def orthonormal_polynomials(self, *max_degree):
'''
Return the max_degree-1 first orthonormal polynomials associated with
the NormalRandomVariable.
Parameters
----------
max_degree : int, optional
The maximum degree of the returned polynomials. The default is
None, choosing the default maximum degree associated with the
constructor of the polynomials.
Returns
-------
poly : tensap.OrthonormalPolynomials
The generated orthonormal polynomials.
'''
poly = tensap.HermitePolynomials(*max_degree)
if self != NormalRandomVariable(0, 1):
# print('ShiftedOrthonormalPolynomials are created.')
poly = tensap.ShiftedOrthonormalPolynomials(poly, self.mu,
self.sigma)
return poly
# %% Projection on polynomial space through quadrature
def FUN(x):
# Function to approximate
return x**2/2
FUN = tensap.UserDefinedFunction(FUN, 1)
FUN.evaluation_at_multiple_points = True
X = tensap.NormalRandomVariable()
# Integration rule
I = X.gauss_integration_rule(5)
# Approximation basis
P = 3
H = tensap.PolynomialFunctionalBasis(tensap.HermitePolynomials(), range(P+1))
# Computation of the approximation
F = H.projection(FUN, I)
# Derivative and second derivative of F through projection
DF = H.projection(lambda x, F=F: F.eval_derivative(1, x), I)
DDF = H.projection(lambda x, DF=DF: DF.eval_derivative(1, x), I)
# Displays and error
print('\nProjection on polynomial space through quadrature')
N = 100
ERR_L2, ERR_L_INF = F.test_error(FUN, N, X)
print('Mean squared error = %2.5e' % ERR_L2)
plt.figure()
X_PLOT = np.linspace(-1, 1, 100)
'''
import sys
import numpy as np
sys.path.insert(0, './../../../')
import tensap
# %% Approximation of a function F(X)
DIM = 6
FUN = tensap.UserDefinedFunction('x0 + x0*x1 + x0*x2**2 + x3**3 + x4 + x5',
DIM)
FUN.evaluationAtMultiplePoints = True
DEGREE = 3
X = tensap.RandomVector(tensap.NormalRandomVariable(0, 1), DIM)
P = tensap.PolynomialFunctionalBasis(tensap.HermitePolynomials(),
range(DEGREE+1))
BASES = tensap.FunctionalBases.duplicate(P, DIM)
GRIDS = X.random(1000)
G = tensap.FullTensorGrid([np.reshape(GRIDS[:, i], [-1, 1]) for
i in range(DIM)])
H = tensap.FullTensorProductFunctionalBasis(BASES)
F, OUTPUT = H.tensor_product_interpolation(FUN, G)
X_TEST = X.random(1000)
F_X_TEST = F(X_TEST)
Y_TEST = FUN(X_TEST)
ERR = np.linalg.norm(Y_TEST-F_X_TEST) / np.linalg.norm(Y_TEST)
print('Mean squared error = %2.5e\n' % ERR)