Binary Operations on Arrays

Tensor

Einstein

Nutils

1

\(\mathbf{a} \in \mathbb{R}^n\)

\(\mathbf{b} \in \mathbb{R}^n\)

\(c = \mathbf{a} \cdot \mathbf{b} \in \mathbb{R}\)

\(c = a_i b_i\)

c = (a*b).sum(-1)

2

\(\mathbf{a} \in \mathbb{R}^n\)

\(\mathbf{b} \in \mathbb{R}^m\)

\(\mathbf{C} = \mathbf{a} \otimes \mathbf{b} \in \mathbb{R}^{n \times m}\)

\(C_{ij} = a_i b_j\)

C = a[:,_]*b[_,:]

C = function.outer(a,b)

3

\(\mathbf{A} \in \mathbb{R}^{m \times n}\)

\(\mathbf{b} \in \mathbb{R}^n\)

\(\mathbf{c} = \mathbf{A}\mathbf{b} \in \mathbb{R}^{m}\)

\(c_{i} = A_{ij} b_j\)

c = (A[:,:]*b[_,:]).sum(-1)

4

\(\mathbf{A} \in \mathbb{R}^{m \times n}\)

\(\mathbf{B} \in \mathbb{R}^{n \times p}\)

\(\mathbf{C} = \mathbf{A} \mathbf{B} \in \mathbb{R}^{m \times p}\)

\(c_{ij} = A_{ik} B_{kj}\)

c = (A[:,:,_]*B[_,:,:]).sum(-2)

5

\(\mathbf{A} \in \mathbb{R}^{m \times n}\)

\(\mathbf{B} \in \mathbb{R}^{p \times n}\)

\(\mathbf{C} = \mathbf{A} \mathbf{B}^T \in \mathbb{R}^{m \times p}\)

\(C_{ij} = A_{ik} B_{jk}\)

C = (A[:,_,:]*B[_,:,:]).sum(-1)

C = function.outer(A,B).sum(-1)

6

\(\mathbf{A} \in \mathbb{R}^{m \times n}\)

\(\mathbf{B} \in \mathbb{R}^{m \times n}\)

\(c = \mathbf{A} : \mathbf{B} \in \mathbb{R}\)

\(c = A_{ij} B_{ij}\)

c = (A*B).sum([-2,-1])

Notes:

  1. In the above table the summation axes are numbered backward. For example, sum(-1) is used to sum over the last axis of an array. Although forward numbering is possible in many situations, backward numbering is generally preferred in Nutils code.

  2. When a summation over multiple axes is performed (#6), these axes are to be listed. In the case of single-axis summations listing is optional (for example sum(-1) is equivalent to sum([-1])). The shorter notation sum(-1) is preferred.

  3. When the numer of dimensions of the two arguments of a binary operation mismatch, singleton axes are automatically prepended to the “shorter” argument. This property can be used to shorten notation. For example, #3 can be written as (A*b).sum(-1). To avoid ambiguities, in general, such abbreviations are discouraged.