# The `sparse-tensor` Package
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`sparse-tensor` is a Haskell tensor algebra library. It defines the usual tensor algebra functions such as
- addition
```haskell
result = t1 &+ t2
```
- scalar multiplication
```haskell
result = s &. t
```
- tensor product
```haskell
result = t1 &* t2
```
- or symmetrizations
```haskell
result = symTens (0,1) t -- symmetrization in first two indices
```
## The `Tensor` type
Tensor types can be defined with any value type and index types. For example, a tensor type with `n` contravariant and `m` covariant 4-d spacetime indices ranging from 0 to 3 and `Rational` values can be defined as
```haskell
type MyTensor n m = AbsTensor2 n m Ind3 (SField Rational)
```
The operations on tensors are **type-safe**, for example it is not possible to add two tensors of different rank,
```haskell
>>> :set -XDataKinds
>>> (undefined :: MyTensor 0 1) &+ (undefined :: MyTensor 0 2)
```
```
:3:33: error:
• Couldn't match type ‘2’ with ‘1’
[...]
```
as this causes a type error at **compile time**.
## Predefined tensors
The package comes with pre-defined tensor types. Basic tensors of these types for applications in mathematical physics are exported by `Math.Tensor.Examples.Gravity`:
```haskell
>>> sequence_ $ fmap print $ toListT2' delta3 -- print assocs of spacetime delta
(([0],[0]),SField (1 % 1))
(([1],[1]),SField (1 % 1))
(([2],[2]),SField (1 % 1))
(([3],[3]),SField (1 % 1))
>>> sequence_ $ fmap print $ toListT2' eta -- print assocs of Minkowski metric
(([],[0,0]),SField ((-1) % 1))
(([],[1,1]),SField (1 % 1))
(([],[2,2]),SField (1 % 1))
(([],[3,3]),SField (1 % 1))
>>> let t = invEta &* epsilon
>>> contrATens1 (0,0) $ contrATens1 (1,1) t -- contraction of inverse eta with epsilon
ZeroTensor
```
It is of course possible to define further custom tensor types and tensors.
`Math.Tensor.LorentzGenerator` exports functionality to generate a basis for the space of Lorentz-invariant tensors of certain rank which obey certain symmetries.
## Automatic differentiation
`sparse-tensor` also supports tensors with **functions** as values. For such tensors, the package also provides the `partial` function for automatic differentiation. `Math.Tensor.Examples.Gravity.Schwarzschild` exports the Einstein tensor for a Schwarzschild spacetime, calculated from the Schwarzschild metric:
```haskell
>>> let e = einstein 2.0 -- Einstein tensor for Schwarzschild metric with r_s = 2.0
>>> e `evalSec` [1.2, 3.1, 1.3, 2.2] -- evaluate at spacetime point
ZeroTensor
```
## Symbolic calculations
The package can also handle **symbolic** tensor values. All manipulations, including differentiation, are then performed on strings which may be passed to a computer algebra engine. `sparse-tensor` itself cannot yet simplify these symbolic values. `Math.Tensor.Examples.Gravity.SchwarzschildSymbolic` exports the Schwarzschild metric with symbolic entries and methods to calculate derived geometric entities:
```haskell
>>> let g = schwarzschildS
>>> let g' = schwarzschildS'
>>> let gamma = christoffelS g g'
>>> let comps = toListT2 gamma -- get assocs
>>> print $ snd $ comps !! 1 -- component gamma^t_tr
SSymbolic "(1 % 2)*((1/(1 - rs/r))*(diff(1 - rs / r, r)))"
```