def hesaff_output():
import sympy as sy
import collections
from sympy.matrices.expressions.factorizations import lu, LofCholesky, qr, svd
import sympy
import sympy.matrices
import sympy.matrices.expressions
import sympy.matrices.expressions.factorizations
#from sympy.mpmath import sqrtm
sqrtm = sympy.mpmath.sqrtm
a, b, c, a11, a12, a21, a22 = sy.symbols('a b c a11 a12 a21 a22', real=True, commutative=True, nonzero=True, imaginary=False, comparable=True)
E = sy.Matrix(((a, b), (b, c)))
A = sy.Matrix(((a11, 0), (a21, a22)))
#x = E.solve(A.T * A)
# A.T * A == E
eq1 = sy.Eq(E, A.T * A)
a_ = sy.solve(eq1, a, check=False)
b_ = sy.solve(eq1, b, check=False)
c_ = sy.solve(eq1, c, check=False)
a11_ = sy.solve(eq1, a11, check=False)
a21_ = sy.solve(eq1, a21, check=False)
a22_ = sy.solve(eq1, a22, check=False)
eq2 = eq1.subs(a11, a11_[0][0])
a21_ = sy.solve(eq2, a21, check=False)
eq3 = eq1.subs(a21, a21_[a21])
a22_ = sy.solve(eq3, a22, check=False)
L1, D1 = E.LDLdecomposition()
U1 = L1.T
E_LDL1 = L1 * D1 * U1
L3 = E.cholesky()
E3 = L3 * L3.T * L3.inv() * L3
E3 = L3 * L3.T * (L3.inv() * L3) * (L3.T.inv() * L3.T )
E3 = L3 * (L3.inv() * L3).T L3.T * (L3.T.inv() * L3.T ) *
L3.inv() * E3 = L3.T
A2 = L3.T
A2.T * A2
print(E_LDL1)
L2, U2, p = E.LUdecomposition()
E_LU2 = L2 * U2
print(E_LU2)
#---------------
def asMatrix(list_):
N = int(len(list_) / 2)
Z = sympy.Matrix([list_[0:N], list_[N:]])
return Z
def simplify_mat(X):
list_ = [_.simplify() for _ in X]
Z = asMatrix(list_)
return Z
def symdot(X, Y):
Z = asMatrix(X.dot(Y))
return Z
Eq = sy.Eq
solve = sy.solve
R = A
M = symdot(A,A)
_b = sy.solve(b, eq1)
# Solve in terms of A
eq1 = sy.Eq(a, M[0])
eq2 = sy.Eq(c, M[2])
eq3 = sy.Eq(d, M[3])
w1 = sy.solve(eq1, w)[1].subs(y,0)
#y1 = sympy.Eq(0, M[1]) # y is 0
x1 = sy.solve(eq2, x)[0]
z1 = sy.solve(eq3, z)[1].subs(y,0)
x2 = x1.subs(w, w1).subs(z, z1)
R_itoA = simplify_mat(sympy.Matrix([(w1, x2), (0, z1)]))
Rinv_itoA = simplify_mat(R_itoA.inv())
print('Given the lower traingular matrix: A=[(a, 0), (c, d)]')
print('Its inverse is: inv(A)')
print(Ainv) # sub to lower triangular
print('--')
print('Its square root is: R = sqrtm(A)')
print(R_itoA)
print('--')
Epinv = E.pinv()
# Left sinuglar vectors are eigenvectors of M * M.H
left_singular_vects = (E * E.T).eigenvects()
# Right sinuglar vectors are eigenvectors of M * M.H
right_singular_vects = (E.T * E).eigenvects()
# Singular values
singular_vals = E.singular_values()
U = sy.Matrix([list(_[2][0]) for _ in left_singular_vects]).T
S = sy.Matrix([(sing_vals[0], 0),(0, singular_vals[1])])
V = sy.Matrix([list(_[2][0]) for _ in right_singular_vects]).T
assert U.shape == S.shape == V.shape == E.shape
assert sy.ask(sy.Q.orthogonal(U))
assert sy.ask(sy.Q.orthogonal(V))
assert sy.ask(sy.Q.diagonal(S))
u,s,v = svd(E)
#n = sy.Symbol('n')
#X = sy.MatrixSymbol('X', n, n)
U, S, V = svd(E)
assert U.shape == S.shape == V.shape == E.shape
assert sy.ask(sy.Q.orthogonal(U))
assert sy.ask(sy.Q.orthogonal(V))
assert sy.ask(sy.Q.diagonal(S))
# L.H is conjugate transpose
# E = L.dot(L.H)
L, U, p = E.LUdecomposition_Simple()
Q, R = E.QRdecomposition()
E.LUdecompositionFF
def SVD(A):
UEM, UEV = (A.T * A).diagonalize(normalize=True, sort=True)
VEM, VEV = (A * A.T).diagonalize(normalize=True, sort=True)
sigma = UEV ** sy.S(1)/2
return UEM, sigma, VEM
U, S, V = SVD(E)
help(E.cholesky)
help(E.LDLdecomposition)
help(E.QRdecomposition)
help(E.LUdecomposition_Simple)
help(E.LUdecompositionFF)
help(E.LUsolve)
L = E.cholesky()
E_2 = L * L.T
M = Matrix(((1,0,0,0,2),(0,0,3,0,0),(0,0,0,0,0),(0,4,0,0,0)))
M = sy.Matrix(2,3, [1,2,3,4,5,6])
N = M.H * (M * M.H) ** -1
N = M.H * (M * M.H) ** -1
U, Sig, V = M.SVD()
assert U * Sig * V.T == M
assert U*U.T == U.T*U == eye(U.cols)
assert V*V.T == V.T*V == eye(V.cols)
#assert S.is_diagonal()
M = Matrix(((0,1),(1,0),(1,1)))
U, Sig, V = M.SVD()
assert U * Sig * V.T == M
assert U*U.T == U.T*U == eye(U.cols)
assert V*V.T == V.T*V == eye(V.cols)
import sympy
import sympy.galgebra.GA as GA
import sympy.galgebra.latex_ex as tex
import sympy.printing as symprint
import sympy.abc
import sympy.mpmath
a, b, c, d = sympy.symbols('a b c d')
theta = sympy.abc.theta
sin = sympy.functions.elementary.trigonometric.sin
cos = sympy.functions.elementary.trigonometric.cos
sqrtm = sympy.mpmath.sqrtm
xc = sympy.Matrix([sin(theta), cos(theta)])
E = sympy.Matrix([(a, b), (b, c)])
def test_svd():
U, S, V = svd(X)
assert U.shape == S.shape == V.shape == X.shape
assert ask(Q.orthogonal(U))
assert ask(Q.orthogonal(V))
assert ask(Q.diagonal(S))
def test_svd():
U, S, V = svd(X)
assert U.shape == S.shape == V.shape == X.shape
assert ask(Q.orthogonal(U))
assert ask(Q.orthogonal(V))
assert ask(Q.diagonal(S))
