def _energy_gradss(self,argnum,max_scf=301,max_d=300,printguess=None,name='Output.molden',output=False,order='first'):
"""This function returns the gradient of args"""
## For the moment it retuns a value at a time
## This is used only by testing functions.
eigen = True
rguess = False
args=[np.log(self.sys.alpha),self.sys.coef,self.sys.xyz,self.sys.l,self.sys.charges,self.sys.atom,self.sys.natoms,self.sys.nbasis,
self.sys.list_contr,self.sys.ne,
max_scf,max_d,log,eigen,None,None,
name,output,self.sys.alpha] # Last term is only used for Algopy
if self.verbose:
self.tape.write(' \n Grad point ...\n')
self.tape.write(' ---Start--- \n')
self.tape.write(' Initial parameters \n')
self.tape.write(' Maximum number of SCF: %d\n'%max_scf)
self.tape.write(' Default SCF tolerance: %f\n'%1e-8)
self.tape.write(' Initial density matrix: %s\n'%str(rguess))
self.sys.printcurrentgeombasis(self.tape)
grad_fun =[]
for i in argnum:
var = UTPM.init_jacobian(args[i])
diff_args = list(args) # We are making a copy of args
diff_args[i] = var
diff_args[-1]= var
t0 = time.clock()
grad = UTPM.extract_jacobian(rhfenergy(*(diff_args)))
timer = time.clock() - t0
self.sys.grad = grad
self.tape.write(' ---End--- \n')
self.tape.write(' Time %3.7f :\n'%timer)
return grad
def test_expm(self):
def f(x):
x = x.reshape((2,2))
return sum(expm(x))
x = numpy.random.random(2*2)
# forward mode
ax = UTPM.init_jacobian(x)
ay = f(ax)
g1 = UTPM.extract_jacobian(ay)
# reverse mode
cg = CGraph()
ax = Function(x)
ay = f(ax)
cg.independentFunctionList = [ax]
cg.dependentFunctionList = [ay]
g2 = cg.gradient(x)
assert_array_almost_equal(g1, g2)
def __test_kineticmatrix__(self, Mol):
''' In computes the kinetics matrix
and compare it with the file benckmark '''
tool = 1E-7
epsilon = 1e-5
T = kineticmatrix(Mol.alpha, Mol.coef, Mol.xyz, Mol.l, Mol.nbasis,
Mol.list_contr, np.float64(1.0))
f = open(Mol.tape + '_kinetics_pyquante.out', 'r')
f_lines = f.read().split('\n')
line = 0
for i in range(Mol.nbasis):
for j in range(Mol.nbasis):
self.assertAlmostEqual(T[i, j],
float(f_lines[line].split()[2]),
msg="Error: Test Kinetic " +
Mol.mol_name,
places=5)
line += 1
f.close()
grad_algo_alpha = UTPM.extract_jacobian(
kineticmatrix(Mol.alpha_algopy, Mol.coef, Mol.xyz, Mol.l,
Mol.nbasis, Mol.list_contr, Mol.alpha_algopy))
### Testing grad alpha:
for i in range(len(Mol.alpha)):
for j in range(len(Mol.alpha)):
alpha_epsilon = np.copy(Mol.alpha)
alpha_epsilon[i] = Mol.alpha[i] + epsilon
Tij_epsilon = kineticmatrix(alpha_epsilon, Mol.coef, Mol.xyz,
Mol.l, Mol.nbasis, Mol.list_contr,
np.float64(1.0))
dTij_da = (Tij_epsilon - T) / epsilon
np.testing.assert_almost_equal(
dTij_da,
grad_algo_alpha[:, :, i],
decimal=3,
verbose=True,
err_msg='Error: Test Overlap Grad')
grad_algo_coef = UTPM.extract_jacobian(
kineticmatrix(Mol.alpha, Mol.coef_algopy, Mol.xyz, Mol.l,
Mol.nbasis, Mol.list_contr, Mol.coef_algopy))
for i in range(len(Mol.alpha)):
for j in range(len(Mol.alpha)):
coef_epsilon = np.copy(Mol.coef)
coef_epsilon[i] = Mol.coef[i] + epsilon
Tij_epsilon = kineticmatrix(Mol.alpha, coef_epsilon, Mol.xyz,
Mol.l, Mol.nbasis, Mol.list_contr,
np.float64(1.0))
dTij_da = (Tij_epsilon - T) / epsilon
np.testing.assert_almost_equal(
dTij_da,
grad_algo_coef[:, :, i],
decimal=3,
verbose=True,
err_msg='Error: Test Overlap Grad')
pass
def nuclear_gradient(mol,center = np.array([0.0,0.0,0.0])):
''' In computes the geometric derivatives of the kinetics matrix '''
'''
grad_algo_xyz = np.zeros((mol.nbasis,mol.nbasis,3))
#for i in range(len(mol.alpha)):
for i in range(1):
xyz = np.array(mol.xyz[i] - center)
print xyz
xyz_algopy = UTPM.init_jacobian(xyz)
charges = [1]
ncharges = 1
grad_algo_xyz = grad_algo_xyz + UTPM.extract_jacobian(nuclearmatrix(mol.alpha,mol.coef,mol.xyz,mol.l,mol.nbasis,charges,xyz_algopy,ncharges,xyz_algopy))
'''
xyz = np.array(center)
xyz_algopy = UTPM.init_jacobian(xyz)
charges = [1]
ncharges = 1
grad_algo_xyz = UTPM.extract_jacobian(nuclearmatrix(mol.alpha,mol.coef,mol.xyz,mol.l,mol.nbasis,charges,xyz_algopy,ncharges,xyz_algopy))
print mol.atom
print grad_algo_xyz[:,:,0]
print grad_algo_xyz[:,:,1]
print grad_algo_xyz[:,:,2]
print grad_algo_xyz.shape
return grad_algo_xyz
def d_f(x):
"""function"""
return x[0] * x[1] * x[2] + exp(x[0]) * x[1] # x[differnce]
# forward AD without building a computational graph
x = UTPM.init_jacobian([3, 5, 7])
y = d_f(x)
algopy_jacobian = UTPM.extract_jacobian(y)
print('jacobian = ', algopy_jacobian)
# reverse mode using a computational graph
# Step 1/2 - trace the evaluation function
cg = algopy.CGraph()
x = algopy.Function([1, 2, 3])
y = d_f(x)
cg.trace_off()
cg.independentFunctionList = [x]
cg.dependentFunctionList = [y]
# Step 2/2 - use the graph to evaluate derivatives
print('gradient =', cg.gradient([3., 5, 7]))
print(
'Jacobian =', cg.jacobian([3., 5, 7])
) # a square matrix of first order partial derivatives, the derivative of f at all possible points wrt x
print(
'Hessian =', cg.hessian([3., 5., 7.])
) # a matrix of second order partial derivatives of the function in question (square), can use optimisation for local min/max/saddle of a critical value.
print('Hessian vector product =', cg.hess_vec([3., 5., 7.], [4, 5, 6]))
def probconv(p, x):
"""1-component BD(NBD)"""
# def BDNBD(p, x):
# return ((p[1] * (((1 - p[3]) / (1 - (p[3] * x))) ** p[2])) + (1. - p[1])) ** p[0]
"""1-component BD(NBD) X NBD"""
# def BDNBD(p, x):
# return (((p[1] * (((1. - p[3]) / (1. - (p[3] * x))) ** p[2])) + (1. - p[1])) ** p[0]) * (((1. - p[5]) / (1. - (p[5] * x))) ** p[4])
"""2-component BD(NBD) """
# def BDNBD(p, x):
# return (p[8] * (((p[1] * (((1 - p[3]) / (1 - (p[3] * x))) ** p[2])) + (1. - p[1])) ** p[0])) + ((1. - p[8]) * ((p[5] * (((1. - p[7]) / (1. - (p[7] * x))) ** p[6]) + (1. - p[5])) ** p[4]))
"""2-component BD(NBD) X NBD"""
def BDNBD(p, x):
return (p[12] *
((((p[1] * (((1. - p[3]) / (1. - (p[3] * x)))**p[2])) +
(1. - p[1]))**p[0]) *
(((1. - p[5]) /
(1. - (p[5] * x)))**p[4]))) + ((1. - p[12]) * ((
((p[7] * (((1. - p[9]) / (1. - (p[9] * x)))**p[8])) +
(1. - p[7]))**p[6]) * (((1. - p[11]) /
(1. - (p[11] * x)))**p[10])))
D = len(x)
P = 1
xderv = UTPM(np.zeros((D, P)))
xderv.data[0, 0] = 0 #the value to evaluate Cj at corresponding to z=0
xderv.data[1, 0] = 1
derv = BDNBD(p, xderv)
prob = derv.data[:, 0]
return prob
def algo_jaco(*args, **kwargs):
var = UTPM.init_hessian(args[narg])
diff_args = list(args) # We are making a copy of args
diff_args[narg] = var
diff_args[-1]= var
diff_args = tuple(diff_args)
return UTPM.extract_hessian(rhfenergy(*(diff_args)))
def magic5():
x = UTPM(derp)
x.data[1,0] = 1
y = algo_f(x)
#print len(y.data)
#print y.data[100],
return y.data[100][0]
def test_utpm_logdet_trace_expm(self):
D, P, N = 3, 5, 4
x = 0.1 * UTPM(numpy.random.randn(D, P, N, N))
x = UTPM.dot(x.T, x)
observed_logdet = UTPM.logdet(expm(x))
desired_logdet = UTPM.trace(x)
assert_allclose(observed_logdet.data, desired_logdet.data)
def alg_grad(x, func, args=()):
if not algopy_avail:
raise ImportError(
"Algopy not installed. Please install algopy or use diff_grad")
x = UTPM.init_jacobian(x)
y = func(x, *args)
#print("Y", y)
return UTPM.extract_jacobian(y)
def gradient(f, theta, x):
a = UTPM.init_hessian(x)
y = f(theta, a)
res = UTPM.extract_hessian(2, y) #UTPM.extract_jacobian(y)
out = res.diagonal(1)
return out[0]
def test_utpm_logdet_trace_expm(self):
D, P, N = 3, 5, 4
x = 0.1 * UTPM(numpy.random.randn(D, P, N, N))
x = UTPM.dot(x.T, x)
observed_logdet = UTPM.logdet(expm(x))
desired_logdet = UTPM.trace(x)
assert_allclose(observed_logdet.data, desired_logdet.data)
def J_fcn(x_new, x, t_new, t, p):
""" computes the Jacobian of F_fcn
all inputs are double arrays
"""
y = UTPM(numpy.zeros((D, N, N)))
y.data[0, :] = x_new
y.data[1, :, :] = numpy.eye(N)
F = F_fcn(y, x, t_new, t, p)
return F.data[1, :, :].T
def J_fcn(x_new, x, t_new, t, p):
""" computes the Jacobian of F_fcn
all inputs are double arrays
"""
y = UTPM(numpy.zeros((D,N,N)))
y.data[0,:] = x_new
y.data[1,:,:] = numpy.eye(N)
F = F_fcn(y, x, t_new, t, p)
return F.data[1,:,:].T
def __call__(self, z0=0):
z = np.atleast_1d(z0).ravel()
x = UTPM(np.zeros((self.n, 1, z.size), dtype=z.dtype))
x.data[0, 0, :] = z
x.data[1, 0, :] = 1
y = self.fun(x)
coefs = np.squeeze(y.data)
return coefs
def __init__(self,mol,shifted=False):
self.mol = select_geom.get(mol)
System_mol.__init__(self,self.mol,select_basis.get(mol),select_e.get(mol),mol)
self.tape = "./test/"+mol
self.energy = select_energy.get(mol)
self.coef = normalization(np.array(self.alpha),self.coef,self.l,self.list_contr)
#coef = normalization(np.array(self.alpha),np.copy(self.coef),self.xyz,self.l,self.list_contr)
self.alpha_algopy = UTPM.init_jacobian(self.alpha)
self.coef_algopy = UTPM.init_jacobian(self.coef)
return
def _forward(self, x, *args, **kwds):
x0 = np.asarray(x)
shape = x0.shape
P = 1
x = UTPM(np.zeros((self.n + 1, P) + shape))
x.data[0, 0] = x0
x.data[1, 0] = 1
z = self.fun(x, *args, **kwds)
y = UTPM.as_utpm(z)
return y.data[self.n, 0] * special.factorial(self.n)
def _forward(self, x, *args, **kwds):
d, n = 2+1, x.size
p = n
y = UTPM(np.zeros((d, p, n)))
y.data[0, :] = x.ravel()
y.data[1, :] = np.eye(n)
z0 = self.fun(y, *args, **kwds)
z = UTPM.as_utpm(z0)
H = z.data[2, ...] * 2
return H
def _example_taylor():
def f(x):
return x*x*x*x # np.sin(np.cos(x) + np.sin(x))
D = 5
P = 1
x = UTPM(np.zeros((D, P)))
x.data[0, 0] = 1.0
x.data[1, 0] = 1
y = f(x)
print('coefficients of y =', y.data[:, 0])
def _forward(self, x, *args, **kwds):
x0 = np.asarray(x)
shape = x0.shape
P = 1
x = UTPM(np.zeros((self.n + 1, P) + shape))
x.data[0, 0] = x0
x.data[1, 0] = 1
z = self.f(x, *args, **kwds)
y = UTPM.as_utpm(z)
return y.data[self.n, 0] * misc.factorial(self.n)
def _forward(self, x, *args, **kwds):
d, n = 2 + 1, x.size
p = n
y = UTPM(np.zeros((d, p, n)))
y.data[0, :] = x.ravel()
y.data[1, :] = np.eye(n)
z0 = self.fun(y, *args, **kwds)
z = UTPM.as_utpm(z0)
H = z.data[2, ...] * 2
return H
def _forward(self, x, *args, **kwds):
# return np.diag(super(Hessdiag, self)._forward(x, *args, **kwds))
D, Nm = 2+1, x.size
P = Nm
y = UTPM(np.zeros((D, P, Nm)))
y.data[0, :] = x.ravel()
y.data[1, :] = np.eye(Nm)
z0 = self.f(y, *args, **kwds)
z = UTPM.as_utpm(z0)
H = z.data[2, ...] * 2
return H
def alg_jac(x, func, args=()):
try:
func(x, *args)[0]
except IndexError:
# If func returns a scalar, the jacobian is just the gradient
return alg_grad(x, func, args)
if not algopy_avail:
raise ImportError(
"Algopy not installed. Please install algopy or use diff_jac")
x = UTPM.init_jacobian(x)
y = func_part(x, func, args)
return UTPM.extract_jacobian(y)
def _forward(self, x, *args, **kwds):
# return np.diag(super(Hessdiag, self)._forward(x, *args, **kwds))
D, Nm = 2 + 1, x.size
P = Nm
y = UTPM(np.zeros((D, P, Nm)))
y.data[0, :] = x.ravel()
y.data[1, :] = np.eye(Nm)
z0 = self.f(y, *args, **kwds)
z = UTPM.as_utpm(z0)
H = z.data[2, ...] * 2
return H
def _jacobian_forward(self, x, *args, **kwds):
x = np.asarray(x, dtype=float)
# shape = x.shape
D, Nm = 2, x.size
P = Nm
y = UTPM(np.zeros((D, P, Nm)))
y.data[0, :] = x.ravel()
y.data[1, :] = np.eye(Nm)
z0 = self.f(y, *args, **kwds)
z = UTPM.as_utpm(z0)
J = z.data[1, :, :, 0]
return J
def init_UTPM_jacobian(x):
# print 'type(x)=', type(x)
if isinstance(x, Function):
return x.init_UTPM_jacobian()
elif isinstance(x, numpy.ndarray):
return UTPM.init_jacobian(x)
elif isinstance(x, UTPM):
# print x.data.shape
return UTPM.init_UTPM_jacobian(x.data[0,0])
else:
raise ValueError('don\'t know what to do with this input!')
def test_expm_jacobian_vector_product(self):
n = 4
x = numpy.random.randn(n, n)
E = numpy.random.randn(n, n)
# use algopy to get the jacobian vector product
ax = UTPM.init_jac_vec(x.flatten(), E.flatten())
ay = expm(ax.reshape((n, n))).reshape((n * n, ))
g1 = UTPM.extract_jac_vec(ay)
# compute the jacobian vector product directly using expm_frechet
M = expm_frechet(x, E, compute_expm=False).flatten()
assert_allclose(g1, M, rtol=1e-6)
def init_UTPM_jacobian(x):
# print 'type(x)=', type(x)
if isinstance(x, Function):
return x.init_UTPM_jacobian()
elif isinstance(x, numpy.ndarray):
return UTPM.init_jacobian(x)
elif isinstance(x, UTPM):
# print x.data.shape
return UTPM.init_UTPM_jacobian(x.data[0, 0])
else:
raise ValueError('don\'t know what to do with this input!')
def algopy_fprime(xk, *args):
""" Evaluates the gradient of the function
Parameters:
xk : array_like
The coordinate vector at which to determine the gradient of `f`.
Returns:
grad : ndarray
The partial derivatives of `f` to `xk`.
"""
var = UTPM.init_jacobian(xk)
grad = UTPM.extract_jacobian(function(*(tuple([var])+args)))
return grad
def test_expm_jacobian_vector_product(self):
n = 4
x = numpy.random.randn(n, n)
E = numpy.random.randn(n, n)
# use algopy to get the jacobian vector product
ax = UTPM.init_jac_vec(x.flatten(), E.flatten())
ay = expm(ax.reshape((n, n))).reshape((n*n,))
g1 = UTPM.extract_jac_vec(ay)
# compute the jacobian vector product directly using expm_frechet
M = expm_frechet(x, E, compute_expm=False).flatten()
assert_allclose(g1, M, rtol=1e-6)
def ones(shape, dtype=float, order='C'):
"""
generic implementation of numpy.ones
"""
if numpy.isscalar(shape):
shape = (shape, )
if isinstance(dtype, type):
return numpy.ones(shape, dtype=dtype, order=order)
elif isinstance(dtype, numpy.ndarray):
return numpy.ones(shape, dtype=dtype.dtype, order=order)
elif isinstance(dtype, UTPM):
D, P = dtype.data.shape[:2]
tmp = numpy.zeros((D, P) + shape, dtype=dtype.data.dtype)
tmp[0, ...] = 1.
return UTPM(tmp)
elif isinstance(dtype, Function):
return dtype.pushforward(ones, [shape, dtype, order])
else:
return numpy.ones(shape, dtype=type(dtype), order=order)
def hessian_forward():
def f(x, *args, **kwds):
return x[0] + x[1] ** 2 + x[2] ** 3
x = np.asarray([1, 2, 3], dtype=float)
# shape = x.shape
D, Nm = 2+1, x.size
P = Nm
y = UTPM(np.zeros((D, P, Nm)))
y.data[0, :] = x.ravel()
y.data[1, :] = np.eye(Nm)
z0 = f(y)
z = UTPM.as_utpm(z0)
J = z.data[2, ...] * 2
return J
def extract_UTPM_jacobian(x):
if isinstance(x, Function):
return x.extract_UTPM_jacobian()
elif isinstance(x, UTPM):
return UTPM.extract_UTPM_jacobian(x)
else:
raise ValueError('don\'t know what to do with this input!')
def test_expm_jacobian(self):
n = 4
x = numpy.random.randn(n, n)
# use algopy to get the jacobian
ax = UTPM.init_jacobian(x)
ay = expm(ax)
g1 = UTPM.extract_jacobian(ay)
# compute the jacobian directly using expm_frechet
M = numpy.zeros((n, n, n*n))
ident = numpy.identity(n*n)
for i in range(n*n):
E = ident[i].reshape(n, n)
M[:, :, i] = expm_frechet(x, E, compute_expm=False)
assert_allclose(g1, M, rtol=1e-6)
def gradient(x):
out = []
if type(x) == float:
i = UTPM.init_jacobian([x])
y = f(i)
algopy_jacobian = UTPM.extract_jacobian(y)
out = algopy_jacobian
else:
for i in x:
i = UTPM.init_jacobian([i])
y = f(i)
algopy_jacobian = UTPM.extract_jacobian(y)
# print('jacobian = ',algopy_jacobian)
out.append(algopy_jacobian[0])
return np.array(out)
def test_expm_jacobian(self):
n = 4
x = numpy.random.randn(n, n)
# use algopy to get the jacobian
ax = UTPM.init_jacobian(x)
ay = expm(ax)
g1 = UTPM.extract_jacobian(ay)
# compute the jacobian directly using expm_frechet
M = numpy.zeros((n, n, n * n))
ident = numpy.identity(n * n)
for i in range(n * n):
E = ident[i].reshape(n, n)
M[:, :, i] = expm_frechet(x, E, compute_expm=False)
assert_allclose(g1, M, rtol=1e-6)
def extract_UTPM_jacobian(x):
if isinstance(x, Function):
return x.extract_UTPM_jacobian()
elif isinstance(x, UTPM):
return UTPM.extract_UTPM_jacobian(x)
else:
raise ValueError('don\'t know what to do with this input!')
def eigh1(A):
"""
generic implementation of eigh1
"""
if isinstance(A, UTPM):
return UTPM.eigh1(A)
elif isinstance(A, Function):
return Function.eigh1(A)
elif isinstance(A, numpy.ndarray):
A = UTPM(A.reshape((1,1) + A.shape))
retval = UTPM.eigh1(A)
return retval[0].data[0,0], retval[1].data[0,0],retval[2]
else:
raise NotImplementedError('don\'t know what to do with this instance')
def eigh1(A):
"""
generic implementation of eigh1
"""
if isinstance(A, UTPM):
return UTPM.eigh1(A)
elif isinstance(A, Function):
return Function.eigh1(A)
elif isinstance(A, numpy.ndarray):
A = UTPM(A.reshape((1,1) + A.shape))
retval = UTPM.eigh1(A)
return retval[0].data[0,0], retval[1].data[0,0],retval[2]
else:
raise NotImplementedError('don\'t know what to do with this instance')
def test_interpolation(self):
def f(x):
return x[0] + x[1] + 3.*x[0]*x[1] + 7.*x[1]*x[1] + 17.*x[0]*x[0]*x[0]
N = 2
D = 5
deg_list = [0,1,2,3,4]
coeff_list = []
for n,deg in enumerate(deg_list):
Gamma, rays = generate_Gamma_and_rays(N,deg)
x = UTPM(numpy.zeros((D,) + rays.shape))
x.data[1,:,:] = rays
y = f(x)
coeff_list.append(numpy.dot(Gamma, y.data[deg]))
assert_array_almost_equal([0], coeff_list[0])
assert_array_almost_equal([1,1], coeff_list[1])
assert_array_almost_equal([0,3,7], coeff_list[2])
assert_array_almost_equal([17,0,0,0], coeff_list[3])
def vecsym(v):
if isinstance(v, UTPM):
return UTPM.vecsym(v)
elif isinstance(v, Function):
return Function.vecsym(v)
elif isinstance(v, numpy.ndarray):
return utils.vecsym(v)
else:
raise NotImplementedError('don\'t know what to do with this instance')
def dot(a,b):
"""
Same as NumPy dot but in UTP arithmetic
"""
if isinstance(a,Function) or isinstance(b,Function):
return Function.dot(a,b)
elif isinstance(a,UTPM) or isinstance(b,UTPM):
return UTPM.dot(a,b)
else:
return numpy.dot(a,b)
def outer(a, b):
"""
Same as NumPy outer but in UTP arithmetic
"""
if isinstance(a, Function) or isinstance(b, Function):
return Function.outer(a, b)
elif isinstance(a, UTPM) or isinstance(b, UTPM):
return UTPM.outer(a, b)
else:
return numpy.outer(a, b)
def dot(a, b):
"""
Same as NumPy dot but in UTP arithmetic
"""
if isinstance(a, Function) or isinstance(b, Function):
return Function.dot(a, b)
elif isinstance(a, UTPM) or isinstance(b, UTPM):
return UTPM.dot(a, b)
else:
return numpy.dot(a, b)
def symvec(A, UPLO='F'):
if isinstance(A, UTPM):
return UTPM.symvec(A, UPLO=UPLO)
elif isinstance(A, Function):
return Function.symvec(A, UPLO=UPLO)
elif isinstance(A, numpy.ndarray):
return utils.symvec(A, UPLO=UPLO)
else:
raise NotImplementedError('don\'t know what to do with this instance')
def vecsym(v):
if isinstance(v, UTPM):
return UTPM.vecsym(v)
elif isinstance(v, Function):
return Function.vecsym(v)
elif isinstance(v, numpy.ndarray):
return utils.vecsym(v)
else:
raise NotImplementedError('don\'t know what to do with this instance')
def IV_algopy_jac (Ee, Tc, Rs, Rsh, Isat1_0, Isat2, Isc0, alpha_Isc, Eg, Vd):
"""
Calculate Jacobian of IV curve using AlgoPy
:param Ee: [suns] effective irradiance
:param Tc: [C] cell temperature
:param Rs: [ohms] series resistance
:param Rsh: [ohms] shunt resistance
:param Isat1_0: [A] saturation current of first diode at STC
:param Isat2: [A] saturation current of second diode
:param Isc0: [A] short circuit current at STC
:param alpha_Isc: [1/K] short circuit current temperature coefficient
:param Eg: [eV] band gap
:param Vd: [V] diode voltages
:return: Jacobian :math:`\\frac{\\partial f_i}{\\partial x_{j,k}}`
where :math:`k` are independent observations of :math:`x`
"""
x = UTPM.init_jacobian([
Ee, Tc, Rs, Rsh, Isat1_0, Isat2, Isc0, alpha_Isc, Eg
])
return UTPM.extract_jacobian(IV_algopy(x, Vd))
def outer(a,b):
"""
Same as NumPy outer but in UTP arithmetic
"""
if isinstance(a,Function) or isinstance(b,Function):
return Function.outer(a,b)
elif isinstance(a,UTPM) or isinstance(b,UTPM):
return UTPM.outer(a,b)
else:
return numpy.outer(a,b)
def symvec(A, UPLO='F'):
if isinstance(A, UTPM):
return UTPM.symvec(A, UPLO=UPLO)
elif isinstance(A, Function):
return Function.symvec(A, UPLO=UPLO)
elif isinstance(A, numpy.ndarray):
return utils.symvec(A, UPLO=UPLO)
else:
raise NotImplementedError('don\'t know what to do with this instance')
def test_interpolation(self):
def f(x):
return x[0] + x[1] + 3. * x[0] * x[1] + 7. * x[1] * x[1] + 17. * x[
0] * x[0] * x[0]
N = 2
D = 5
deg_list = [0, 1, 2, 3, 4]
coeff_list = []
for n, deg in enumerate(deg_list):
Gamma, rays = generate_Gamma_and_rays(N, deg)
x = UTPM(numpy.zeros((D, ) + rays.shape))
#print x
#print type(x)
x.data[1, :, :] = rays
y = f(x)
coeff_list.append(numpy.dot(Gamma, y.data[deg]))
assert_array_almost_equal([0], coeff_list[0])
assert_array_almost_equal([1, 1], coeff_list[1])
assert_array_almost_equal([0, 3, 7], coeff_list[2])
assert_array_almost_equal([17, 0, 0, 0], coeff_list[3])
def fft(a, n=None, axis=-1):
"""
equivalent to numpy.fft.fft(a, n=None, axis=-1)
"""
if isinstance(a, UTPM):
return UTPM.fft(a, n=n, axis=axis)
elif isinstance(a, Function):
return Function.fft(a, n=n, axis=axis)
elif isinstance(a, numpy.ndarray):
return numpy.fft.fft(a, n=n, axis=axis)
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')
# compute Taylor series # # Jx( 1. + 2.*t + 3.*t**2 + 4.*t**3 + 5.*t**5, # 6. + 7.*t + 8.*t**2 + 9.*t**3 + 10.*t**5 ) # Jy( 1. + 2.*t + 3.*t**2 + 4.*t**3 + 5.*t**5, # 6. + 7.*t + 8.*t**2 + 9.*t**3 + 10.*t**5 ) # # where # # Jx = dg/dx # Jy = dg/dy # setup input Taylor polynomials D,P = 5, 3 # order D=5, number of directions P ax = UTPM(numpy.zeros((D, P))) ay = UTPM(numpy.zeros((D, P))) ax.data[:, :] = numpy.array([1., 2. ,3., 4. ,5.]).reshape((5,1)) # input Taylor polynomial ay.data[:, :] = numpy.array([6., 7. ,8., 9. ,10.]).reshape((5,1)) # input Taylor polynomial # forward sweep cg.pushforward([ax, ay]) azbar = UTPM(numpy.zeros((D, P, 3))) azbar.data[0, ...] = numpy.eye(3) # reverse sweep cg.pullback([azbar]) # get results Jx = cg.independentFunctionList[0].xbar
import numpy
from algopy import CGraph, Function, UTPM, dot, qr, eigh, inv, solve
# 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 A
# N number of cols of A
D,P,M,N = 2,1,5,2
# generate badly conditioned matrix A
A = UTPM(numpy.zeros((D,P,M,N)))
x = UTPM(numpy.zeros((D,P,M,1)))
y = UTPM(numpy.zeros((D,P,M,1)))
x.data[0,0,:,0] = [1,1,1,1,1]
x.data[1,0,:,0] = [1,1,1,1,1]
y.data[0,0,:,0] = [1,2,1,2,1]
y.data[1,0,:,0] = [1,2,1,2,1]
alpha = 10**-5
A = dot(x,x.T) + alpha*dot(y,y.T)
A = A[:,:2]
# Method 1: Naive approach
Apinv = dot(inv(dot(A.T,A)),A.T)
print('naive approach: A Apinv A - A = 0 \n', dot(dot(A, Apinv),A) - A)
import numpy; from numpy import sin,cos; from algopy import UTPM, zeros
D,P = 4,1
x = UTPM(numpy.zeros((D,P,2)))
x.data[0,:,0] = 1
p = UTPM(numpy.zeros((D,P)))
p.data[0,:] = 3; p.data[1,:] = 1
def f(t, x, p):
retval = x.copy()
retval[0] = x[1]
retval[1] = -p* x[0]
return retval
def implicit_euler(f_fcn, x0, ts, p):
""" implicit euler with fixed stepsizes, using Newton's method to solve
the occuring implicit system of nonlinear equations
"""
def F_fcn(x_new, x, t_new, t, p):
""" implicit function to solve: 0 = F(x_new, x, t_new, t_old)"""
return (t_new - t) * f_fcn(t_new, x_new, p) - x_new + x
def J_fcn(x_new, x, t_new, t, p):
""" computes the Jacobian of F_fcn
all inputs are double arrays
"""
y = UTPM(numpy.zeros((D,N,N)))
y.data[0,:] = x_new
y.data[1,:,:] = numpy.eye(N)
F = F_fcn(y, x, t_new, t, p)
return F.data[1,:,:].T
print eval_jac_g_forward(x)
print eval_g(x)
def f(x):
nobs = x.shape[1:]
f0 = x[0]**2 * sin(x[1])**2
f1 = x[0]**2 * cos(x[1])**2
out = zeros((2,) + nobs, dtype=x)
out[0,:] = f0
out[1,:] = f1
return out
x = np.array([(1, 2, 3, 4),(5, 6, 7, 8)],dtype=float)
y = f(x)
xj = UTPM.init_jacobian(x)
j = UTPM.extract_jacobian(f(xj))
print "x =\n%r\n" % x
print "f =\n%r\n" % y
print "j =\n%r\n" % j
# time it
jaca = nda.Jacobian(f)
x = np.array([np.arange(100),np.random.rand(100)])
%timeit jaca(x)
# x =
# array([[ 1., 2., 3., 4.],
# [ 5., 6., 7., 8.]])
import numpy; from algopy import UTPM
# symmetric eigenvalue decomposition, forward UTPM
D,P,M,N = 3,1,4,4
Q,R = UTPM.qr(UTPM(numpy.random.rand(D,P,M,N)))
l = UTPM(numpy.random.rand(*(D,P,N)))
l.data[0,0,:4] = [1,1,2,3]
l.data[1,0,:4] = [0,0,3,4]
l.data[2,0,:4] = [1,2,5,6]
L = UTPM.diag(l)
B = UTPM.dot(Q,UTPM.dot(L,Q.T))
print('B = \n', B)
l2,Q2 = UTPM.eigh(B)
print('l2 - l =\n',l2 - l)
In the reverse mode of AD one computes M adjoint derivatives, i.e. Q = M.
"""
import numpy
from algopy import CGraph, Function, UTPM, dot, qr, eigh, inv, zeros
def f(y):
retval = zeros((3,1),dtype=y)
retval[0,0] = numpy.log(dot(y.T,y))
retval[1,0] = numpy.exp(dot(y.T,y))
retval[2,0] = numpy.exp(dot(y.T,y)) - numpy.log(dot(y.T,y))
return retval
D,Nm = 2,40
P = Nm
y = UTPM(numpy.zeros((2,P,Nm)))
y.data[0,:] = numpy.random.rand(Nm)
y.data[1,:] = numpy.eye(Nm)
# print f(y)
J = f(y).data[1,:,:,0]
print('Jacobian J(y) = \n', J)
C_epsilon = 0.3*numpy.eye(Nm)
print(J.shape)
C = dot(J.T, dot(C_epsilon,J))
def eval_jac_g_forward(x):
x = UTPM.init_jacobian(x)
return UTPM.extract_jacobian(eval_g(x))