tensor-contraction

implements tensor contraction on a single mixed tensor

Install

With npm do

npm install tensor-contraction --save

Usage

Signature is (addition, indicesPair, tensorDim, tensorData) where

• addition is a function that defines the scalar operator used
• indicesPair is an array of two elements that indicates which indices will be used for contraction
• tensorDim is an array that defines tensor indices set
• tensorData is an array that defines the tensor components

It returns the contractedTensorData array given by the tensors contraction.

It throws a TypeError: Contraction indices does not have the same dimension. See indices pair check example.

Examples

All code in the examples below is intended to be contained into a single file.

Let’s use common real addition.

var tensorContraction = require('tensor-contraction')

function addition (a, b) { return a + b }

var contraction = tensorContraction.bind(null, addition)

Matrix trace

The trace of a matrix is the sum of the components in its diagonal. It is the simplest example of tensor contraction. In the following example, the trace is 1 + 5 + 9 = 15

contraction([0, 1], [3, 3], [1, 2, 3,
                             4, 5, 6,
                             7, 8, 9]).should.be.eql(15)

Indices pair check

It is required that indices chosen for contraction have the same dimension. The following example

contraction([0, 1], [3, 2], [1, 2, 3,
                             4, 5, 6])

will throw with message

Contraction indices does not have the same dimension: 0-th index = 3 but 1-th index = 2.

Order 3

Consider a tensor of order 3 and contract pairing first and last index. A multidimensional array t is used to point out which component corresponds to a given combination of indices, but keep in mind that tensors are always given as a monodimensional array. Note that contracted tensor is a vector with two elements, where:

1. the first one is given by the sum of the tensor components with middle index equal to 0 and with first and last index equals;
2. the second one is given by the sum of tensor components with middle index equal to 1 and with first and last index equals.

var t = [[[0, 1], [2, 3]], [[4, 5], [6, 7]]] // t[0][0][0] = 0
                                             // t[0][0][1] = 1
                                             // t[0][1][0] = 2
                                             // t[0][1][1] = 3
                                             // t[1][0][0] = 4
                                             // t[1][0][1] = 5
                                             // t[1][1][0] = 6
                                             // t[1][1][1] = 7

contraction([0, 2], [2, 2, 2], [t[0][0][0], t[0][0][1], t[0][1][0], t[0][1][1],  // [5, 9] = [0 + 5, 2 + 7] =
                                t[1][0][0], t[1][0][1], t[1][1][0], t[1][1][1]]) //        = [t[0][0][0] + t[1][0][1],
                                                                                 //           t[0][1][0] + t[1][1][1]]

Follows calculation of other 2 possible contractions, where the tensorData is given by explicit numeric values.

contraction([0, 1], [2, 2, 2], [0, 1, 2, 3, 4, 5, 6, 7]) // [6, 8]

contraction([1, 2], [2, 2, 2], [0, 1, 2, 3, 4, 5, 6, 7]) // [3, 11]

License

MIT