--------------------------------------------------------------------------------- -- | -- Module : Math.ConjugateGradient -- Copyright : (c) Levent Erkok -- License : BSD3 -- Maintainer : erkokl@gmail.com -- Stability : stable -- -- (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 -- >>> import System.Random -- >>> 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 --------------------------------------------------------------------------------- module Math.ConjugateGradient( -- * Types -- $typeInfo SV, SM -- * Sparse operations , sMulSV, sMulSM, addSV, subSV, dotSV, mulSMV, normSV -- * Conjugate-Gradient solver , solveCG -- * Displaying solutions , showSolution ) where import Data.List (intercalate) import Data.Maybe (fromMaybe) import qualified Data.IntMap as IM (IntMap, lookup, map, unionWith, intersectionWith, fold, fromList) import System.Random (Random, RandomGen, randomRs) import Numeric (showFFloat) -- | A sparse vector containing elements of type 'a'. For our purposes, the elements will be either 'Float's or 'Double's. 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.) type SV a = IM.IntMap 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. -- -- * 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. type SM a = (Int, IM.IntMap (SV a)) --------------------------------------------------------------------------------- -- Sparse vector/matrix operations --------------------------------------------------------------------------------- -- | Look-up a value in a sparse-vector. vLookup :: Num a => SV a -> Int -> a vLookup m k = fromMaybe 0 (k `IM.lookup` m) -- | Look-up a value in a sparse-matrix. mLookup :: Num a => SM a -> (Int, Int) -> a mLookup (_, m) (i, j) = maybe 0 (`vLookup` j) (i `IM.lookup` m) -- | Multiply a sparse-vector by a scalar. sMulSV :: RealFloat a => a -> SV a -> SV a sMulSV s = IM.map (s *) -- | Multiply a sparse-matrix by a scalar. sMulSM :: RealFloat a => a -> SM a -> SM a sMulSM s (n, m) = (n, IM.map (s `sMulSV`) m) -- | Add two sparse vectors. addSV :: RealFloat a => SV a -> SV a -> SV a addSV = IM.unionWith (+) -- | Subtract two sparse vectors. subSV :: RealFloat a => SV a -> SV a -> SV a subSV v1 v2 = addSV v1 (IM.map ((-1)*) v2) -- | Dot product of two sparse vectors. dotSV :: RealFloat a => SV a -> SV a -> a dotSV v1 v2 = IM.fold (+) 0 $ IM.intersectionWith (*) v1 v2 -- | Multiply a sparse matrix (nxn) with a sparse vector (nx1), obtaining a sparse vector (nx1). mulSMV :: RealFloat a => SM a -> SV a -> SV a mulSMV (_, m) v = IM.map (`dotSV` v) m -- | Norm of a sparse vector. (Square-root of its dot-product with itself.) normSV :: RealFloat a => SV a -> a normSV = sqrt . IM.fold (\e s -> e*e + s) 0 -- | 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. -- -- 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 symmetric positive definite. solveCG :: (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@. solveCG g a@(n, _) b = cg a b x0 where rs = take n (randomRs (0, 1) g) x0 = IM.fromList [p | p@(_, j) <- zip [0..] rs, j /= 0] -- | The Conjugate-gradient algorithm. Our implementation closely follows the -- one given here: <http://en.wikipedia.org/wiki/Conjugate_gradient_method#Example_code_in_Matlab> cg :: RealFloat a => SM a -> SV a -> SV a -> SV a cg a b x0 = cgIter (1000000 :: Int) (norm r0) r0 r0 x0 where r0 = b `subSV` (a `mulSMV` x0) cgIter 0 _ _ _ _ = error "Conjugate Gradient: No convergence after 10^6 iterations. Make sure the input matrix is symmetric positive-definite!" cgIter i eps p r x -- Stop if the square of the error is less than 1e-20, i.e., -- if the error itself is less than 1e-10. | eps' < 1e-20 = x' | True = cgIter (i-1) eps' p' r' x' where ap = a `mulSMV` p alpha = eps / ap `dotSV` p x' = x `addSV` (alpha `sMulSV` p) r' = r `subSV` (alpha `sMulSV` ap) eps' = norm r' p' = r' `addSV` ((eps' / eps) `sMulSV` p) norm = IM.fold (\e s -> e*e + s) 0 -- square of normSV, but no need for expensive square-root -- | 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. showSolution :: 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 showSolution prec ma@(n, _) vb vx = intercalate "\n" $ header ++ res where res = zipWith3 row a x b range = [0..n-1] sf d = showFFloat (Just prec) d "" a = [[sf (ma `mLookup` (i, j)) | j <- range] | i <- range] x = [sf (vx `vLookup` i) | i <- range] b = [sf (vb `vLookup` i) | i <- range] cellWidth = maximum (0 : map length (concat a ++ x ++ b)) row as xv bv = unwords (map pad as) ++ " | " ++ pad xv ++ " = " ++ pad bv pad s = reverse $ take (length s `max` cellWidth) $ reverse s ++ repeat ' ' center l s = let extra = l - length s (left, right) = (extra `div` 2, extra - left) in replicate left ' ' ++ s ++ replicate right ' ' header = case res of [] -> ["Empty matrix"] (r:_) -> let l = length (takeWhile (/= '|') r) h = center (l-1) "A" ++ " | " ++ center cellWidth "x" ++ " = " ++ center cellWidth "b" s = replicate l '-' ++ "+" ++ replicate (length r - l - 1) '-' in [h, s] {- $typeInfo We represent sparse matrices and vectors using 'IM.IntMap's. 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. Note that all non-@0@ rows should be present in the sparse matrix: Even if we only use symmetric matrices, the algorithm still requires all rows to be present, not just the upper (or the lower)-triangle. Indexing starts at @0@, and is assumed to be non-negative, corresponding to the row numbers. -}