
return SMatrix(n,n,lambda i,j:0)
def test_SMatrix():
M = SMatrix([[x**+1, 1], [y, x + y]])
assert str(M) == sstr(M) == "[x, 1]\n[y, x + y]"
def eye(n):
tmp = SMatrix(n,n,lambda i,j:0)
for i in range(tmp.rows):
tmp[i,i] = 1
return tmp
def test_sparse_matrix():
return
def eye(n):
tmp = SMatrix(n,n,lambda i,j:0)
for i in range(tmp.rows):
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.rows
assert m.cols == 2
assert m[:] == [x,0,0,0]
b = SMatrix(2,2, [x, 0, 0, 0])
m = b
assert m.cols == m.rows
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, {})
def test_SMatrix_add():
assert SMatrix(((1,0), (0,1))) + SMatrix(((0,1), (1,0))) == SMatrix(((1,1), (1,1)))
a = SMatrix(100, 100, lambda i, j : int(j != 0 and i % j == 0))
b = SMatrix(100, 100, lambda i, j : int(i != 0 and j % i == 0))
assert (len(a.mat) + len(b.mat) - len((a+b).mat) > 0)
def test_SMatrix_CL_RL():
assert SMatrix((1,2),(3,4)).row_list() == [(0, 0, 1), (0, 1, 2), (1, 0, 3), (1, 1, 4)]
assert SMatrix((1,2),(3,4)).col_list() == [(0, 0, 1), (1, 0, 3), (0, 1, 2), (1, 1, 4)]
def test_SMatrix_transpose():
assert SMatrix((1,2),(3,4)).transpose() == SMatrix((1,3),(2,4))
