Macaulay2 » Documentation
Packages » PathSignatures :: tensorLog
next | previous | forward | backward | up | index | toc

tensorLog -- Compute a component of the logarithm of a tensor.

Description

Let $T((\mathbb{R}^d))$ denote the dual of the tensor algebra on $\mathbb{R}^d$, i.e., the space $\prod_k (\mathbb{R}^d)^{\otimes k}$. Given $x \in T(\mathbb{R}^d)$ with constant term $1$, its logarithm is $$\log(x) := - \sum_{k \geq 0} \frac{1}{k} (1-x)^{\otimes k} \in T((\mathbb{R}^d)). $$ $\texttt{tensorLog(x,k)}$ computes the degree $k$ component of $\log(x)$.

i1 : A2 = wordAlgebra(2);
 
i2 : x = 1 + [1]_A2 + 1/2 * [1,1]_A2 + 1/6 * [1,1,1]_A2;
 
i3 : sum(0..3, i -> tensorLog(x, i))

o3 = Lt
       1

o3 : A2

Caveat

The method is implemented only for tensors with constant term $1$.

See also

Ways to use tensorLog:

  • tensorLog(NCRingElement,ZZ)

For the programmer

The object tensorLog is a method function.

The source of this document is in PathSignatures/documentation.m2:918:0.