Math.ConjugateGradient

(The linear equation solver library is hosted at http://github.com/LeventErkok/conjugateGradient. Comments, bug reports, and patches are always welcome.)

Sparse matrix linear-equation solver, using the conjugate gradient algorithm. Note that the technique only applies to matrices that are symmetric and positive definite. See http://en.wikipedia.org/wiki/Conjugate_gradient_method for details.

The conjugate gradient method can handle very large sparse matrices, where direct methods (such as LU decomposition) are way too expensive to be useful in practice. Such large sparse matrices arise naturally in many engineering problems, such as in ASIC placement algorithms and when solving partial differential equations.

Here's an example usage, for the simple system:

       4x +  y = 1
        x + 3y = 2

>>> import Data.IntMap.Strict
>>> import System.Random
>>> import Math.ConjugateGradient
>>> let a = SM (2, fromList [(0, SV (fromList [(0, 4), (1, 1)])), (1, SV (fromList [(0, 1), (1, 3)]))]) :: SM Double
>>> let b = SV (fromList [(0, 1), (1, 2)]) :: SV Double
>>> let g = mkStdGen 12345
>>> let x = solveCG g a b
>>> putStrLn $ showSolution 4 a b x
      A       |   x    =   b   
--------------+----------------
4.0000 1.0000 | 0.0909 = 1.0000
1.0000 3.0000 | 0.6364 = 2.0000

Types

newtype SV a

A sparse vector containing elements of type a. Only the indices that contain non-0 elements should be given for efficiency purposes. (Nothing will break if you put in elements that are 0's, it's just not as efficient.)

Constructors

SV (IntMap a)

newtype SM a

A sparse matrix is essentially an int-map containing sparse row-vectors:

• The first element, n, is the number of rows in the matrix, including those with all 0 elements.
• The matrix is implicitly assumed to be nxn, indexed by keys (0, 0) to (n-1, n-1).
• When constructing a sparse-matrix, only put in rows that have a non-0 element in them for efficiency.
• Note that you have to give all the non-0 elements: Even though the matrix must be symmetric for the algorithm to work, the matrix should contain all the non-0 elements, not just the upper (or the lower)-triangle.
• Make sure the keys of the int-map is a subset of [0 .. n-1], both for the row-indices and the indices of the vectors representing the sparse-rows.

Constructors

SM (Int, IntMap (SV a))

Sparse operations

lookupSV :: Num a => SV a -> Int -> a

Look-up a value in a sparse-vector.

lookupSM :: Num a => SM a -> (Int, Int) -> a

Look-up a value in a sparse-matrix.

addSV :: Num a => SV a -> SV a -> SV a

Add two sparse vectors.

subSV :: Num a => SV a -> SV a -> SV a

Subtract two sparse vectors.

dotSV :: Num a => SV a -> SV a -> a

Dot product of two sparse vectors.

normSV :: RealFloat a => SV a -> a

Norm of a sparse vector. (Square-root of its dot-product with itself.)

sMulSV :: Num a => a -> SV a -> SV a

Multiply a sparse-vector by a scalar.

sMulSM :: Num a => a -> SM a -> SM a

Multiply a sparse-matrix by a scalar.

mulSMV :: Num a => SM a -> SV a -> SV a

Multiply a sparse matrix (nxn) with a sparse vector (nx1), obtaining a sparse vector (nx1).

Conjugate-Gradient solver

solveCG

Arguments

:: (RandomGen g, RealFloat a, Random a)

=> g

The seed for the random-number generator.

-> SM a

The A sparse matrix (nxn).

-> SV a

The b sparse vector (nx1).

-> SV a

The x sparse matrix (nx1), such that Ax = b.

Conjugate Gradient Solver for the system Ax=b. See: http://en.wikipedia.org/wiki/Conjugate_gradient_method.

NB. Assumptions on the input:

• The A matrix is symmetric and positive definite.
• All non-0 rows are present. (Even if the input is assumed symmetric, all rows must be present.)
• The indices start from 0 and go consecutively up-to n-1. (Only non-0 value/row indices has to be present, of course.)

For efficiency reasons, we do not check that these properties hold of the input. (If these assumptions are violated, the algorithm will still produce a result, but not the one you expected!)

We perform either 10^6 iterations of the Conjugate-Gradient algorithm, or until the error factor is less than 1e-10. The error factor is defined as the difference of the norm of the current solution from the last one, as we go through the iterative solver. See http://en.wikipedia.org/wiki/Conjugate_gradient_method#Convergence_properties_of_the_conjugate_gradient_method for a discussion on the convergence properties of this algorithm.

The solver can throw an error if it does not converge by 10^6 iterations. This is typically an indication that the input matrix is not well formed, i.e., not symmetric positive-definite.

Displaying solutions

showSolution

Arguments

:: RealFloat a

=> Int

Precision: Use this many digits after the decimal point.

-> SM a

The A matrix, nxn

-> SV a

The b matrix, nx1

-> SV a

The x matrix, nx1, as returned by solveCG, for instance.

-> String

Display a solution in a human-readable form. Needless to say, only use this method when the system is small enough to fit nicely on the screen.