The sparse-tensor Package

sparse-tensor is a Haskell tensor algebra library. It defines the usual tensor algebra functions such as

• addition

result = t1 &+ t2

• scalar multiplication

result = s &. t

• tensor product

result = t1 &* t2

• or symmetrizations

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

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,

>>> :set -XDataKinds
>>> (undefined :: MyTensor 0 1) &+ (undefined :: MyTensor 0 2)

<interactive>: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:

>>> 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:

>>> 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:

>>> 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)))"