Function Loading

The SeeMPS library provides several methods to load univariate and multivariate functions in MPS and MPO structures. In the following, the most important are listed.

Tensorized operations

These methods are useful to construct MPS corresponding to domain discretizations, and compose them using tensor products and sums to construct multivariate domains.

RegularInterval(start, stop, size[, ...])	Equispaced discretization between start and stop with size points.
ChebyshevInterval(start, stop, size[, endpoints])	Irregular discretization between start and stop given by the zeros or extrema of a Chebyshev polynomial of order size or size-1 respectively.
mps_interval(interval[, strategy])	Returns an MPS corresponding to a specific type of interval.
mps_tensor_product(mps_list[, mps_order, ...])	Returns the tensor product of a list of MPS, with the sites arranged according to the specified MPS order.
mps_tensor_sum(mps_list[, mps_order, ...])	Returns the tensor sum of a list of MPS, with the sites arranged according to the specified MPS order.

Tensor cross-interpolation (TT-Cross)

These methods are useful to compose MPS or MPO representations of black-box functions using tensor-train cross-interpolation (TT-Cross). See Tensor-train cross-interpolation (TT-Cross)

BlackBoxLoadMPS(func, domain[, base, mps_order])	Black-box representing a multivariate scalar function discretized on an Interval or Mesh object.
BlackBoxLoadTT(func, mesh)	Black-box representing a multivariate scalar function discretized on a Mesh object following the tensor-train structure.
BlackBoxLoadMPO(func, mesh[, base_mpo, ...])	Black-box representing a 2-dimensional function discretized on a 2D Mesh and quantized in a MPO with physical dimensions given by base_mpo.
BlackBoxComposeMPS(func, mps_list)	Black-box representing the composition of a multivariate scalar function with a collection of MPS objects.
cross_maxvol(black_box[, cross_strategy, ...])	Computes the MPS representation of a black-box function using the tensor cross-approximation (TCI) algorithm based on one-site optimizations using the rectangular maxvol decomposition.
cross_dmrg(black_box[, cross_strategy, ...])	Computes the MPS representation of a black-box function using the tensor cross-approximation (TCI) algorithm based on two-site optimizations in a DMRG-like manner.
cross_greedy(black_box[, cross_strategy, ...])	Computes the MPS representation of a black-box function using the tensor cross-approximation (TCI) algorithm based on two-site optimizations following greedy updates of the pivot matrices.

Chebyshev expansions

These methods are useful to compose univariate function on generic initial MPS or MPO and compute MPS approximations of functions. See Chebyshev Approximation.

cheb2mps(coefficients[, initial_mps, ...])	Composes a function on an initial MPS by expanding it on the basis of Chebyshev polynomials.
cheb2mpo(coefficients, initial_mpo[, ...])	Composes a function on an initial MPO by expanding it on the basis of Chebyshev polynomials.
interpolation_coefficients(func[, order, ...])	Returns the coefficients for the Chebyshev interpolation of a function on a given set of nodes and on a specified interval.
projection_coefficients(func[, order, ...])	Returns the coefficients for the Chebyshev projection of a function using Chebyshev-Gauss integration.
estimate_order(func[, start, stop, domain, ...])	Returns an estimation of the number of Chebyshev coefficients required to achieve a given accuracy such that the last pair of coefficients fall below a given tolerance, as they theoretically bound the maximum error of the expansion.

Multiscale interpolative constructions

These methods are useful to construct polynomial interpolants of univariate functions in MPS using the Lagrange interpolation framework. See Multiscale interpolative constructions.

lagrange_basic(func, order, sites[, start, ...])	Performs a basic Lagrange MPS Chebyshev interpolation of a function.
lagrange_rank_revealing(func, order, sites)	Performs a Lagrange rank-revealing MPS Chebyshev interpolation of a function.
lagrange_local_rank_revealing(func, order, ...)	Performs a local rank-revealing Lagrange MPS Chebyshev interpolation of a function.

Generic polynomial constructions

These methods are useful to construct generic polynomials in the monomial basis from a collection of coefficients.

mps_from_polynomial(p, domain[, first, strategy])

Construct a tensor representation of a polynomial.