tensap.tools package¶
Submodules¶
tensap.tools.chebyshev_points module¶
Module chebyshev_points.
-
tensap.tools.chebyshev_points.chebyshev_points(n, s=None)¶ Return the first n Chebyshev points in [s[0], s[1]].
- Parameters
- nint
The number of Chebyshev points.
- slist or numpy.ndarray, optional
The interval on which the points are computed. The default is [-1, 1].
- Returns
- xnumpy.ndarray
The first n Chebyshev points in [s[0], s[1]].
tensap.tools.grids module¶
Module grids.
-
class
tensap.tools.grids.FullTensorGrid(grids, dim=None)¶ Bases:
tensap.tools.grids.TensorGridClass FullTensorGrid: tensor product grid.
- Attributes
sizeReturn the number of grid points.
Methods
array([ind])Return an array of shape (n, d) where n is the number of grid points and d is the dimension.
Return a set of multi-indices for indexing the grid.
ndim()Return the dimension of the underlying space.
plot(y, *args)Plot the grid.
plot_grid(*args, **kwargs)Plot the grid.
random(X, n)Generate a random FullTensorGrid from a RandomVector.
shape([d])Return the shape of the grid, along the dimension d if provided.
eval_at_indices
-
array(ind=None)¶ Return an array of shape (n, d) where n is the number of grid points and d is the dimension.
- Returns
- numpy.ndarray
The TensorGrid as an array.
-
eval_at_indices(ind)¶
-
multi_indices()¶ Return a set of multi-indices for indexing the grid.
- Returns
- tensap.MultiIndices
Aset of multi-indices for indexing the grid.
-
plot(y, *args)¶ Plot the grid.
- Parameters
- ylist or numpy.ndarray
The values of the function on the grid.
- *argstuple
Parameters of the plot, see matplotlib.pyplot.plot for more information.
- Returns
- None.
-
static
random(X, n)¶ Generate a random FullTensorGrid from a RandomVector.
- Parameters
- Xtensap.RandomVector
The RandomVector used to generate the FullTensorGrid.
- nlist or numpy.ndarray
Array of size d (or an integer for isotropic grid) containing the sizes of the grids in each dimension.
- Returns
- tensap.FullTensorGrid
The random FullTensorGrid.
-
property
size¶ Return the number of grid points.
- Returns
- int
The number of grid points.
-
class
tensap.tools.grids.SparseTensorGrid(grids, indices, dim=None)¶ Bases:
tensap.tools.grids.TensorGridClass SparseTensorGrid: sparse tensor product grid.
- Attributes
sizeReturn the number of grid points.
Methods
array()Return an array of shape (n, d) where n is the number of grid points and d is the dimension.
ndim()Return the dimension of the underlying space.
plot_grid(*args, **kwargs)Plot the grid.
shape([d])Return the shape of the grid, along the dimension d if provided.
-
array()¶ Return an array of shape (n, d) where n is the number of grid points and d is the dimension.
- Returns
- numpy.ndarray
The TensorGrid as an array.
-
property
size¶ Return the number of grid points.
- Returns
- int
The number of grid points.
-
class
tensap.tools.grids.TensorGrid¶ Bases:
abc.ABCClass TensorGrid: grids in product sets.
- Attributes
- dimint
The dimension.
- grids: list
List containing the grids.
- shape: numpy.ndarray
The shapes of the grids.
Methods
array()Return an array of shape (n, d) where n is the number of grid points and d is the dimension.
ndim()Return the dimension of the underlying space.
plot_grid(*args, **kwargs)Plot the grid.
shape([d])Return the shape of the grid, along the dimension d if provided.
-
abstract
array()¶ Return an array of shape (n, d) where n is the number of grid points and d is the dimension.
- Returns
- numpy.ndarray
The TensorGrid as an array.
-
ndim()¶ Return the dimension of the underlying space.
- Returns
- int
The dimension of the underlying space.
-
plot_grid(*args, **kwargs)¶ Plot the grid.
- Parameters
- *argstuple
Parameters of the plot, see matplotlib.pyplot.plot for more information.
- Returns
- None.
- Raises
- ValueError
If the dimension is greater than 3.
-
shape(d=None)¶ Return the shape of the grid, along the dimension d if provided.
Equivalent to self.shape[d] if d is provided, self.shape otherwise.
- Parameters
- dint, optional
The dimension for which the shape is asked. The default is None.
- Returns
- numpy.ndarray
The shape of the grid.
-
abstract property
size¶ Return the number of grid points.
- Returns
- int
The number of grid points.
tensap.tools.multi_indices module¶
Module multi_indices.
-
class
tensap.tools.multi_indices.MultiIndices(array=None)¶ Bases:
objectClass MultiIndices.
- Attributes
- arraynumpy.ndarray
An array containing n multi-indices in N^d.
Methods
add_indices(J)Return the union of multi-indices of self and J.
bounded_by(m[, m0])Create the set of multi-indices bounded by m.
cardinal()Return the cardinal of the MultiIndices object.
envelope(u)Compute the monotone envelope (or monotone majorant) of a bounded sequence u.
get_indices(k)Return the multi-indices k in self.
Return the margin of the multi-index set self defined by the set of multi-indices i not in self such that it exists k in N^* s.t.
Return the set of maximal multi-indices contained in the downward closed multi-index set self.
Return the reduced margin of the multi-index set self defined by the set of multi-indices i not in self such that for all k in N^* s.t.
ind2sub(shape, ind)Create the set of multi-indices with array [I1, …, Id] such that (I1, …, Id) = np.unravel_index(np.ravel(ind), shape).
Return the intersection of the multi-indices of self and J.
is_downward_closed([m])Check whether or not the multi-index set is downward closed (or lower or monotone).
keep_dims(dims)Keep the dimensions dims in self.
keep_indices(k)Keep the multi-indices k in self.
ndim()Return the dimension of the multi-indices.
norm([p, k])Compute the p-norm of multi-indices k in the object.
plot(*args)Plot the multi-index set self.
product_set(L[, d])Create the set of multi-indices obtained by a product of sets of indices.
remove_dims(dims)Remove the dimensions in ind in the MultiIndices.
Remove multi-indices J from self.
sort([columns, mode])Sort multi-indices column-wise using the column order provided in columns.
sort_by_norm(p[, mode])Sort the multi-indices by increasing or decreasing p-norm.
sort_by_weighted_norm(p, w[, mode])Sort the multi-indices by increasing or decreasing weighted p-norm.
sub2ind(shape)Convert the indices of the MultiIndices object into flat indices.
to_list()Convert the MultiIndices’ array into a list of arrays.
weighted_norm(p, w[, k])Compute the weighted p-norm of multi-indices k in the object.
with_bounded_norm(d, p, m)Create the set of multi-indices in N^d with p-norm bounded by m, p>0.
with_bounded_weighted_norm(d, p, m, w)Create the set of multi-indices in N^d with weighted p-norm bounded by m, p>0.
-
add_indices(J)¶ Return the union of multi-indices of self and J.
- Parameters
- Jtensap.MultiIndices
The second MultiIndices object.
- Returns
- tensap.MultiIndices
The union of multi-indices of self and J.
-
static
bounded_by(m, m0=0)¶ Create the set of multi-indices bounded by m.
- Parameters
- mlist or numpy.ndarray
List or array of length d containing the highest indices in each dimension.
- m0int, optional
The lowest index for all dimensions. The default is 0.
- Returns
- tensap.MultiIndices
MultiIndices object containing the product set (m0:m[0]) x … x (m0:m[d-1]).
-
cardinal()¶ Return the cardinal of the MultiIndices object.
- Returns
- int
The cardinal of the MultiIndices object.
-
envelope(u)¶ Compute the monotone envelope (or monotone majorant) of a bounded sequence u.
- Parameters
- ulist or numpy.ndarray
The bounded sequence.
- Returns
- envnumpy.ndarray
The monotone envelope corresponding to the sequence defined by \(env_i = max_{j >= i} |u_j|\)
-
get_indices(k)¶ Return the multi-indices k in self.
- Parameters
- kint or list or numpy.ndarray
The numbers of the multi-indices to select.
- Returns
- numpy.ndarray
The multi-indices k in self.
-
get_margin()¶ Return the margin of the multi-index set self defined by the set of multi-indices i not in self such that it exists k in N^* s.t. i_k != 0 implies i - e_k in self where e_k is the k-th Kronecker sequence.
- Returns
- tensap.MultiIndices
The margin of self.
-
get_maximal_indices()¶ Return the set of maximal multi-indices contained in the downward closed multi-index set self.
- Returns
- tensap.MultiIndices
The set of maximal multi-indices contained in the downward closed multi-index set self.
-
get_reduced_margin()¶ Return the reduced margin of the multi-index set self defined by the set of multi-indices i not in self such that for all k in N^* s.t. i_k != 0 implies i - e_k in self where e_k is the k-th Kronecker sequence.
- Returns
- tensap.MultiIndices
The reduced margin of self.
-
static
ind2sub(shape, ind)¶ Create the set of multi-indices with array [I1, …, Id] such that (I1, …, Id) = np.unravel_index(np.ravel(ind), shape).
- Parameters
- shapelist or numpy.ndarray
The shape of the array.
- indlist or numpy.ndarray
The indices into the flattened version of an array of dimensions shape.
- Returns
- tensap.MultiIndices
The MultiIndices created using the flat indices ind and shape.
-
intersect_indices(J)¶ Return the intersection of the multi-indices of self and J.
- Parameters
- Jtensap.MultiIndices
The second multi-indices.
- Returns
- tensap.MultiIndices
The intersection of the multi-indices of self and J.
- ind_Inumpy.array
The indices of the multi-indices in self common to self and J.
- ind_Jnumpy.array
The indices of the multi-indices in J common to self and J.
-
is_downward_closed(m=0)¶ Check whether or not the multi-index set is downward closed (or lower or monotone).
- Parameters
- mint, optional
The lowest index for all dimensions. The default is 0.
- Returns
- condboolean
Boolean indicating whether or not the multi-index set is downward closed.
-
keep_dims(dims)¶ Keep the dimensions dims in self.
- Parameters
- dimsint or list or numpy.ndarray
The dimension(s) to keep.
- Returns
- tensap.MultiIndices
The MultiIndices with retained dimensions.
-
keep_indices(k)¶ Keep the multi-indices k in self.
- Parameters
- kint or list or numpy.ndarray
The number(s) of the multi-indices to keep.
- Returns
- tensap.MultiIndices
The MultiIndices with retained indices.
-
ndim()¶ Return the dimension of the multi-indices.
- Returns
- int
The dimension of the multi-indices.
-
norm(p=2, k=None)¶ Compute the p-norm of multi-indices k in the object.
- Parameters
- pint or numpy.inf, optional
The positive real scalar p of the p-norm, or numpy.inf. The default is 2.
- klist or numpy.ndarray, optional
The multi-indices of which the norm is to be computed. The default is all the multi-indices of the object.
- Returns
- normnumpy.ndarray
The p-norm of the selected multi-indices.
-
plot(*args)¶ Plot the multi-index set self.
See also the function plot_multi_indices.
- Parameters
- *argstuple
Parameters used in tensap’s function plot_multi_indices.
- Returns
- None.
-
static
product_set(L, d=None)¶ Create the set of multi-indices obtained by a product of sets of indices.
If ndim(L) is 1 and L contains a set of indices, the method returns a MultiIndices containing the product set L x … x L (d times).
If ndim(L) is 2 and L contains arrays of integers of size m_k, 0 <= k <= d-1, the method returns a MultiIndices containing the product set L[0] x … x L[d-1].
- Parameters
- Llist or numpy.ndarray
The grid or grids used to create the product set.
- dint, optional
The dimension of the set of multi-indices. The default is None, indicating to infer it from L.
- Returns
- tensap.MultiIndices
The set of multi-indices obtained by a product of sets of indices.
- Raises
- ValueError
If the provided arguments are wrong.
-
remove_dims(dims)¶ Remove the dimensions in ind in the MultiIndices.
- Parameters
- dimsint or list or numpy.ndarray
The dimension(s) to be removed.
- Returns
- tensap.MultiIndices
The MultiIndices object with removed dimenions in dims.
-
remove_indices(J)¶ Remove multi-indices J from self.
- Parameters
- Jtensap.MultiIndices or int or list or numpy.ndarray
The multi-indices to remove in a tensap.MultiIndices object, or their numbers as an int or in a list or numpy.ndarray.
- Returns
- tensap.MultiIndices
The MultiIndices object with removed multi-indices in J.
-
sort(columns=None, mode='ascend')¶ Sort multi-indices column-wise using the column order provided in columns.
The method first sorts according to the column columns[0], then sorts the equal coefficients of column columns[1] and so on.
- Parameters
- columnslist or numpy.ndarray, optional
The column order used to sort the multi-indices. The default is last to first column.
- modestring, optional
The sorting mode, equal to ‘ascend’ or ‘descend’. The default is ‘ascend’.
- Returns
- tensap.MultiIndices
The MultiIndices object with sorted multi-indices.
-
sort_by_norm(p, mode='ascend')¶ Sort the multi-indices by increasing or decreasing p-norm.
- Parameters
- pint
The positive real scalar p of the p-norm, or numpy.inf.
- modestring, optional
The sorting mode, equal to ‘ascend’ or ‘descend’. The default is ‘ascend’.
- Returns
- tensap.MultiIndices
The MultiIndices object with multi-indices sorted by increasing or decreasing p-norm.
-
sort_by_weighted_norm(p, w, mode='ascend')¶ Sort the multi-indices by increasing or decreasing weighted p-norm.
- Parameters
- pint
The positive real scalar p of the p-norm, or numpy.inf.
- wlist or numpy.ndarray
The self.cardinal() weights used in the computation of the norm.
- modestring, optional
The sorting mode, equal to ‘ascend’ or ‘descend’. The default is ‘ascend’.
- Returns
- tensap.MultiIndices
The MultiIndices object with multi-indices sorted by increasing or decreasing weighted p-norm.
-
sub2ind(shape)¶ Convert the indices of the MultiIndices object into flat indices.
- Parameters
- shapelist or numpy.ndarray
The shape of the array.
- Returns
- numpy.ndarray
The flat indices associated with the MultiIndices object and shape.
-
to_list()¶ Convert the MultiIndices’ array into a list of arrays.
- Returns
- list
The MultiIndices’ array as a list of arrays.
-
weighted_norm(p, w, k=None)¶ Compute the weighted p-norm of multi-indices k in the object.
- Parameters
- pint or numpy.inf
The positive real scalar p of the p-norm, or numpy.inf.
- wlist or numpy.ndarray
The self.cardinal() weights used in the computation of the norm.
- klist or numpy.ndarray, optional
The multi-indices of which the norm is to be computed. The default is all the multi-indices of the object.
- Returns
- normnumpy.ndarray
The p-norm of the selected multi-indices.
-
static
with_bounded_norm(d, p, m)¶ Create the set of multi-indices in N^d with p-norm bounded by m, p>0.
- Parameters
- dint
The dimension, a positive integer.
- pint or numpy.inf
The p of the p-norm, either a positive real scalar or numpy.inf.
- mint
The bound of the norm, a positive real scalar.
- Returns
- indtensap.MultiIndices
The set of multi-indices in N^d with p-norm bounded by m.
-
static
with_bounded_weighted_norm(d, p, m, w)¶ Create the set of multi-indices in N^d with weighted p-norm bounded by m, p>0.
- Parameters
- dint
The dimension, a positive integer.
- pint or numpy.inf
The p of the p-norm, either a positive real scalar or numpy.inf.
- mint
The bound of the norm, a positive real scalar.
- wlist or numpy.ndarray
The d weights defining the weighted p-norm of a multi-index i. \(|i|_{p, w} = (sum_{k=1}^d w_k^p i_k^p)^(1/p) for 0 < p < inf\) \(|i|_{Inf, w} = max_k w_k i_k\)
- Returns
- indtensap.MultiIndices
The set of multi-indices in N^d with weighted p-norm bounded by m.
tensap.tools.utils module¶
Module utils.
-
tensap.tools.utils.baseb2integer(I, b)¶ Return the integers with given representations in base b.
- Parameters
- Inumpy.ndarray or list
A row of I contains d integers [i_1, …, i_d] in {0, …, b-1} associated with an integer i = sum_{k=1}^d i_k b^(d-k) in [0, b^d-1].
- bint
The base.
- Returns
- numpy.ndarray
The integers with given representations in base b.
-
tensap.tools.utils.fast_intersect(a, b)¶
-
tensap.tools.utils.fast_setdiff(a, b)¶
-
tensap.tools.utils.integer2baseb(i, b, d=None)¶ Returns the representation [i_1, …, i_d] in base b of a set of non negative integers i = sum_{k=1}^d i_k b^(d-k) in [0, b^d-1].
- Parameters
- ilist or numpy.ndarray
The integers to be convertes in base b.
- bint
The base.
- dint, optional
The dimension. The default is the minimal integer allowing the representation of max(i).
- Returns
- numpy.ndarray
The representation [i_1, …, i_d] in base b of the integers.