Function `spraycross()` function in the `spray` package

spraycross

function (S, ...) 
{
    if (nargs() < 3) {
        spraycross2(S, ...)
    }
    else {
        spraycross2(S, Recall(...))
    }
}

spraycross2

function (S1, S2) 
{
    M1 <- index(S1)
    M2 <- index(S2)
    jj <- as.matrix(expand.grid(seq_len(nrow(M1)), seq_len(nrow(M2))))
    f <- function(i) {
        c(M1[jj[i, 1], ], M2[jj[i, 2], ])
    }
    spray(t(sapply(seq_len(nrow(jj)), f)), c(outer(coeffs(S1), 
        coeffs(S2))))
}

To cite the spray package in publications, please use Hankin (2022). Function spraycross() returns the tensor cross product of any number of spray objects (interpreted as tensors); spraycross2() is a helper function that returns the product of two such spray objects.

The tensor cross product

In a memorable passage, Spivak (1965) states:

Integration on chains

If $V$ is a vector space over $\mathbb{R}$ , we denote the $k$ -fold product $V\times\cdots\times V$ by $V^k$ . A function $T\colon V^k\longrightarrow\mathbb{R}$ is called multilinear if for each $i$ with $1\leqslant i\leqslant k$ we have

$$ T\left(v_1,\ldots, v_i + {v'}_i,\ldots, v_k\right)= T\left(v_1,\ldots,v_i,\ldots,v_k\right)+ T\left(v_1,\ldots,{v'}_i,\ldots,v_k\right),\\ T\left(v_1,\ldots,av_i,\ldots,v_k\right)=aT\left(v_1,\ldots,v_i,\ldots,v_k\right) $$

A multilinear function $T\colon v^k\longrightarrow\mathbb{R}$ is called a $k$ -tensor on $V$ and the set of all $k$ -tensors, denoted by $\mathcal{J}^k(V)$ , becomes a vector space (over $\mathbb{R}$ ) if for $S,T\in\mathcal{J}^k(V)$ and $a\in\mathbb{R}$ we define

$(S+T)(v_1,\ldots,v_k) = S(v_1,\ldots,v_k) + T(v_1,\ldots,v_k) (aS)(v_1,\ldots,v_k) = a\cdot S(v_1,\ldots,v_k)$

There is also an operation connecting the various spaces $\mathcal{J}(V)$ . If $S\in\mathcal{J}^k(V)$ and $T\in\mathcal{J}^l(V)$ , we define the tensor product $S\otimes T\in\mathcal{J}^{k+l}(V)$ by

$S\otimes T(v_1,\ldots,v_k,v_{k+1},\ldots,v_{k+l})= S(v_1,\ldots,v_k)\cdot T(v_{k+1},\ldots,v_{k+l}).$

- Michael Spivak, 1969 (Calculus on Manifolds, Perseus books). Page 75

Spivak goes on to observe that the tensor product is distributive and associative but not commutative. He then proves that the set of all $k$ -fold tensor products

$\phi_{i_1}\otimes\cdots\otimes\phi_{i_k}\qquad 1\leqslant i_1,\ldots,i_k\leqslant n$

[where $\phi_i(v_j)=\delta_{ij}$ , $v_1,\ldots,v_k$ being a basis for $V$ ] is a basis for $\mathcal{J}^k(V)$ , which therefore has dimension $n^k$ . Function spraycross2() evaluates the tensor product and I give examples here.

References

Hankin, Robin K. S. 2022. “Sparse Arrays in : The Package.” arXiv. https://doi.org/10.48550/ARXIV.2210.10848.

Spivak, M. 1965. Calculus on Manifolds. Addison-Wesley.

Robin K. S. Hankin

The tensor cross product

References