LU factorization for the accelerate backend

sparse-linear-algebra

Numerical computation in native Haskell

This library provides common numerical analysis functionality, without requiring any external bindings. It aims to serve as an experimental platform for scientific computation in a purely functional setting.

State of the library

Mar 14, 2018: Mostly functional, but there are still a few (documented) bugs. Complex number support is still incomplete, so the users are advised to not rely on that for the time being. The issues related to Complex number handling are tracked in #51, #12, #30.

News

Oct 7, 2017: The library is evolving in a number of ways, to reflect performance observations and user requests:

typeclasses and instances for primitive types will become sparse-linear-algebra-core, along with a typeclass-oriented reformulation of the numerical algorithms that used to depend on the nested IntMap representation. This will let other developers build on top of this library, in the spirit of vector-space and linear.
The vector-based backend is being reworked.
An accelerate-based backend is under development [6, 7].

Iterative linear solvers (<\>)
- Generalized Minimal Residual (GMRES) (non-Hermitian systems)
- BiConjugate Gradient (BCG)
- Conjugate Gradient Squared (CGS)
- BiConjugate Gradient Stabilized (BiCGSTAB) (non-Hermitian systems)
- Moore-Penrose pseudoinverse (pinv) (rectangular systems)
Direct linear solvers
- LU-based (luSolve); forward and backward substitution (triLowerSolve, triUpperSolve)
Matrix factorization algorithms
- QR (qr)
- LU (lu)
- Cholesky (chol)
- Arnoldi iteration (arnoldi)
Eigenvalue algorithms
- QR (eigsQR)
- QR-Arnoldi (eigsArnoldi)
Utilities : Vector and matrix norms, matrix condition number, Givens rotation, Householder reflection
Predicates : Matrix orthogonality test (A^T A ~= I)

Under development

Eigenvalue algorithms
- Rayleigh quotient iteration (eigRayleigh)
Matrix factorization algorithms
- Golub-Kahan-Lanczos bidiagonalization (gklBidiag)
- Singular value decomposition (SVD)
Iterative linear solvers
- Transpose-Free Quasi-Minimal Residual (TFQMR)

Examples

The module Numeric.LinearAlgebra.Sparse contains the user interface.

Creation of sparse data

The fromListSM function creates a sparse matrix from a collection of its entries in (row, column, value) format. This is its type signature:

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

and, in case you have a running GHCi session (the terminal is denoted from now on by λ>), you can try something like this:

λ> amat = fromListSM (3,3) [(0,0,2),(1,0,4),(1,1,3),(1,2,2),(2,2,5)] :: SpMatrix Double

Similarly, fromListSV is used to create sparse vectors:

fromListSV :: Int -> [(Int, a)] -> SpVector a

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:

λ> prd amat

( 3 rows, 3 columns ) , 5 NZ ( density 55.556 % )

2.00   , _      , _      
4.00   , 3.00   , 2.00   
_      , _      , 5.00

Note (sparse storage): sparse data should only contain non-zero entries not to waste memory and computation.

Note (approximate output): prd rounds the results to two significant digits, and switches to scientific notation for large or small values. Moreover, values which are indistinguishable from 0 (see the Numeric.Eps module) are printed as _.

Matrix factorizations, matrix product

There are a few common matrix factorizations available; in the following example we compute the LU factorization of matrix amat and verify it with the matrix-matrix product ## of its factors :

λ> (l, u) <- lu amat
λ> prd $ l ## u

( 3 rows, 3 columns ) , 9 NZ ( density 100.000 % )

2.00   , _      , _      
4.00   , 3.00   , 2.00   
_      , _      , 5.00

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 Floats and 10^-12 for Doubles).

λ> prd $ l #~# u

( 3 rows, 3 columns ) , 5 NZ ( density 55.556 % )

2.00   , _      , _      
4.00   , 3.00   , 2.00   
_      , _      , 5.00

A matrix is transposed using the transpose function.

Sometimes we need to compute matrix-matrix transpose products, which is why the library offers the infix operators #^# (i.e. matrix transpose * matrix) and ##^ (matrix * matrix transpose):

λ> amat' = amat #^# amat
λ> prd amat'

( 3 rows, 3 columns ) , 9 NZ ( density 100.000 % )

20.00  , 12.00  , 8.00   
12.00  , 9.00   , 6.00   
8.00   , 6.00   , 29.00      


λ> lc <- chol amat'
λ> prd $ lc ##^ lc

( 3 rows, 3 columns ) , 9 NZ ( density 100.000 % )

20.00  , 12.00  , 8.00   
12.00  , 9.00   , 10.80  
8.00   , 10.80  , 29.00

In the last example we have also shown the Cholesky decomposition (M = L L^T where L is a lower-triangular matrix), which is only defined 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 <\> (inspired by Matlab's "backslash" function. Currently this method uses GMRES as default) :

λ> b = fromListDenseSV 3 [3,2,5] :: SpVector Double
λ> x <- amat <\> b
λ> prd x

( 3 elements ) ,  3 NZ ( density 100.000 % )

1.50   , -2.00  , 1.00

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 :

λ> prd $ amat #> x

( 3 elements ) ,  3 NZ ( density 100.000 % )

3.00   , 2.00   , 5.00

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 LU factors defined previously we can cross-verify the two solution methods:

λ> x' <- luSolve l u b
λ> prd x'

( 3 elements ) ,  3 NZ ( density 100.000 % )

1.50   , -2.00  , 1.00

License

GPL3, see LICENSE

Credits

Inspired by

linear : https://hackage.haskell.org/package/linear
vector-space : https://hackage.haskell.org/package/vector-space
sparse-lin-alg : https://github.com/laughedelic/sparse-lin-alg

References

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

[2] G.H. Golub and C.F. Van Loan, Matrix Computations, 3rd ed., 1996

[3] T.A. Davis, Direct Methods for Sparse Linear Systems, 2006

[4] L.N. Trefethen, D. Bau, Numerical Linear Algebra, SIAM, 1997

[5] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in Fortran 77, 2nd ed., 1992

[6] M. M. T. Chakravarty, et al., Accelerating Haskell array codes with multicore GPUs - DAMP'11

[7] accelerate

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sparse-linear-algebra's Introduction

sparse-linear-algebra

State of the library

News

Contents

Under development

Examples

Creation of sparse data

Displaying sparse data

Matrix factorizations, matrix product

Linear systems

License

Credits

References

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