# sparse-linear-algebra
Numerical computation in native Haskell
TravisCI : [](https://travis-ci.org/ocramz/sparse-linear-algebra)
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.
Contents :
* Iterative linear solvers (`linSolve`)
* Generalized Minimal Residual (GMRES) (non-Hermitian systems)
* BiConjugate Gradient (BCG)
* Conjugate Gradient Squared (CGS)
* BiConjugate Gradient Stabilized (BiCGSTAB) (non-Hermitian systems)
* Transpose-Free Quasi-Minimal Residual (TFQMR)
* Direct linear solvers
* LU-based (`luSolve`)
* Matrix factorization algorithms
* QR (`qr`)
* LU (`lu`)
* Cholesky (`chol`)
* Eigenvalue algorithms
* Arnoldi iteration (`arnoldi`)
* QR (`eigsQR`)
* Rayleigh quotient iteration (`eigRayleigh`)
* 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 of sparse data
The `fromListSM` function creates a sparse matrix from an array of its entries we use :
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
fromListSV :: Int -> [(Int, a)] -> SpVector a
can be used to create sparse vectors.
Alternatively, the user can copy the contents of a list to a (dense) SpVector using
fromListDenseSV :: Int -> [a] -> SpVector a
### Displaying sparse data
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 printing time; sparse vectors and matrices should only contain non-zero entries.
### Matrix operations
There are a few common matrix factorizations available; in the following example we compute the LU factorization of a matrix and verify it with the matrix-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`) depends on the numerical type used (e.g. it is 10^-6 for `Float`s and 10^-12 for `Double`s).
> 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]
A matrix is transposed using `transposeSM`.
Sometimes we need to compute matrix-matrix transpose products, which is why the library offers the infix operators `#^#` (M^T N) and `##^` (M N^T):
> amat' = amat #^# amat
> prd amat'
( 3 rows, 3 columns ) , 9 NZ ( sparsity 1.0 )
[20.0,12.0,8.0]
[12.0,9.0,6.0]
[8.0,6.0,29.0]
> l = chol amat'
> prd $ l ##^ l
( 3 rows, 3 columns ) , 9 NZ ( sparsity 1.0 )
[20.000000000000004,12.0,8.0]
[12.0,9.0,10.8]
[8.0,10.8,29.0]
In the above example we have also shown the Cholesky decomposition (M = L L^T where L is a lower-triangular matrix), which is only possible for symmetric positive-definite matrices.
### Linear systems
Large sparse linear systems are best solved with iterative methods. `sparse-linear-algebra` provides a selection of these via the `linSolve` function, or alternatively `<\>` (which uses GMRES as default solver method) :
> b = fromListDenseSV 3 [3,2,5]
> 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]
The library also provides a forward-backward substitution solver (`luSolve`) based on a triangular factorization of the system matrix (usually LU). This should be the preferred for solving smaller, dense systems. Using the data defined above we can cross-verify the two solution methods:
> x' = luSolve l u b
> prd x'
( 3 elements ) , 3 NZ ( sparsity 1.0 )
[1.5,-2.0,1.0]
----------
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, 2]
## License
GPL3, see LICENSE
## Credits
Inspired by
* `linear` : https://hackage.haskell.org/package/linear
* `sparse-lin-alg` : https://github.com/laughedelic/sparse-lin-alg
## References
[1] : Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., 2000
[2] : L. N. Trefethen, D. Bau, Numerical Linear Algebra, SIAM, 1997