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
• 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
>>> let a = (2, fromList [(0, fromList [(0, 4), (1, 1)]), (1, fromList [(0, 1), (1, 3)])]) :: SM Double
>>> let b = 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

We represent sparse matrices and vectors using IntMaps. In a sparse vector, we only populate those elements that are non-0. In a sparse matrix, we only populate those rows that contain a non-0 element. This leads to an efficient representation for sparse matrices and vectors, where the space usage is proportional to number of non-0 elements. Strictly speaking, putting non-0 elements would not break the algorithms we use, but clearly they would be less efficient.

Indexing starts at 0, and is assumed to be non-negative, corresponding to the row numbers.

type SV a = IntMap a

A sparse vector containing elements of type a. (For our purposes, the elements will be either Floats or Doubles.)

type SM a = (Int, IntMap (SV a))

A sparse matrix is 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. (Nothing will break if you put in rows that have all 0's in them, it's just not as efficient.)
• 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.

Sparse operations

sMulSV :: RealFloat a => a -> SV a -> SV a

Multiply a sparse-vector by a scalar.

sMulSM :: RealFloat a => a -> SM a -> SM a

Multiply a sparse-matrix by a scalar.

addSV :: RealFloat a => SV a -> SV a -> SV a

Add two sparse vectors.

subSV :: RealFloat a => SV a -> SV a -> SV a

Subtract two sparse vectors.

dotSV :: RealFloat a => SV a -> SV a -> a

Dot product of two sparse vectors.

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

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

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

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

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).

-> (a, SV a)

The final error factor, and 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.
• 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 for either property. (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 iteration. 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.

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.