# sparse-linear-algebra

Numerical computation in native Haskell

TravisCI :

This library provides common numerical analysis functionality, without requiring any external bindings. It is not optimized for performance (yet), but it serves as an experimental platform for scientific computation in a purely functional setting.

Algorithms :

Iterative linear solvers

BiConjugate Gradient (BCG)

Conjugate Gradient Squared (CGS)

BiConjugate Gradient Stabilized (BiCGSTAB) (non-Hermitian systems)

Transpose-Free Quasi-Minimal Residual (TFQMR)

Matrix decompositions

QR factorization

LU factorization

Eigenvalue algorithms

Utilities : Vector and matrix norms, matrix condition number, Givens rotation, Householder reflection

Predicates : Matrix orthogonality test (A^T A ~= I)

## Examples

The module `Numeric.LinearAlgebra.Sparse`

contains the user interface.

### Creation and pretty-printing

To create a sparse matrix from an array of its entries we use `fromListSM`

:

```
fromListSM :: Foldable t => (Int, Int) -> t (IxRow, IxCol, a) -> SpMatrix a
```

e.g.

```
> amat = fromListSM (3,3) [(0,0,2),(1,0,4),(1,1,3),(1,2,2),(2,2,5)]
```

And similarly for sparse vectors : `fromListSV :: Int -> [(Int, a)] -> SpVector a`

.

Both sparse vectors and matrices can be pretty-printed using `prd`

:

```
> prd amat
( 3 rows, 3 columns ) , 5 NZ ( sparsity 0.5555555555555556 )
[2,0,0]
[4,3,2]
[0,0,5]
```

The zeros are just added at pretty printing time; sparse vectors and matrices should only contain non-zero entries.

### Matrix operations

Matrix factorizations are available as `lu`

and `qr`

respectively, and are straightforward to verify by using the matrix product `##`

:

```
> (l, u) = lu amat
> prd $ l ## u
( 3 rows, 3 columns ) , 9 NZ ( sparsity 1.0 )
[2.0,0.0,0.0]
[4.0,3.0,2.0]
[0.0,0.0,5.0]
```

Notice that the result is *dense*, i.e. certain entries are numerically zero but have been inserted into the result along with all the others (thus taking up memory!).
To preserve sparsity, we can use a sparsifying matrix-matrix product `#~#`

, which filters out all the elements x for which `|x| <= eps`

, where `eps`

(defined) in `Numeric.Eps`

, is fixed at 10^-8.

```
> prd $ l #~# u
( 3 rows, 3 columns ) , 5 NZ ( sparsity 0.5555555555555556 )
[2.0,0.0,0.0]
[4.0,3.0,2.0]
[0.0,0.0,5.0]
```

### Linear systems

Linear systems can be solved with either `linSolve`

(which also requires choosing a method) or with `<\>`

(which uses BiCGSTAB as default) :

```
> b = fromListSV 3 [(0,3),(1,2),(2,5)]
> x = amat <\> b
> prd x
( 3 elements ) , 3 NZ ( sparsity 1.0 )
[1.4999999999999998,-1.9999999999999998,0.9999999999999998]
```

The result can be verified by computing the matrix-vector action `amat #> x`

, which should (ideally) be very close to the right-hand side `b`

:

```
> prd $ amat #> x
( 3 elements ) , 3 NZ ( sparsity 1.0 )
[2.9999999999999996,1.9999999999999996,4.999999999999999]
```

This is also an experiment in principled scientific programming :

set the stage by declaring typeclasses and some useful generic operations (normed linear vector spaces, i.e. finite-dimensional spaces equipped with an inner product that induces a distance function),

define appropriate data structures, and how they relate to those properties (sparse vectors and matrices, defined internally via `Data.IntMap`

, are made instances of the VectorSpace and Additive classes respectively). This allows to decouple the algorithms from the actual implementation of the backend,

implement the algorithms, following 1:1 the textbook [1]

## License

GPL3, see LICENSE

## Credits

Inspired by

## References

[1] : Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., 2000