"""Package teneva, module func_full: functional full format with interpolation.
This module contains the functions for construction of the functional
representation in the full format with Chebyshev interpolation, as well as
calculating the values of the function using the constructed interpolation
coefficients. See module "func" with the same functions in the TT-format. The
functions presented in this module are especially relevant for the
one-dimensional (and two-dimensional) case, when the TT-decomposition cannot be
applied.
"""
import numpy as np
import teneva
[docs]def func_get_full(X, A, a, b, z=0., skip_out=True):
"""Compute the Chebyshev approximation in given points (approx. f(X)).
Args:
X (np.ndarray): spatial points of interest (it is 2D array of the shape
[samples, d], where d is the number of dimensions).
A (list): TT-tensor of the interpolation coefficients (it has d
dimensions).
a (float, list, np.ndarray): grid lower bounds for each dimension (list
or np.ndarray of length d). It may be also float, then the lower
bounds for each dimension will be the same.
b (float, list, np.ndarray): grid upper bounds for each dimension (list
or np.ndarray of length d). It may be also float, then the upper
bounds for each dimension will be the same.
z (float): the value for points, which are outside the spatial grid.
skip_out (bool): if flag is set, then the values outside the spatial
grid will be set to z values.
Returns:
np.ndarray: approximated function values in X points (it is 1D array of
the shape [samples]).
Note:
See also the same function ("func_get") in the TT-format.
"""
d = len(A.shape)
n = A.shape
m = X.shape[0]
a, b, n = teneva.grid_prep_opts(a, b, n, d)
# TODO: check if this operation is effective. It may be more profitable to
# generate polynomials for each tensor mode separately:
T = teneva.func_basis(teneva.poi_scale(X, a, b, 'cheb'), max(n))
Y = np.ones(m) * z
for i in range(m):
if skip_out:
if np.max(a - X[i, :]) > 1.E-99 or np.max(X[i, :] - b) > 1.E-99:
# We skip the points outside the grid bounds:
continue
Q = A.copy()
for j in range(d):
Q = np.tensordot(Q, T[:n[j], i, j], axes=([0], [0]))
Y[i] = Q
return Y
[docs]def func_gets_full(A, a, b, m=None):
"""Compute the Chebyshev approximation all over the new grid.
Args:
A (np.ndarray): d-dimensional tensor of the interpolation coefficients.
a (float, list, np.ndarray): grid lower bounds for each dimension (list
or np.ndarray of length d). It may be also float, then the lower
bounds for each dimension will be the same.
b (float, list, np.ndarray): grid upper bounds for each dimension (list
or np.ndarray of length d). It may be also float, then the upper
bounds for each dimension will be the same.
m (int, float, list, np.ndarray): tensor size for each dimension of the
new grid (list or np.ndarray of length d). It may be also
int/float, then the size for each dimension will be the same. If it
is not set, then original grid size (from the interpolation) will be
used.
Returns:
np.ndarray: array of the approximated function values on the full new
grid. This is the d-dimensional array of the shape m.
Note:
This function is not efficient in the full format, and corresponds to a
simple explicit calculation of all values at the nodes of the new grid.
At the same time, in the TT-format ("func_gets"), the corresponding
operation is carried out in an implicit efficient form. Accordingly,
this function is provided only for uniformity of presentation.
"""
d = len(A.shape)
n = A.shape
m = n if m is None else teneva.grid_prep_opt(m, d, int)
I = teneva.grid_flat(m)
X = teneva.ind_to_poi(I, -1., +1., m, 'cheb')
Z = func_get_full(X, A, -1., +1.)
Z = Z.reshape(m, order='F')
return Z
[docs]def func_int_full(Y):
"""Compute the tensor for Chebyshev interpolation coefficients.
Args:
Y (np.ndarray): d-dimensional array with function values on the
Chebyshev grid.
Returns:
np.ndarray: array that collects interpolation coefficients. It has the
same shape as the given tensor Y.
Note:
See also the same function ("func_int") in the TT-format.
"""
d = len(Y.shape)
A = Y.copy()
for k in range(d):
n = np.array(Y.shape, dtype=int)
m = n[k]
n[[0, k]] = n[[k, 0]]
A = np.swapaxes(A, 0, k)
A = A.reshape((m, -1), order='F')
A = np.vstack([A, A[m-2 : 0 : -1, :]])
A = np.fft.fft(A, axis=0).real
A = A[:m, :] / (m - 1)
A[0, :] /= 2.
A[m-1, :] /= 2.
A = A.reshape(n, order='F')
A = np.swapaxes(A, 0, k)
return A
[docs]def func_sum_full(A, a, b):
"""Integrate the function from its Chebyshev approximation.
Args:
A (np.ndarray): d-dimensional tensor of the interpolation coefficients.
a (float, list, np.ndarray): grid lower bounds for each dimension (list
or np.ndarray of length d). It may be also float, then the lower
bounds for each dimension will be the same.
b (float, list, np.ndarray): grid upper bounds for each dimension (list
or np.ndarray of length d). It may be also float, then the upper
bounds for each dimension will be the same.
Returns:
float: the value of the integral.
Note:
This function works only for symmetric grids! See also the same
function ("func_sum") in the TT-format.
"""
d = len(A.shape)
n = A.shape
a, b, n = teneva.grid_prep_opts(a, b, n, d)
for k in range(d):
if abs(abs(b[k]) - abs(a[k])) > 1.E-16:
raise ValueError('This function works only for symmetric grids')
v = A.copy()
for k in range(d):
v = v.reshape(n[k], -1)
p = np.arange(n[k])[::2]
p = np.repeat(p.reshape(-1, 1), v.shape[1], axis=1)
v = np.sum(v[::2, :] * 2. / (1. - p**2), axis=0)
v *= (b[k] - a[k]) / 2.
return v[0]