def qr_full(A):
"""
Q,R = qr_full(A)
This function is merely a wrapper of
UTPM.qr_full, Function.qr_full, scipy.linalg.qr
Parameters
----------
A: algopy.UTPM or algopy.Function or numpy.ndarray
A.shape = (M,N), M >= N
Returns
--------
Q: same type as A
Q.shape = (M,M)
R: same type as A
R.shape = (M,N)
"""
if isinstance(A, UTPM):
return UTPM.qr_full(A)
elif isinstance(A, Function):
return Function.qr_full(A)
elif isinstance(A, numpy.ndarray):
return scipy.linalg.qr(A)
else:
raise NotImplementedError('don\'t know what to do with this instance')
def qr_full(A):
"""
Q,R = qr_full(A)
This function is merely a wrapper of
UTPM.qr_full, Function.qr_full, scipy.linalg.qr
Parameters
----------
A: algopy.UTPM or algopy.Function or numpy.ndarray
A.shape = (M,N), M >= N
Returns
--------
Q: same type as A
Q.shape = (M,M)
R: same type as A
R.shape = (M,N)
"""
if isinstance(A, UTPM):
return UTPM.qr_full(A)
elif isinstance(A, Function):
return Function.qr_full(A)
elif isinstance(A, numpy.ndarray):
return scipy.linalg.qr(A)
else:
raise NotImplementedError('don\'t know what to do with this instance')
# first order derivatives, one directional derivative
# D - 1 is the degree of the Taylor polynomial
# P directional derivatives at once
# M number of rows of J1
# N number of cols of J1
# K number of rows of J2 (must be smaller than N)
D, P, M, N, K, Nx = 2, 1, 100, 3, 1, 1
# METHOD 1: nullspace method
cg1 = CGraph()
J1 = Function(UTPM(numpy.random.rand(*(D, P, M, N))))
J2 = Function(UTPM(numpy.random.rand(*(D, P, K, N))))
Q, R = Function.qr_full(J2.T)
Q2 = Q[:, K:].T
J1_tilde = dot(J1, Q2.T)
Q, R = qr(J1_tilde)
V = solve(R.T, Q2)
C = dot(V.T, V)
cg1.trace_off()
cg1.independentFunctionList = [J1, J2]
cg1.dependentFunctionList = [C]
print('covariance matrix: C =\n', C)
print('check that Q2.T spans the nullspace of J2:\n', dot(J2, Q2.T))
# METHOD 2: image space method (potentially numerically unstable)
# D - 1 is the degree of the Taylor polynomial
# P directional derivatives at once
# M number of rows of J1
# N number of cols of J1
# K number of rows of J2 (must be smaller than N)
D,P,M,N,K,Nx = 2,1,100,3,1,1
# METHOD 1: nullspace method
cg1 = CGraph()
J1 = Function(UTPM(numpy.random.rand(*(D,P,M,N))))
J2 = Function(UTPM(numpy.random.rand(*(D,P,K,N))))
Q,R = Function.qr_full(J2.T)
Q2 = Q[:,K:].T
J1_tilde = dot(J1,Q2.T)
Q,R = qr(J1_tilde)
V = solve(R.T, Q2)
C = dot(V.T,V)
cg1.trace_off()
cg1.independentFunctionList = [J1, J2]
cg1.dependentFunctionList = [C]
print('covariance matrix: C =\n',C)
print('check that Q2.T spans the nullspace of J2:\n', dot(J2,Q2.T))
# METHOD 2: image space method (potentially numerically unstable)
