-- 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)
+ => Mat (S m) (S n) a
+ -> (Mat (S m) (S m) a, Mat (S m) (S n) a)
qr matrix =
ifoldl col_function initial_qr columns
where
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 _ -- 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 :: Mat (S m) (S m) a
rotator = givens_rotator col_idx idx (r !!! (col_idx, col_idx)) y
-- Examples:
--
-- >>> import Linear.Matrix ( Col2, Col3, Mat2, Mat3 )
--- >>> import Linear.Matrix ( column', frobenius_norm, fromList )
+-- >>> import Linear.Matrix ( column, frobenius_norm, fromList )
-- >>> import Linear.Matrix ( identity_matrix, vec3d )
-- >>> import Normed ( Normed(..) )
--
-- >>> 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
+-- >>> frobenius_norm ((column vecs 0) - v0) < 1e-12
-- True
--- >>> frobenius_norm ((column' vecs 1) - v1) < 1e-12
+-- >>> frobenius_norm ((column vecs 1) - v1) < 1e-12
-- True
--- >>> frobenius_norm ((column' vecs 2) - v2) < 1e-12
+-- >>> frobenius_norm ((column vecs 2) - v2) < 1e-12
-- True
--
eigenvectors_symmetric :: forall m a. (Arity m, Algebraic.C a, Eq a)