tensor: A completely type-safe library for linear algebra

This library defines data types and classes for fixed dimension vectors and tensors. The main objects are:

Data.Ordinal.Ordinal

A totally ordered set with fixed size. The Data.Ordinal.Ordinal type Data.Ordinal.One contains 1 element, Data.Ordinal.Succ Data.Ordinal.One contains 2 elements, Data.Ordinal.Succ Data.Ordinal.Succ Data.Ordinal.One contains 3 elements, and so on (see Data.Ordinal for more details). The type Data.Ordinal.Two is an alias for Data.Ordinal.Succ Data.Ordinal.One, Data.Ordinal.Three is an alias for Data.Ordinal.Succ Data.Ordinal.Succ Data.Ordinal.One, and so on.

Data.TypeList.MultiIndex.MultiIndex

The index set. It can be linear, rectangular, parallelepipedal, etc. The dimensions of the sides are expressed using Data.Ordinal.Ordinal types and the type constructor Data.TypeList.MultiIndex.:|:, e.g. (Data.Ordinal.Two Data.TypeList.MultiIndex.:|: (Data.Ordinal.Three Data.TypeList.MultiIndex.:|: Data.TypeList.MultiIndex.Nil)) is a rectangular index set with 2 rows and 3 columns. The index set also contains elements, for example (Data.Ordinal.Two Data.TypeList.MultiIndex.:|: (Data.Ordinal.Three Data.TypeList.MultiIndex.:|: Data.TypeList.MultiIndex.Nil)) contains all the pairs (i Data.TypeList.MultiIndex.:|: (j Data.TypeList.MultiIndex.:|: Nil)) where i is in Data.Ordinal.Two and j is in Data.Ordinal.Three. See Data.TypeList.MultiIndex for more details.

Data.Tensor.Tensor

It is an assignment of elements to each element of its Data.TypeList.MultiIndex.MultiIndex.

Objects like vectors and matrices are special cases of tensors. Most of the functions to manipulate tensors are grouped into type classes. This allow the possibility of having different internal representations (backends) of a tensor, and act on these with the same functions. At the moment we only provide one backend based on http://hackage.haskell.org/package/vector, which is accessible by importing the module Data.Tensor.Vector. More backends will be provided in future releases.

Here is a usage example:

>>> :m Data.Ordinal Data.TypeList.MultiIndex Data.Tensor.Vector
>>> fromList [2,3,5,1,3,6,0,5,4,2,1,3] :: Tensor (Four :|: (Three :|: Nil)) Int
[[2,3,5],[1,3,6],[0,5,4],[2,1,3]]

The above defines a tensor with 4 rows and 3 columns (a matrix) and Int coefficients. The entries of this matrix are taken from a list using Data.Tensor.fromList which is a method of the class Data.Tensor.FromList. Notice the output: the Show instance is defined in such a way to give a readable representation as list of lists. The is equivalent but slightly more readable code:

>>> fromList [2,3,5,1,3,6,0,5,4,2,1,3] :: Matrix Four Three Int
[[2,3,5],[1,3,6],[0,5,4],[2,1,3]]

Analogously

>>> fromList [7,3,-6] :: Tensor (Three :|: Nil) Int
[7,3,-6]

and

>>> fromList [7,3,-6] :: Vector Three Int
[7,3,-6]

are the same. In order to access an entry of a Data.Tensor.Tensor we use the Data.Tensor.! operator, which takes the same Data.TypeList.MultiIndex.MultiIndex of the Data.Tensor.Tensor as its second argument:

>>> let a = fromList [2,3,5,1,3,6,0,5,4,2,1,3] :: Matrix Four Three Int
>>> let b = fromList [7,3,-6] :: Vector Three Int
>>> a ! (toMultiIndex [1,3] :: (Four :|: (Three :|: Nil)))
5
>>> b ! (toMultiIndex [2] :: (Three :|: Nil))
3

it returns the element at the coordinate (1,3) of the matrix a, and the element at the coordinate 2 of the vector b. In fact, thanks to type inference, we could simply write

>>> a ! toMultiIndex [1,3]
5
>>> b ! toMultiIndex [2]
2

And now a couple of examples of algebraic operations (requires adding Data.Tensor.LinearAlgebra.Vector to the import list):

>>> :m Data.Ordinal Data.TypeList.MultiIndex Data.Tensor.Vector Data.Tensor.LinearAlgebra.Vector
>>> let a = fromList [2,3,5,1,3,6,0,5,4,2,1,3] :: Matrix Four Three Int
>>> let b = fromList [7,3,-6] :: Vector Three Int
>>> a .*. b
[-7,-20,-9,-1]

is the product of matrix a and vector b, while

>>> let c = fromList [3,4,0,-1,4,5,6,2,1] :: Matrix Three Three Int
>>> c
[[3,4,0],[-1,4,5],[6,2,1]]
>>> charPoly c
[106,13,8]

gives the coefficients of the characteristic polynomial of the matrix c.

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