diff --git a/Math/ConjugateGradient.hs b/Math/ConjugateGradient.hs --- a/Math/ConjugateGradient.hs +++ b/Math/ConjugateGradient.hs @@ -20,6 +20,8 @@ -- -- 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: -- @@ -28,11 +30,12 @@ -- x + 3y = 2 -- @ -- --- >>> let a = IM.fromList [(0, IM.fromList [(0,4::Double), (1,1)]), (1, IM.fromList [(0, 1), (1, 3)])] --- >>> let b = IM.fromList [(0,1), (1,2::Double)] +-- >>> 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 2 a b --- >>> putStrLn $ showSolution 6 4 2 a b x +-- >>> let (_, x) = solveCG g a b +-- >>> putStrLn $ showSolution 4 a b x -- A | x = b -- --------------+---------------- -- 4.0000 1.0000 | 0.0909 = 1.0000 @@ -60,8 +63,17 @@ -- | A sparse vector containing elements of type 'a'. (For our purposes, the elements will be either 'Float's or 'Double's.) type SV a = IM.IntMap a --- | A sparse matrix is an int-map containing sparse row-vectors. Again, only put in rows that have a non-@0@ element in them for efficiency. -type SM a = IM.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. +type SM a = (Int, IM.IntMap (SV a)) --------------------------------------------------------------------------------- -- Sparse vector/matrix operations @@ -73,7 +85,7 @@ -- | 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) +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 @@ -81,7 +93,7 @@ -- | Multiply a sparse-matrix by a scalar. sMulSM :: RealFloat a => a -> SM a -> SM a -sMulSM s = IM.map (s `sMulSV`) +sMulSM s (n, m) = (n, IM.map (s `sMulSV`) m) -- | Add two sparse vectors. addSV :: RealFloat a => SV a -> SV a -> SV a @@ -97,7 +109,7 @@ -- | 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 (_, 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 @@ -122,11 +134,10 @@ -- for a discussion on the convergence properties of this algorithm. solveCG :: (RandomGen g, RealFloat a, Random a) => g -- ^ The seed for the random-number generator. - -> Int -- ^ Number of variables. -> 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@. -solveCG g n a b = cg a b x0 +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] @@ -152,22 +163,21 @@ -- | 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 -- ^ Cell-width. Each value will occupy this many characters at least. - -> Int -- ^ Precision: Use this many digits after the decimal point. - -> Int -- ^ Number of variables, @n@ - -> SM a -- ^ The @A@ matrix, @nxn@ - -> SV a -- ^ The @b@ matrix, @nx1@ - -> SV a -- ^ The @x@ matrix, @nx1@, as returned by 'solveCG', for instance. + => 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 padLen prec n ma vb vx = intercalate "\n" $ header ++ res +showSolution prec ma@(n, _) vb vx = intercalate "\n" $ header ++ res where res = zipWith3 row a x b range = [0..n-1] - a = [[ma `mLookup` (i, j) | j <- range] | i <- range] - x = [vx `vLookup` i | i <- range] - b = [vb `vLookup` i | i <- range] - row as xv bv = unwords (map sh as) ++ " | " ++ sh xv ++ " = " ++ sh bv - sh d = pad $ showFFloat (Just prec) d "" - pad s = reverse $ take (length s `max` padLen) $ reverse s ++ repeat ' ' + 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 ' ' @@ -175,7 +185,7 @@ [] -> ["Empty matrix"] (r:_) -> let l = length (takeWhile (/= '|') r) h = center (l-1) "A" ++ " | " - ++ center padLen "x" ++ " = " ++ center padLen "b" + ++ center cellWidth "x" ++ " = " ++ center cellWidth "b" s = replicate l '-' ++ "+" ++ replicate (length r - l - 1) '-' in [h, s] @@ -185,5 +195,5 @@ 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. -Indexings starts at @0@, and is assumed to be non-negative, corresponding to the row numbers. +Indexing starts at @0@, and is assumed to be non-negative, corresponding to the row numbers. -} diff --git a/conjugateGradient.cabal b/conjugateGradient.cabal --- a/conjugateGradient.cabal +++ b/conjugateGradient.cabal @@ -1,5 +1,5 @@ Name: conjugateGradient -Version: 1.1 +Version: 1.2 Category: Math Synopsis: Sparse matrix linear-equation solver Description: Sparse matrix linear-equation solver, using the conjugate gradient algorithm. Note that the @@ -13,6 +13,8 @@ . 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. Copyright: Levent Erkok, 2013 License: BSD3 License-file: LICENSE