-- | QR factorization via Givens rotations.
--
module Linear.QR (
+ eigenvalues,
+ eigenvectors_symmetric,
givens_rotator,
qr )
where
import qualified Algebra.Ring as Ring ( C )
import qualified Algebra.Algebraic as Algebraic ( C )
-import Data.Vector.Fixed ( ifoldl )
+import Control.Arrow ( first )
+import Data.Vector.Fixed ( S, ifoldl )
import Data.Vector.Fixed.Cont ( Arity )
import NumericPrelude hiding ( (*) )
import Linear.Matrix (
+ Col,
Mat(..),
(*),
(!!!),
construct,
+ diagonal,
identity_matrix,
+ symmetric,
transpose )
--
-- Examples (Watkins, p. 193):
--
--- >>> import Linear.Matrix ( Mat2, fromList )
+-- >>> import Normed ( Normed(..) )
+-- >>> import Linear.Vector ( Vec2, Vec3 )
+-- >>> import Linear.Matrix ( Mat2, Mat3, fromList, frobenius_norm )
+-- >>> import qualified Data.Vector.Fixed as V ( map )
+--
-- >>> let m = givens_rotator 0 1 1 1 :: Mat2 Double
-- >>> let m2 = fromList [[1, -1],[1, 1]] :: Mat2 Double
-- >>> m == (1 / (sqrt 2) :: Double) *> m2
-- True
--
-givens_rotator :: forall m a. (Ring.C a, Algebraic.C a, Arity m)
+-- >>> let m = fromList [[2,3],[5,7]] :: Mat2 Double
+-- >>> let rot = givens_rotator 0 1 2.0 5.0 :: Mat2 Double
+-- >>> ((transpose rot) * m) !!! (1,0) < 1e-12
+-- True
+-- >>> let (Mat rows) = rot
+-- >>> let (Mat cols) = transpose rot
+-- >>> V.map norm rows :: Vec2 Double
+-- fromList [1.0,1.0]
+-- >>> V.map norm cols :: Vec2 Double
+-- fromList [1.0,1.0]
+--
+-- >>> let m = fromList [[12,-51,4],[6,167,-68],[-4,24,-41]] :: Mat3 Double
+-- >>> let rot = givens_rotator 1 2 6 (-4) :: Mat3 Double
+-- >>> let ex_rot_r1 = [1,0,0] :: [Double]
+-- >>> let ex_rot_r2 = [0,0.83205,-0.55470] :: [Double]
+-- >>> let ex_rot_r3 = [0, 0.55470, 0.83205] :: [Double]
+-- >>> let ex_rot = fromList [ex_rot_r1, ex_rot_r2, ex_rot_r3] :: Mat3 Double
+-- >>> frobenius_norm ((transpose rot) - ex_rot) < 1e-4
+-- True
+-- >>> ((transpose rot) * m) !!! (2,0) == 0
+-- True
+-- >>> let (Mat rows) = rot
+-- >>> let (Mat cols) = transpose rot
+-- >>> V.map norm rows :: Vec3 Double
+-- fromList [1.0,1.0,1.0]
+-- >>> V.map norm cols :: Vec3 Double
+-- fromList [1.0,1.0,1.0]
+--
+givens_rotator :: forall m a. (Eq a, Ring.C a, Algebraic.C a, Arity m)
=> Int -> Int -> a -> a -> Mat m m a
givens_rotator i j xi xj =
construct f
where
xnorm = sqrt $ xi^2 + xj^2
- c = xi / xnorm
- s = xj / xnorm
+ c = if xnorm == (fromInteger 0) then (fromInteger 1) else xi / xnorm
+ s = if xnorm == (fromInteger 0) then (fromInteger 0) else xj / xnorm
f :: Int -> Int -> a
f y z
-- factorization. We keep the pair updated by multiplying @q@ and
-- @r@ by the new rotator (or its transpose).
--
-qr :: forall m n a. (Arity m, Arity n, Algebraic.C a, Ring.C a)
+-- We do not require that the diagonal elements of R are positive,
+-- so our factorization is a little less unique than usual.
+--
+-- Examples:
+--
+-- >>> import Linear.Matrix
+--
+-- >>> let m = fromList [[1,2],[1,3]] :: Mat2 Double
+-- >>> let (q,r) = qr m
+-- >>> let c = (1 / (sqrt 2 :: Double))
+-- >>> let ex_q = c *> (fromList [[1,-1],[1,1]] :: Mat2 Double)
+-- >>> let ex_r = c *> (fromList [[2,5],[0,1]] :: Mat2 Double)
+-- >>> frobenius_norm (q - ex_q) == 0
+-- True
+-- >>> frobenius_norm (r - ex_r) == 0
+-- True
+-- >>> let m' = q*r
+-- >>> frobenius_norm (m - m') < 1e-10
+-- True
+-- >>> is_upper_triangular' 1e-10 r
+-- True
+--
+-- >>> let m = fromList [[2,3],[5,7]] :: Mat2 Double
+-- >>> let (q,r) = qr m
+-- >>> frobenius_norm (m - (q*r)) < 1e-12
+-- True
+-- >>> is_upper_triangular' 1e-10 r
+-- True
+--
+-- >>> let m = fromList [[12,-51,4],[6,167,-68],[-4,24,-41]] :: Mat3 Double
+-- >>> let (q,r) = qr m
+-- >>> frobenius_norm (m - (q*r)) < 1e-12
+-- True
+-- >>> is_upper_triangular' 1e-10 r
+-- True
+--
+qr :: forall m n a. (Arity m, Arity n, Eq a, Algebraic.C a, Ring.C a)
=> Mat m n a -> (Mat m m a, Mat m n a)
qr matrix =
ifoldl col_function initial_qr columns
ifoldl (f col_idx) (q,r) col
-- | Process the entries in a column, doing basically the same
- -- thing as col_dunction does. It updates the QR factorization,
+ -- thing as col_function does. It updates the QR factorization,
-- maybe, and returns the current one.
- f col_idx (q,r) idx x
- | idx <= col_idx = (q,r) -- leave it alone.
- | otherwise =
- (q*rotator, (transpose rotator)*r)
+ f col_idx (q,r) idx _ -- ignore the current element
+ | idx <= col_idx = (q,r) -- leave it alone
+ | otherwise = (q*rotator, (transpose rotator)*r)
where
+ y = r !!! (idx, col_idx)
rotator :: Mat m m a
- rotator = givens_rotator col_idx idx (r !!! (idx, col_idx)) x
+ rotator = givens_rotator col_idx idx (r !!! (col_idx, col_idx)) y
+
+
+
+-- | Determine the eigenvalues of the given @matrix@ using the
+-- iterated QR algorithm (see Golub and Van Loan, \"Matrix
+-- Computations\").
+--
+-- Warning: this may not converge if there are repeated eigenvalues
+-- (in magnitude).
+--
+-- Examples:
+--
+-- >>> import Linear.Matrix ( Col2, Col3, Mat2, Mat3 )
+-- >>> import Linear.Matrix ( frobenius_norm, fromList, identity_matrix )
+--
+-- >>> let m = fromList [[1,1],[-2,4]] :: Mat2 Double
+-- >>> let actual = eigenvalues 1000 m
+-- >>> let expected = fromList [[3],[2]] :: Col2 Double
+-- >>> frobenius_norm (actual - expected) < 1e-12
+-- True
+--
+-- >>> let m = identity_matrix :: Mat2 Double
+-- >>> let actual = eigenvalues 10 m
+-- >>> let expected = fromList [[1],[1]] :: Col2 Double
+-- >>> frobenius_norm (actual - expected) < 1e-12
+-- True
+--
+-- >>> let m = fromList [[0,1,0],[0,0,1],[1,-3,3]] :: Mat3 Double
+-- >>> let actual = eigenvalues 1000 m
+-- >>> let expected = fromList [[1],[1],[1]] :: Col3 Double
+-- >>> frobenius_norm (actual - expected) < 1e-2
+-- True
+--
+eigenvalues :: forall m a. (Arity m, Algebraic.C a, Eq a)
+ => Int
+ -> Mat (S m) (S m) a
+ -> Col (S m) a
+eigenvalues iterations matrix
+ | iterations < 0 = error "negative iterations requested"
+ | iterations == 0 = diagonal matrix
+ | otherwise =
+ diagonal (ut_approximation (iterations - 1))
+ where
+ ut_approximation :: Int -> Mat (S m) (S m) a
+ ut_approximation 0 = matrix
+ ut_approximation k = ut_next
+ where
+ ut_prev = ut_approximation (k-1)
+ (qk,rk) = qr ut_prev
+ ut_next = rk*qk
+
+
+
+-- | Compute the eigenvalues and eigenvectors of a symmetric matrix
+-- using an iterative QR algorithm. This is similar to what we do in
+-- 'eigenvalues' except we also return the product of all \"Q\"
+-- matrices that we have generated. This turns out to me the matrix
+-- of eigenvectors when the original matrix is symmetric. For
+-- references see Goluv and Van Loan, \"Matrix Computations\", or
+-- \"Calculation of Gauss Quadrature Rules\" by Golub and Welsch.
+--
+-- Warning: this may not converge if there are repeated eigenvalues
+-- (in magnitude).
+--
+-- Examples:
+--
+-- >>> import Linear.Matrix ( Col2, Col3, Mat2, Mat3 )
+-- >>> import Linear.Matrix ( column, frobenius_norm, fromList )
+-- >>> import Linear.Matrix ( identity_matrix, vec3d )
+-- >>> import Normed ( Normed(..) )
+--
+-- >>> let m = identity_matrix :: Mat3 Double
+-- >>> let (vals, vecs) = eigenvectors_symmetric 100 m
+-- >>> let expected_vals = fromList [[1],[1],[1]] :: Col3 Double
+-- >>> let expected_vecs = m
+-- >>> vals == expected_vals
+-- True
+-- >>> vecs == expected_vecs
+-- True
+--
+-- >>> let m = fromList [[3,2,4],[2,0,2],[4,2,3]] :: Mat3 Double
+-- >>> let (vals, vecs) = eigenvectors_symmetric 1000 m
+-- >>> let expected_vals = fromList [[8],[-1],[-1]] :: Col3 Double
+-- >>> let v0' = vec3d (2, 1, 2) :: Col3 Double
+-- >>> let v0 = (1 / (norm v0') :: Double) *> v0'
+-- >>> let v1' = vec3d (-1, 2, 0) :: Col3 Double
+-- >>> let v1 = (1 / (norm v1') :: Double) *> v1'
+-- >>> let v2' = vec3d (-4, -2, 5) :: Col3 Double
+-- >>> let v2 = (1 / (norm v2') :: Double) *> v2'
+-- >>> frobenius_norm ((column vecs 0) - v0) < 1e-12
+-- True
+-- >>> frobenius_norm ((column vecs 1) - v1) < 1e-12
+-- True
+-- >>> frobenius_norm ((column vecs 2) - v2) < 1e-12
+-- True
+--
+eigenvectors_symmetric :: forall m a. (Arity m, Algebraic.C a, Eq a)
+ => Int
+ -> Mat (S m) (S m) a
+ -> (Col (S m) a, Mat (S m) (S m) a)
+eigenvectors_symmetric iterations matrix
+ | iterations < 0 = error "negative iterations requested"
+ | iterations == 0 = (diagonal matrix, identity_matrix)
+ | not $ symmetric matrix = error "argument is not symmetric"
+ | otherwise =
+ (values, vectors)
+ where
+ -- | We think of \"T\" as an approximation to an
+ -- upper-triangular matrix from which we get our
+ -- eigenvalues. The matrix \"P\" is the product of all
+ -- previous \"Q\"s and its columns approximate the
+ -- eigenvectors.
+ tp_pair :: Int -> (Mat (S m) (S m) a, Mat (S m) (S m) a)
+ tp_pair 0 = (matrix, identity_matrix)
+ tp_pair k = (tk,pk)
+ where
+ (t_prev, p_prev) = tp_pair (k-1)
+ (qk,rk) = qr t_prev
+ pk = p_prev*qk
+ tk = rk*qk
+
+ (values, vectors) = (first diagonal) (tp_pair iterations)
projects / numerical-analysis.git / blobdiff