Multi-index to tensor

Contents

function [T] = multi_index_to_tensor(C,I)
        

MULTI_INDEX_TO_TENSOR

This function returns the multi-dimensional array for given multi-index coefficients and the corresponding exponents

Inputs:

Outputs:

Other variables:

Assuming the tensor array $\mathbf{T}$ acts on the vector $\mathbf{q}\in\bf{R}^{r}$ along the dimensions $2,\dots,d+1$ . We have the following relationship between the input and output

$\sum_{j_1,\dots,j_d=1}^r T_{ij_1\dots j_d} q_{j_1} q_{j_2} \dots q_{j_d} = \sum_{\mathbf{p}\in\bf{N}_0^{r}, |\mathbf{p}|=d} C_{i,\mathbf{p}}\mathbf{q}^{\mathbf{p}},\quad \mathbf{q}\in\bf{R}^{r} ,\quad i\in \{1,\dots,n\},\quad d = \textrm{dim}(\textrm{T}) - 1$

or

$\mathbf{T}\mathbf{q}^{\otimes d} = \mathbf{C}\mathbf{q}^{\mathbf{I}}$ ,

where the different arrays are organized in the following form:

$\mathbf{C} = \left[\begin{array}{c}\mathbf{c}_1\\\vdots\\\mathbf{c}_{n}\end{array}\right]$ , where the i-th row represents the coefficients $\mathbf{c}_i = [C_{i,\mathbf{p}_1},\dots,C_{i,\mathbf{p}_{n_I}}]$ ,

$\mathbf{I} = \left[\begin{array}{c}\mathbf{p}_1\\\vdots\\\mathbf{p}_{n_I}\end{array}\right]$ is a matrix whose rows contain the multi-indices $\mathbf{p}\in\bf{N}_0^{r}$ respresented by the tensor $\mathbf{T}$ ,

$\mathbf{q}^{\mathbf{I}} = \left[\begin{array}{c}\mathbf{q}^{\mathbf{p}_1}\\\vdots\\\mathbf{q}^{\mathbf{p}_{n_I}}\end{array}\right]$ .

n = size(C,1);
d = sum(I(1,:));
r = size(I,2);
nSubs = size(I,1);
SIZE = [n r*ones(1, d)];
SUBS = zeros(nSubs,d);
for j = 1:nSubs
    p = I(j,:);
    SUBS(j,:) = multi_index_to_sub(p);
end
[i,j,VALS] = find(sparse(C));
T = sptensor([i, SUBS(j,:)],VALS,SIZE);
        

end
        

function subs = multi_index_to_sub(p)
    d = sum(p);
    subs = zeros(1,d);
    [~, i, s] = find(p);
    m = 1;
    for k = 1:d
        subs(k) = i(m);
        if s(m) == 1
            m = m + 1;
        else
            s(m) = s(m) - 1;
        end
    end
end
        

Published with MATLAB® R2023a