-BIN = dist/build/numerical-analysis/numerical-analysis
+# There's only one '$' in the awk script, but we have to double-money
+# it for make.
+PN = $(shell grep 'name:' *.cabal | awk '{ print $$2 }')
+BIN = dist/build/$(PN)/$(PN)
DOCTESTS_BIN = dist/build/doctests/doctests
+SRCS = $(shell find src/ -name '*.hs')
.PHONY : test publish_doc doc dist hlint
-$(BIN): src/*.hs
+$(BIN): $(SRCS)
runghc Setup.hs clean
runghc Setup.hs configure --user --flags=${FLAGS}
runghc Setup.hs build
-$(DOCTESTS_BIN): src/*.hs test/Doctests.hs
+$(DOCTESTS_BIN): $(SRCS) test/Doctests.hs
runghc Setup.hs configure --user --flags=${FLAGS} --enable-tests
runghc Setup.hs build
-profile: src/*.hs
+profile: $(SRCS)
runghc Setup.hs configure --user --enable-executable-profiling
runghc Setup.hs build
-hpc: src/*.hs
+hpc: $(SRCS)
FLAGS="hpc" make
clean:
runghc Setup.hs configure
runghc Setup.hs sdist
-# Neither 'haddock' nor 'hscolour' seem to work properly.
-doc: src/*.hs
+
+doc: $(SRCS)
runghc Setup.hs configure --user --flags=${FLAGS}
runghc Setup.hs hscolour --executables
runghc Setup.hs haddock --internal \
--- /dev/null
+-- | The Roots.Fast module contains faster implementations of the
+-- 'Roots.Simple' algorithms. Generally, we will pass precomputed
+-- values to the next iteration of a function rather than passing
+-- the function and the points at which to (re)evaluate it.
+
+module Roots.Fast
+where
+
+has_root :: (Fractional a, Ord a, Ord b, Num b)
+ => (a -> b) -- ^ The function @f@
+ -> a -- ^ The \"left\" endpoint, @a@
+ -> a -- ^ The \"right\" endpoint, @b@
+ -> Maybe a -- ^ The size of the smallest subinterval
+ -- we'll examine, @epsilon@
+ -> Maybe b -- ^ Precoumpted f(a)
+ -> Maybe b -- ^ Precoumpted f(b)
+ -> Bool
+has_root f a b epsilon f_of_a f_of_b =
+ if not ((signum (f_of_a')) * (signum (f_of_b')) == 1) then
+ -- We don't care about epsilon here, there's definitely a root!
+ True
+ else
+ if (b - a) <= epsilon' then
+ -- Give up, return false.
+ False
+ else
+ -- If either [a,c] or [c,b] have roots, we do too.
+ (has_root f a c (Just epsilon') (Just f_of_a') Nothing) ||
+ (has_root f c b (Just epsilon') Nothing (Just f_of_b'))
+ where
+ -- If the size of the smallest subinterval is not specified,
+ -- assume we just want to check once on all of [a,b].
+ epsilon' = case epsilon of
+ Nothing -> (b-a)
+ Just eps -> eps
+
+ -- Compute f(a) and f(b) only if needed.
+ f_of_a' = case f_of_a of
+ Nothing -> f a
+ Just v -> v
+
+ f_of_b' = case f_of_b of
+ Nothing -> f b
+ Just v -> v
+
+ c = (a + b)/2
+
+
+
+bisect :: (Fractional a, Ord a, Num b, Ord b)
+ => (a -> b) -- ^ The function @f@ whose root we seek
+ -> a -- ^ The \"left\" endpoint of the interval, @a@
+ -> a -- ^ The \"right\" endpoint of the interval, @b@
+ -> a -- ^ The tolerance, @epsilon@
+ -> Maybe b -- ^ Precomputed f(a)
+ -> Maybe b -- ^ Precomputed f(b)
+ -> Maybe a
+bisect f a b epsilon f_of_a f_of_b
+ -- We pass @epsilon@ to the 'has_root' function because if we want a
+ -- result within epsilon of the true root, we need to know that
+ -- there *is* a root within an interval of length epsilon.
+ | not (has_root f a b (Just epsilon) (Just f_of_a') (Just f_of_b')) = Nothing
+ | f_of_a' == 0 = Just a
+ | f_of_b' == 0 = Just b
+ | (b - c) < epsilon = Just c
+ | otherwise =
+ -- Use a 'prime' just for consistency.
+ let f_of_c' = f c in
+ if (has_root f a c (Just epsilon) (Just f_of_a') (Just f_of_c'))
+ then bisect f a c epsilon (Just f_of_a') (Just f_of_c')
+ else bisect f c b epsilon (Just f_of_c') (Just f_of_b')
+ where
+ -- Compute f(a) and f(b) only if needed.
+ f_of_a' = case f_of_a of
+ Nothing -> f a
+ Just v -> v
+
+ f_of_b' = case f_of_b of
+ Nothing -> f b
+ Just v -> v
+
+ c = (a + b) / 2
--- | The Roots module contains root-finding algorithms. That is,
--- procedures to (numerically) find solutions to the equation,
+-- | The Roots.Simple module contains root-finding algorithms. That
+-- is, procedures to (numerically) find solutions to the equation,
--
-- > f(x) = 0
--
-- where f is assumed to be continuous on the interval of interest.
--
-module Roots
+module Roots.Simple
where
+import qualified Roots.Fast as F
+
-- | Does the (continuous) function @f@ have a root on the interval
-- [a,b]? If f(a) <] 0 and f(b) ]> 0, we know that there's a root in
-- we'll examine, @epsilon@
-> Bool
has_root f a b epsilon =
- if not ((signum (f a)) * (signum (f b)) == 1) then
- -- We don't care about epsilon here, there's definitely a root!
- True
- else
- if (b - a) <= epsilon' then
- -- Give up, return false.
- False
- else
- -- If either [a,c] or [c,b] have roots, we do too.
- (has_root f a c (Just epsilon')) || (has_root f c b (Just epsilon'))
- where
- -- If the size of the smallest subinterval is not specified,
- -- assume we just want to check once on all of [a,b].
- epsilon' = case epsilon of
- Nothing -> (b-a)
- Just eps -> eps
- c = (a + b)/2
+ F.has_root f a b epsilon Nothing Nothing
+
-> a -- ^ The \"right\" endpoint of the interval, @b@
-> a -- ^ The tolerance, @epsilon@
-> Maybe a
-bisect f a b epsilon
- -- We pass @epsilon@ to the 'has_root' function because if we want a
- -- result within epsilon of the true root, we need to know that
- -- there *is* a root within an interval of length epsilon.
- | not (has_root f a b (Just epsilon)) = Nothing
- | f a == 0 = Just a
- | f b == 0 = Just b
- | (b - c) < epsilon = Just c
- | otherwise =
- if (has_root f a c (Just epsilon)) then bisect f a c epsilon
- else bisect f c b epsilon
- where
- c = (a + b) / 2
+bisect f a b epsilon =
+ F.bisect f a b epsilon Nothing Nothing
projects / numerical-analysis.git / commitdiff