Documentation

mutable struct MatrixCrossInterpolation{T}

Represents a cross interpolation of a matrix - or, to be more precise, the data necessary to evaluate a cross interpolation. This data can be fed into various methods such as evaluate(c, i, j) to get interpolated values, and be improved by dynamically adding pivots.

AtimesBinv(A::Matrix, B::Matrix)

Calculates the matrix product $A^{-1} B$, given a square matrix $A$ and a rectangular matrix $B$ in a numerically stable way using QR decomposition. This is useful in case $A^{-1}$ is ill-conditioned.

AtimesBinv(A::Matrix, B::Matrix)

Calculates the matrix product $A B^{-1}$, given a rectangular matrix $A$ and a square matrix $B$ in a numerically stable way using QR decomposition. This is useful in case $B^{-1}$ is ill-conditioned.

function addpivot(
    a::AbstractMatrix{T},
    ci::MatrixCrossInterpolation{T},
    pivotindices::AbstractArray{T, 2})

Add a new pivot given by pivotindices to the cross interpolation ci of the matrix a.

function addpivot(
    a::AbstractMatrix{T},
    ci::MatrixCrossInterpolation{T})

Add a new pivot that maximizes the error to the cross interpolation ci of the matrix a.

function crossinterpolate(
    a::AbstractMatrix{T};
    tolerance=1e-6,
    maxiter=200,
    firstpivot=argmax(abs.(a))
) where {T}

Cross interpolate a $m\times n$ matrix $a$, i.e. find a set of indices $I$ and $J$, such that $a(1\ldots m, 1\ldots n) \approx a(1\ldots m, I) (a(I, J))^{-1} a(J, 1\ldots m)$.

findnewpivot(
    a::AbstractMatrix{T},
    ci::MatrixCrossInterpolation{T},
    row_indices::Union{Vector{Int}},
    col_indices::Union{Vector{Int}})

Finds the pivot that maximizes the local error across all components of a and ci within the subset given by rowindices and colindices. By default, all avalable rows of ci will be considered.

localerror(
    a::AbstractMatrix{T},
    ci::MatrixCrossInterpolation{T},
    row_indices::Union{AbstractVector{Int},Colon,Int}=Colon(),
    col_indices::Union{AbstractVector{Int},Colon,Int}=Colon())

Returns all local errors of the cross interpolation ci with respect to matrix a. To find local errors on a submatrix, specify rowindices or colindices.

function addpivot!(a::AbstractMatrix{T}, aca::MatrixACA{T})

Find and add a new pivot according to the ACA algorithm in Kumar 2016 ()

Compute u_k(x) for all x

Compute v_k(y) for all y

Override solving Ax = b using LU decomposition

Correct estimate of the last pivot error A special care is taken for a full-rank matrix: the last pivot error is set to zero.

function rrlu!(
    A::AbstractMatrix{T};
    maxrank::Int=typemax(Int),
    reltol::Number=1e-14,
    abstol::Number=0.0,
    leftorthogonal::Bool=true
)::rrLU{T} where {T}

Rank-revealing LU decomposition.

function rrlu(
    A::AbstractMatrix{T};
    maxrank::Int=typemax(Int),
    reltol::Number=1e-14,
    abstol::Number=0.0,
    leftorthogonal::Bool=true
)::rrLU{T} where {T}

Rank-revealing LU decomposition.

Solve (LU) x = b

L: lower triangular matrix U: upper triangular matrix b: right-hand side vector

Return x

Note: Not optimized for performance

abstract type AbstractTensorTrain{V} <: Function end

Abstract type that is a supertype to all tensor train types found in this module. The main purpose of this type is for the definition of functions such as rank and linkdims that are shared between different tensor train classes.

When iterated over, the tensor train will return each of the tensors in order.

Implementations: TensorTrain, TensorCI2, TensorCI1

function (+)(lhs::AbstractTensorTrain{V}, rhs::AbstractTensorTrain{V}) where {V}

Addition of two tensor trains. If c = a + b, then c(v) ≈ a(v) + b(v) at each index set v. Note that this function increases the bond dimension, i.e. $\chi_{\text{result}} = \chi_1 + \chi_2$ if the original tensor trains had bond dimensions $\chi_1$ and $\chi_2$. Can be combined with automatic recompression by calling add.

function (-)(lhs::AbstractTensorTrain{V}, rhs::AbstractTensorTrain{V}) where {V}

Subtraction of two tensor trains. If c = a - b, then c(v) ≈ a(v) - b(v) at each index set v. Note that this function increases the bond dimension, i.e. $\chi_{\text{result}} = \chi_1 + \chi_2$ if the original tensor trains had bond dimensions $\chi_1$ and $\chi_2$. Can be combined with automatic recompression by calling subtract (see documentation for add).

function length(tt::AbstractTensorTrain{V}) where {V}

Length of the tensor train, i.e. the number of tensors in the tensor train.

function sum(tt::TensorTrain{V}) where {V}

Evaluates the sum of the tensor train approximation over all lattice sites in an efficient factorized manner.

Frobenius norm of a tensor train.

Squared Frobenius norm of a tensor train.

function rank(tt::AbstractTensorTrain{V}) where {V}

Rank of the tensor train, i.e. the maximum link dimension.

See also: linkdims, linkdim

function add(
    lhs::AbstractTensorTrain{V}, rhs::AbstractTensorTrain{V};
    factorlhs=one(V), factorrhs=one(V),
    tolerance::Float64=0.0, maxbonddim::Int=typemax(Int)
) where {V}

Addition of two tensor trains. If C = add(A, B), then C(v) ≈ A(v) + B(v) at each index set v. Note that this function increases the bond dimension, i.e. $\chi_{\text{result}} = \chi_1 + \chi_2$ if the original tensor trains had bond dimensions $\chi_1$ and $\chi_2$.

Arguments:

• lhs, rhs: Tensor trains to be added.
• factorlhs, factorrhs: Factors to multiply each tensor train by before addition.
• tolerance, maxbonddim: Parameters to be used for the recompression step.

Returns: A new TensorTrain representing the function factorlhs * lhs(v) + factorrhs * rhs(v).

See also: +

function evaluate(
    tt::TensorTrain{V},
    indexset::Union{AbstractVector{LocalIndex}, NTuple{N, LocalIndex}}
)::V where {N, V}

Evaluates the tensor train tt at indices given by indexset.

function evaluate(tt::TensorTrain{V}, indexset::CartesianIndex) where {V}

Evaluates the tensor train tt at indices given by indexset.

function linkdim(tt::AbstractTensorTrain{V}, i::Int)::Int where {V}

Bond dimensions at the link between tensor $T_i$ and $T_{i+1}$ in the tensor train.

function linkdims(tt::AbstractTensorTrain{V})::Vector{Int} where {V}

Bond dimensions along the links between $T$ tensors in the tensor train.

See also: rank

function linkdim(tt::AbstractTensorTrain{V}, i::Int)::Int where {V}

Dimension of the ith site index along the tensor train.

function sitedims(tt::AbstractTensorTrain{V})::Vector{Vector{Int}} where {V}

Dimensions of the site indices (local indices, physical indices) of the tensor train.

sitetensors(tt::AbstractTensorTrain{V}, i) where {V}

The tensor at site i of the tensor train.

sitetensors(tt::AbstractTensorTrain{V}) where {V}

The tensors that make up the tensor train.

function subtract(
    lhs::AbstractTensorTrain{V}, rhs::AbstractTensorTrain{V};
    tolerance::Float64=0.0, maxbonddim::Int=typemax(Int)
)

Subtract two tensor trains lhs and rhs. See add.

struct TTCache{ValueType}

Cached evalulation of a tensor train. This is useful when the same TT is evaluated multiple times with the same indices. The number of site indices per tensor core can be arbitray irrespective of the number of site indices of the original tensor train.

struct TensorTrain{ValueType}

Represents a tensor train, also known as MPS.

The tensor train can be evaluated using standard function call notation:

    tt = TensorTrain(...)
    value = tt([1, 2, 3, 4])

The corresponding function is:

function (tt::TensorTrain{V})(indexset) where {V}

Evaluates the tensor train tt at indices given by indexset.

function TensorTrain(sitetensors::Vector{Array{V, 3}}) where {V}

Create a tensor train out of a vector of tensors. Each tensor should have links to the previous and next tensor as dimension 1 and 3, respectively; the local index ("physical index" for MPS in physics) is dimension 2.

function TensorTrain{V2,N}(tci::AbstractTensorTrain{V}) where {V,V2,N}

Convert a tensor-train-like object into a tensor train.

Arguments:

• tt::AbstractTensorTrain{V}: a tensor-train-like object.
• localdims: a vector of local dimensions for each tensor in the tensor train. A each element of localdims should be an array-like object of N-2 integers.

function TensorTrain(tci::AbstractTensorTrain{V}) where {V}

Convert a tensor-train-like object into a tensor train. This includes TCI1 and TCI2 objects.

See also: TensorCI1, TensorCI2.

Fitting data with a TensorTrain object. This may be useful when the interpolated function is noisy.

projector: 0 means no projection, otherwise the index of the projector

function compress!(
    tt::TensorTrain{V, N},
    method::Symbol=:LU;
    tolerance::Float64=1e-12,
    maxbonddim=typemax(Int)
) where {V, N}

Compress the tensor train tt using LU, CI or SVD decompositions.

Contraction of two TTOs Optionally, the contraction can be done with a function applied to the result.

function contract(
    A::TensorTrain{V1,4},
    B::TensorTrain{V2,4};
    algorithm::Symbol=:TCI,
    tolerance::Float64=1e-12,
    maxbonddim::Int=typemax(Int),
    f::Union{Nothing,Function}=nothing,
    kwargs...
) where {V1,V2}

Contract two tensor trains A and B.

Currently, two implementations are available:

1. algorithm=:TCI constructs a new TCI that fits the contraction of A and B.
2. algorithm=:naive uses a naive tensor contraction and subsequent SVD recompression of the tensor train.
3. algorithm=:zipup uses a naive tensor contraction with on-the-fly LU decomposition.

Arguments:

• A and B are the tensor trains to be contracted.
• algorithm chooses the algorithm used to evaluate the contraction.
• tolerance is the tolerance of the TCI or SVD recompression.
• maxbonddim sets the maximum bond dimension of the resulting tensor train.
• f is a function to be applied elementwise to the result. This option is only available with algorithm=:TCI.
• method chooses the method used for the factorization in the algorithm=:zipup case (:SVD or :LU).
• kwargs... are forwarded to crossinterpolate2 if algorithm=:TCI.

See SVD version: https://tensornetwork.org/mps/algorithms/zipupmpo/

mutable struct TensorCI1{ValueType} <: AbstractTensorTrain{ValueType}

Type that represents tensor cross interpolations created using the TCI1 algorithm. Users may want to create these using crossinterpolate1 rather than calling a constructor directly.

function addpivot!(tci::TensorCI1{V}, p::Int, tolerance::1e-12) where {V,F}

Add a pivot to the TCI at site p. Do not add a pivot if the error is below tolerance.

function addpivotcol!(tci::TensorCI1{V}, cross::MatrixCI{V}, p::Int, newj::Int, f) where {V}

Add the column with index newj as a pivot row to bond p.

function addpivotrow!(tci::TensorCI1{V}, cross::MatrixCI{V}, p::Int, newi::Int, f) where {V}

Add the row with index newi as a pivot row to bond p.

function crossinterpolate(
    ::Type{ValueType},
    f,
    localdims::Union{Vector{Int},NTuple{N,Int}},
    firstpivot::MultiIndex=ones(Int, length(localdims));
    tolerance::Float64=1e-8,
    maxiter::Int=200,
    sweepstrategy::Symbol=:backandforth,
    pivottolerance::Float64=1e-12,
    verbosity::Int=0,
    additionalpivots::Vector{MultiIndex}=MultiIndex[],
    normalizeerror::Bool=true
) where {ValueType, N}

Deprecated, and only included for backward compatibility. Please use crossinterpolate1 instead.

function crossinterpolate1(
    ::Type{ValueType},
    f,
    localdims::Union{Vector{Int},NTuple{N,Int}},
    firstpivot::MultiIndex=ones(Int, length(localdims));
    tolerance::Float64=1e-8,
    maxiter::Int=200,
    sweepstrategy::Symbol=:backandforth,
    pivottolerance::Float64=1e-12,
    verbosity::Int=0,
    additionalpivots::Vector{MultiIndex}=MultiIndex[],
    normalizeerror::Bool=true
) where {ValueType, N}

Cross interpolate a function $f(\mathbf{u})$, where $\mathbf{u} \in [1, \ldots, d_1] \times [1, \ldots, d_2] \times \ldots \times [1, \ldots, d_{\mathscr{L}}]$ and $d_1 \ldots d_{\mathscr{L}}$ are the local dimensions.

Arguments:

• ValueType is the return type of f. Automatic inference is too error-prone.
• f is the function to be interpolated. f should have a single parameter, which is a vector of the same length as localdims. The return type should be ValueType.
• localdims::Union{Vector{Int},NTuple{N,Int}} is a Vector (or Tuple) that contains the local dimension of each index of f.
• firstpivot::MultiIndex is the first pivot, used by the TCI algorithm for initialization. Default: [1, 1, ...].
• tolerance::Float64 is a float specifying the tolerance for the interpolation. Default: 1e-8.
• maxiter::Int is the maximum number of iterations (i.e. optimization sweeps) before aborting the TCI construction. Default: 200.
• sweepstrategy::Symbol specifies whether to sweep forward (:forward), backward (:backward), or back and forth (:backandforth) during optimization. Default: :backandforth.
• pivottolerance::Float64 specifies the tolerance below which no new pivot will be added to each tensor. Default: 1e-12.
• verbosity::Int can be set to >= 1 to get convergence information on standard output during optimization. Default: 0.
• additionalpivots::Vector{MultiIndex} is a vector of additional pivots that the algorithm should add to the initial pivot before starting optimization. This is not necessary in most cases.
• normalizeerror::Bool determines whether to scale the error by the maximum absolute value of f found during sampling. If set to false, the algorithm continues until the absolute error is below tolerance. If set to true, the algorithm uses the absolute error divided by the maximum sample instead. This is helpful if the magnitude of the function is not known in advance. Default: true.

Notes:

• Convergence may depend on the choice of first pivot. A good rule of thumb is to choose firstpivot close to the largest structure in f, or on a maximum of f. If the structure of f is not known in advance, optfirstpivot may be helpful.
• By default, no caching takes place. Use the CachedFunction wrapper if your function is expensive to evaluate.

See also: optfirstpivot, CachedFunction, crossinterpolate2

function evaluate(tci::TensorCI1{V}, indexset::AbstractVector{LocalIndex}) where {V}

Evaluate the TCI at a specific set of indices. This method is inefficient; to evaluate many points, first convert the TCI object into a tensor train using tensortrain(tci).

buildPiAt(tci::TensorCrossInterpolation{V}, p::Int)

Build a 4-legged $\Pi$ tensor at site p. Indices are in the order $i, u_p, u_{p + 1}, j$, as in the TCI paper.

Return if a is nested in b ($a < b$). If row_or_col is :row/:col, the last/first element of each index in b is ignored, respectively.

mutable struct TensorCI2{ValueType} <: AbstractTensorTrain{ValueType}

Type that represents tensor cross interpolations created using the TCI2 algorithm. Users may want to create these using crossinterpolate2 rather than calling a constructor directly.

Add global pivots to index sets

Add global pivots to index sets and perform a 1site sweep

Add global pivots to index sets and perform a 2site sweep. Retry until all pivots are added or until ntry iterations are reached.

function crossinterpolate2(
    ::Type{ValueType},
    f,
    localdims::Union{Vector{Int},NTuple{N,Int}},
    initialpivots::Vector{MultiIndex}=[ones(Int, length(localdims))];
    tolerance::Float64=1e-8,
    pivottolerance::Float64=tolerance,
    maxbonddim::Int=typemax(Int),
    maxiter::Int=200,
    sweepstrategy::Symbol=:backandforth,
    pivotsearch::Symbol=:full,
    verbosity::Int=0,
    loginterval::Int=10,
    normalizeerror::Bool=true,
    ncheckhistory=3,
    maxnglobalpivot::Int=5,
    nsearchglobalpivot::Int=5,
    tolmarginglobalsearch::Float64=10.0,
    strictlynested::Bool=false
) where {ValueType,N}

Cross interpolate a function $f(\mathbf{u})$ using the TCI2 algorithm. Here, the domain of $f$ is $\mathbf{u} \in [1, \ldots, d_1] \times [1, \ldots, d_2] \times \ldots \times [1, \ldots, d_{\mathscr{L}}]$ and $d_1 \ldots d_{\mathscr{L}}$ are the local dimensions.

Arguments:

• ValueType is the return type of f. Automatic inference is too error-prone.
• f is the function to be interpolated. f should have a single parameter, which is a vector of the same length as localdims. The return type should be ValueType.
• localdims::Union{Vector{Int},NTuple{N,Int}} is a Vector (or Tuple) that contains the local dimension of each index of f.
• initialpivots::Vector{MultiIndex} is a vector of pivots to be used for initialization. Default: [1, 1, ...].
• tolerance::Float64 is a float specifying the target tolerance for the interpolation. Default: 1e-8.
• pivottolerance::Float64 is a float that specifies the tolerance for adding new pivots, i.e. the truncation of tensor train bonds. It should be <= tolerance, otherwise convergence may be impossible. Default: tolerance.
• maxbonddim::Int specifies the maximum bond dimension for the TCI. Default: typemax(Int), i.e. effectively unlimited.
• maxiter::Int is the maximum number of iterations (i.e. optimization sweeps) before aborting the TCI construction. Default: 200.
• sweepstrategy::Symbol specifies whether to sweep forward (:forward), backward (:backward), or back and forth (:backandforth) during optimization. Default: :backandforth.
• pivotsearch::Symbol determins how pivots are searched (:full or :rook). Default: :full.
• verbosity::Int can be set to >= 1 to get convergence information on standard output during optimization. Default: 0.
• loginterval::Int can be set to >= 1 to specify how frequently to print convergence information. Default: 10.
• normalizeerror::Bool determines whether to scale the error by the maximum absolute value of f found during sampling. If set to false, the algorithm continues until the absolute error is below tolerance. If set to true, the algorithm uses the absolute error divided by the maximum sample instead. This is helpful if the magnitude of the function is not known in advance. Default: true.
• ncheckhistory::Int is the number of history points to use for convergence checks. Default: 3.
• maxnglobalpivot::Int can be set to >= 0. Default: 5.
• nsearchglobalpivot::Int can be set to >= 0. Default: 5.
• tolmarginglobalsearch can be set to >= 1.0. Seach global pivots where the interpolation error is larger than the tolerance by tolmarginglobalsearch. Default: 10.0.
• strictlynested::Bool=false determines whether to preserve partial nesting in the TCI algorithm. Default: true.
• checkbatchevaluatable::Bool Check if the function f is batch evaluatable. Default: false.

Notes:

• Set tolerance to be > 0 or maxbonddim to some reasonable value. Otherwise, convergence is not reachable.
• By default, no caching takes place. Use the CachedFunction wrapper if your function is expensive to evaluate.

See also: optimize!, optfirstpivot, CachedFunction, crossinterpolate1

Invalidate the site tensor at bond b.

Return if site tensors are available

function optimize!(
    tci::TensorCI2{ValueType},
    f;
    tolerance::Float64=1e-8,
    maxbonddim::Int=typemax(Int),
    maxiter::Int=200,
    sweepstrategy::Symbol=:backandforth,
    pivotsearch::Symbol=:full,
    verbosity::Int=0,
    loginterval::Int=10,
    normalizeerror::Bool=true,
    ncheckhistory=3,
    maxnglobalpivot::Int=5,
    nsearchglobalpivot::Int=5,
    tolmarginglobalsearch::Float64=10.0,
    strictlynested::Bool=false
) where {ValueType}

Perform optimization sweeps using the TCI2 algorithm. This will sucessively improve the TCI approximation of a function until it fits f with an error smaller than tolerance, or until the maximum bond dimension (maxbonddim) is reached.

Arguments:

• tci::TensorCI2{ValueType}: The TCI to optimize.
• f: The function to fit.
• f is the function to be interpolated. f should have a single parameter, which is a vector of the same length as localdims. The return type should be ValueType.
• localdims::Union{Vector{Int},NTuple{N,Int}} is a Vector (or Tuple) that contains the local dimension of each index of f.
• tolerance::Float64 is a float specifying the target tolerance for the interpolation. Default: 1e-8.
• maxbonddim::Int specifies the maximum bond dimension for the TCI. Default: typemax(Int), i.e. effectively unlimited.
• maxiter::Int is the maximum number of iterations (i.e. optimization sweeps) before aborting the TCI construction. Default: 200.
• sweepstrategy::Symbol specifies whether to sweep forward (:forward), backward (:backward), or back and forth (:backandforth) during optimization. Default: :backandforth.
• pivotsearch::Symbol determins how pivots are searched (:full or :rook). Default: :full.
• verbosity::Int can be set to >= 1 to get convergence information on standard output during optimization. Default: 0.
• loginterval::Int can be set to >= 1 to specify how frequently to print convergence information. Default: 10.
• normalizeerror::Bool determines whether to scale the error by the maximum absolute value of f found during sampling. If set to false, the algorithm continues until the absolute error is below tolerance. If set to true, the algorithm uses the absolute error divided by the maximum sample instead. This is helpful if the magnitude of the function is not known in advance. Default: true.
• ncheckhistory::Int is the number of history points to use for convergence checks. Default: 3.
• maxnglobalpivot::Int can be set to >= 0. Default: 5.
• nsearchglobalpivot::Int can be set to >= 0. Default: 5.
• tolmarginglobalsearch can be set to >= 1.0. Seach global pivots where the interpolation error is larger than the tolerance by tolmarginglobalsearch. Default: 10.0.
• strictlynested::Bool determines whether to preserve partial nesting in the TCI algorithm. Default: false.
• checkbatchevaluatable::Bool Check if the function f is batch evaluatable. Default: false.

Notes:

• Set tolerance to be > 0 or maxbonddim to some reasonable value. Otherwise, convergence is not reachable.
• By default, no caching takes place. Use the CachedFunction wrapper if your function is expensive to evaluate.

See also: crossinterpolate2, optfirstpivot, CachedFunction, crossinterpolate1

function printnestinginfo(tci::TensorCI2{T}) where {T}

Print information about fulfillment of the nesting criterion ($I_\ell < I_{\ell+1}$ and $J_\ell < J_{\ell+1}$) on each pair of bonds $\ell, \ell+1$ of tci to stdout.

function printnestinginfo(io::IO, tci::TensorCI2{T}) where {T}

Print information about fulfillment of the nesting criterion ($I_\ell < I_{\ell+1}$ and $J_\ell < J_{\ell+1}$) on each pair of bonds $\ell, \ell+1$ of tci to io.

Search global pivots where the interpolation error exceeds abstol.

Perform 2site sweeps on a TCI2.

Update pivots at bond b of tci using the TCI2 algorithm. Site tensors will be invalidated.

function integrate(
    ::Type{ValueType},
    f,
    a::Vector{ValueType},
    b::Vector{ValueType};
    tolerance=1e-8,
    GKorder::Int=15
) where {ValueType}

Integrate the function f using TCI and Gauss–Kronrod quadrature rules.

Arguments:

• ValueType: return type off`.
• a`: Vector of lower bounds in each dimension. Effectively, the lower corner of the hypercube that is being integrated over.
• b`: Vector of upper bounds in each dimension.
• tolerance: tolerance of the TCI approximation for the values of f.
• GKorder: Order of the Gauss–Kronrod rule, e.g. 15.

Represents a function that maps ArgType -> ValueType and automatically caches function values. A CachedFunction object can be called in the same way as the original function using the usual function call syntax. The type K denotes the type of the keys used to cache function values, which could be an integer type. This defaults to UInt128. A safer but slower alternative is BigInt, which is better suited for functions with a large number of arguments. CachedFunction does not support batch evaluation of function values.

function CachedFunction{ValueType}(
    f::Function,
    localdims::Vector{Int}
) where {ValueType}

Constructor for a cached function that avoids repeated evaluation of the same values.

Arguments:

• ValueType is the return type of f.
• K is the type for the keys used to cache function values. This defaults to UInt128.
• f is the function to be wrapped. It should take a vector of integers as its sole

argument, and return a value of type ValueType.

• localdims is a Vector that describes the local dimensions of each argument of f.

Get all cached data of a CachedFunction object.

Wrap any function to support batch evaluation.

ThreadedBatchEvaluator{T} is a wrapper for a function that supports batch evaluation parallelized over the index sets using threads. The function to be wrapped must be thread-safe.

This file contains functions for evaluating a function on a batch of indices mainly for TensorCI2. If the function supports batch evaluation, then it should implement the BatchEvaluator interface. Otherwise, the function is evaluated on each index individually using the usual function call syntax and loops.

Calling batchevaluate on a function that supports batch evaluation will dispatch to this function.

function optfirstpivot(
    f,
    localdims::Union{Vector{Int},NTuple{N,Int}},
    firstpivot::MultiIndex=ones(Int, length(localdims));
    maxsweep=1000
) where {N}

Optimize the first pivot for a tensor cross interpolation.

Arguments:

• f is function to be interpolated.
• localdims::Union{Vector{Int},NTuple{N,Int}} determines the local dimensions of the function parameters (see crossinterpolate1).
• fistpivot::MultiIndex=ones(Int, length(localdims)) is the starting point for the optimization. It is advantageous to choose it close to a global maximum of the function.
• maxsweep is the maximum number of optimization sweeps. Default: 1000.

See also: crossinterpolate1

Construct slice for the site indces of one tensor core Returns a slice and the corresponding shape for resize

function estimatetrueerror(
    tt::TensorTrain{ValueType,3}, f;
    nsearch::Int = 100,
    initialpoints::Union{Nothing,AbstractVector{MultiIndex}} = nothing,
)::Vector{Tuple{MultiIndex,Float64}} where {ValueType}

Estimate the global error by comparing the exact function value and the TT approximation using a greedy algorithm and returns a unique list of pairs of the pivot and the error (error is the absolute difference between the exact function value and the TT approximation). On return, the list is sorted in descending order of the error.

Arguments:

• tt::TensorTrain{ValueType,3}: The tensor train to be compared with f
• f: The function to be compared with tt.
• nsearch::Int: The number of initial points to be used in the search (defaults to 100).
• initialpoints::Union{Nothing,AbstractVector{MultiIndex}}: The initial points to be used in the search (defaults to nothing). If initialpoints is not nothing, nsearch is ignored.