diff --git a/Math/ConjugateGradient.hs b/Math/ConjugateGradient.hs --- a/Math/ConjugateGradient.hs +++ b/Math/ConjugateGradient.hs @@ -10,13 +10,8 @@ -- 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 for details. +-- applies to matrices that are symmetric and positive definite. See +-- 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. @@ -32,8 +27,9 @@ -- -- >>> 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 +-- >>> 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 @@ -45,10 +41,9 @@ module Math.ConjugateGradient( -- * Types - -- $typeInfo - SV, SM + SV(..), SM(..) -- * Sparse operations - , sMulSV, sMulSM, addSV, subSV, dotSV, mulSMV, normSV + , svLookup, smLookup, addSV, subSV, dotSV, normSV, sMulSV, sMulSM, mulSMV -- * Conjugate-Gradient solver , solveCG -- * Displaying solutions @@ -61,12 +56,11 @@ 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 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.) +newtype SV a = SV (IM.IntMap a) --- | A sparse matrix is an int-map containing sparse row-vectors: +-- | 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. -- @@ -78,47 +72,47 @@ -- 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)) +newtype SM a = SM (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) +svLookup :: Num a => SV a -> Int -> a +svLookup (SV v) k = fromMaybe 0 (k `IM.lookup` v) -- | 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) +smLookup :: Num a => SM a -> (Int, Int) -> a +smLookup (SM (_, m)) (i, j) = maybe 0 (`svLookup` 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 *) +sMulSV :: Num a => a -> SV a -> SV a +sMulSV s (SV v) = SV (IM.map (s *) v) -- | 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) +sMulSM :: Num a => a -> SM a -> SM a +sMulSM s (SM (n, m)) = SM (n, IM.map (s `sMulSV`) m) -- | Add two sparse vectors. -addSV :: RealFloat a => SV a -> SV a -> SV a -addSV = IM.unionWith (+) +addSV :: Num a => SV a -> SV a -> SV a +addSV (SV v1) (SV v2) = SV (IM.unionWith (+) v1 v2) -- | Subtract two sparse vectors. -subSV :: RealFloat a => SV a -> SV a -> SV a -subSV v1 v2 = addSV v1 (IM.map ((-1)*) v2) +subSV :: Num a => SV a -> SV a -> SV a +subSV v1 (SV v2) = addSV v1 (SV (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 +dotSV :: Num a => SV a -> SV a -> a +dotSV (SV v1) (SV 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 +mulSMV :: Num a => SM a -> SV a -> SV a +mulSMV (SM (_, m)) v = SV (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 +normSV (SV v) = sqrt . IM.fold (\e s -> e*e + s) 0 $ v -- | Conjugate Gradient Solver for the system @Ax=b@. See: . -- @@ -126,28 +120,30 @@ -- -- * 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 for either property. (If these assumptions are +-- 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 iteration. See +-- the current solution from the last one, as we go through the iterative solver. See -- -- 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. +-- that the input matrix is not well formed, i.e., 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 +solveCG g a@(SM (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] + x0 = SV $ IM.fromList [p | p@(_, j) <- zip [0..] rs, j /= 0] -- | The Conjugate-gradient algorithm. Our implementation closely follows the -- one given here: @@ -161,12 +157,12 @@ | eps' < 1e-20 = x' | True = cgIter (i-1) eps' p' r' x' where ap = a `mulSMV` p - alpha = eps / ap `dotSV` 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 + norm (SV v) = IM.fold (\e s -> e*e + s) 0 v -- 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. @@ -176,13 +172,13 @@ -> 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 +showSolution prec ma@(SM (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] + a = [[sf (ma `smLookup` (i, j)) | j <- range] | i <- range] + x = [sf (vx `svLookup` i) | i <- range] + b = [sf (vb `svLookup` 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 ' ' @@ -196,13 +192,3 @@ ++ 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. --} diff --git a/README b/README --- a/README +++ b/README @@ -6,3 +6,5 @@ Sparse matrix linear equation solver, using the Conjugate Gradient algorithm: http://en.wikipedia.org/wiki/Conjugate_gradient_method. The method is applicable to matrices that are symmetric and positive definite. + +On hackage: http://hackage.haskell.org/package/conjugateGradient diff --git a/RELEASENOTES b/RELEASENOTES --- a/RELEASENOTES +++ b/RELEASENOTES @@ -1,30 +1,27 @@ Hackage: GitHub: -Latest Hackage released version: 1.4 +Latest Hackage released version: 2.0 -Version 1.4, 2013-04-16 +Version 2.0, 2013-04-17 ====================================================================== + - First "official" release + - Add newtype wrappers for safety + - Export all sparse matrix operations + +Versions 1.4-1.1, Early April 2013 +====================================================================== - Fix github source location - Clarify that the entire matrix should be given: Even though we assume it's symmetric, the algorithm needs all non-0 elements to be present; not just the upper (or the lower)-triangle. - -====================================================================== -Version 1.3, 2013-04-16 - Instead of returning an error-bound, throw an error if no convergence is reached after 10^6 iterations. This is more practical, as returning a result after that many iterations typically indicates the input matrix is not symmetric and positive-definite. - Tighten import lists and the example - -====================================================================== -Version 1.2, 2013-04-15 - Simplify types, clean-up example. - -====================================================================== -Version 1.1, 2013-04-15 - Add an example demonstating the usage over a simple example. ====================================================================== diff --git a/conjugateGradient.cabal b/conjugateGradient.cabal --- a/conjugateGradient.cabal +++ b/conjugateGradient.cabal @@ -1,5 +1,5 @@ Name: conjugateGradient -Version: 1.4 +Version: 2.0 Category: Math Synopsis: Sparse matrix linear-equation solver Description: Sparse matrix linear-equation solver, using the conjugate gradient algorithm. Note that the