In Diderot, vectors (rank-1 tensors), matrices (rank-2 tensors), and
tensors generally are objects with some number of ordered indices.
This program demonstrates tensor construction, indexing, slicing with
:, and contracting with either • or :.
The type of a tensor is determined by its shape: a []-enclosed
list of how many values each of its indices can take on. The real
type for scalars can also be expressed as tensor[] (no indices). A
3-vector has type tensor[3]: one index can take on 3 values, for the
three possible dimensions. Diderot also supports the vec3 synonym.
A 3x3 matrix has type tensor[3,3]. Each of the shape dimensions must
be 2 or greater. Note that Diderot does not currently support
distinguishing between covariant and contravariant indices; the
assumption is that all tensor coefficients are measured with respect
to an orthonormal coordinate frame.
Like Mathematica, Diderot doesn't enforce a semantic distinction between
row and column vectors; tensor indices are ordered, and the semantics of
tensor operations depends on that order. Still, we can assume for the sake
of familiarity that in a rank-2 tensor, i.e. a matrix, the first index
selects the row, and the second index selects the column. A matrix created
by mm = [[a,b,c],[d,e,f],[g,h,i]] would conventionally be written as:
a b c
d e f
g h i
The first row is mm[0,:], the first column is mm[:,0], and
b == mm[0,1] (row 0 and column 1).
Reading through this program's source alongside what it prints out can help build understanding of how tensors in Diderot work, which is basically the same way they work in other languages that support higher-order tensors. The program ends with a demonstration of how, when saved out to files, tensor indices are ordered slow to fast.