| Copyright | (c) 2011 Bryan O'Sullivan 2018 Alexey Khudyakov |
|---|---|
| License | BSD3 |
| Maintainer | bos@serpentine.com |
| Stability | experimental |
| Portability | portable |
| Safe Haskell | None |
| Language | Haskell2010 |
Numeric.RootFinding
Description
Haskell functions for finding the roots of real functions of real arguments. These algorithms are iterative so we provide both function returning root (or failure to find root) and list of iterations.
Synopsis
-
data
Root
a
- = NotBracketed
- | SearchFailed
- | Root !a
- fromRoot :: a -> Root a -> a
- data Tolerance
- withinTolerance :: Tolerance -> Double -> Double -> Bool
- class IterationStep a where
- findRoot :: IterationStep a => Int -> Tolerance -> [a] -> Root Double
-
data
RiddersParam
=
RiddersParam
{
- riddersMaxIter :: ! Int
- riddersTol :: ! Tolerance
- ridders :: RiddersParam -> ( Double , Double ) -> ( Double -> Double ) -> Root Double
- riddersIterations :: ( Double , Double ) -> ( Double -> Double ) -> [ RiddersStep ]
-
data
RiddersStep
- = RiddersStep ! Double ! Double
- | RiddersBisect ! Double ! Double
- | RiddersRoot ! Double
- | RiddersNoBracket
-
data
NewtonParam
=
NewtonParam
{
- newtonMaxIter :: ! Int
- newtonTol :: ! Tolerance
- newtonRaphson :: NewtonParam -> ( Double , Double , Double ) -> ( Double -> ( Double , Double )) -> Root Double
- newtonRaphsonIterations :: ( Double , Double , Double ) -> ( Double -> ( Double , Double )) -> [ NewtonStep ]
-
data
NewtonStep
- = NewtonStep ! Double ! Double
- | NewtonBisection ! Double ! Double
- | NewtonRoot ! Double
- | NewtonNoBracket
Data types
The result of searching for a root of a mathematical function.
Constructors
| NotBracketed |
The function does not have opposite signs when evaluated at the lower and upper bounds of the search. |
| SearchFailed |
The search failed to converge to within the given error tolerance after the given number of iterations. |
| Root !a |
A root was successfully found. |
Instances
| Monad Root Source # | |
| Functor Root Source # | |
| Applicative Root Source # | |
| Foldable Root Source # | |
|
Defined in Numeric.RootFinding Methods fold :: Monoid m => Root m -> m Source # foldMap :: Monoid m => (a -> m) -> Root a -> m Source # foldMap' :: Monoid m => (a -> m) -> Root a -> m Source # foldr :: (a -> b -> b) -> b -> Root a -> b Source # foldr' :: (a -> b -> b) -> b -> Root a -> b Source # foldl :: (b -> a -> b) -> b -> Root a -> b Source # foldl' :: (b -> a -> b) -> b -> Root a -> b Source # foldr1 :: (a -> a -> a) -> Root a -> a Source # foldl1 :: (a -> a -> a) -> Root a -> a Source # toList :: Root a -> [a] Source # null :: Root a -> Bool Source # length :: Root a -> Int Source # elem :: Eq a => a -> Root a -> Bool Source # maximum :: Ord a => Root a -> a Source # minimum :: Ord a => Root a -> a Source # |
|
| Traversable Root Source # | |
| Alternative Root Source # | |
| MonadPlus Root Source # | |
| Eq a => Eq ( Root a) Source # | |
| Data a => Data ( Root a) Source # | |
|
Defined in Numeric.RootFinding Methods gfoldl :: ( forall d b. Data d => c (d -> b) -> d -> c b) -> ( forall g. g -> c g) -> Root a -> c ( Root a) Source # gunfold :: ( forall b r. Data b => c (b -> r) -> c r) -> ( forall r. r -> c r) -> Constr -> c ( Root a) Source # toConstr :: Root a -> Constr Source # dataTypeOf :: Root a -> DataType Source # dataCast1 :: Typeable t => ( forall d. Data d => c (t d)) -> Maybe (c ( Root a)) Source # dataCast2 :: Typeable t => ( forall d e. ( Data d, Data e) => c (t d e)) -> Maybe (c ( Root a)) Source # gmapT :: ( forall b. Data b => b -> b) -> Root a -> Root a Source # gmapQl :: (r -> r' -> r) -> r -> ( forall d. Data d => d -> r') -> Root a -> r Source # gmapQr :: forall r r'. (r' -> r -> r) -> r -> ( forall d. Data d => d -> r') -> Root a -> r Source # gmapQ :: ( forall d. Data d => d -> u) -> Root a -> [u] Source # gmapQi :: Int -> ( forall d. Data d => d -> u) -> Root a -> u Source # gmapM :: Monad m => ( forall d. Data d => d -> m d) -> Root a -> m ( Root a) Source # gmapMp :: MonadPlus m => ( forall d. Data d => d -> m d) -> Root a -> m ( Root a) Source # gmapMo :: MonadPlus m => ( forall d. Data d => d -> m d) -> Root a -> m ( Root a) Source # |
|
| Read a => Read ( Root a) Source # | |
| Show a => Show ( Root a) Source # | |
| Generic ( Root a) Source # | |
| NFData a => NFData ( Root a) Source # | |
|
Defined in Numeric.RootFinding |
|
| type Rep ( Root a) Source # | |
|
Defined in Numeric.RootFinding
type
Rep
(
Root
a) =
D1
('
MetaData
"Root" "Numeric.RootFinding" "math-functions-0.3.4.2-8cxCYxAIZeU8rpmBedV7Yk" '
False
) (
C1
('
MetaCons
"NotBracketed" '
PrefixI
'
False
) (
U1
::
Type
->
Type
)
:+:
(
C1
('
MetaCons
"SearchFailed" '
PrefixI
'
False
) (
U1
::
Type
->
Type
)
:+:
C1
('
MetaCons
"Root" '
PrefixI
'
False
) (
S1
('
MetaSel
('
Nothing
::
Maybe
Symbol
) '
NoSourceUnpackedness
'
SourceStrict
'
DecidedStrict
) (
Rec0
a))))
|
|
Arguments
| :: a |
Default value. |
| -> Root a |
Result of search for a root. |
| -> a |
Returns either the result of a search for a root, or the default value if the search failed.
Error tolerance for finding root. It describes when root finding algorithm should stop trying to improve approximation.
Constructors
| RelTol ! Double |
Relative error tolerance. Given
|
| AbsTol ! Double |
Absolute error tolerance. Given
|
Instances
| Eq Tolerance Source # | |
| Data Tolerance Source # | |
|
Defined in Numeric.RootFinding Methods gfoldl :: ( forall d b. Data d => c (d -> b) -> d -> c b) -> ( forall g. g -> c g) -> Tolerance -> c Tolerance Source # gunfold :: ( forall b r. Data b => c (b -> r) -> c r) -> ( forall r. r -> c r) -> Constr -> c Tolerance Source # toConstr :: Tolerance -> Constr Source # dataTypeOf :: Tolerance -> DataType Source # dataCast1 :: Typeable t => ( forall d. Data d => c (t d)) -> Maybe (c Tolerance ) Source # dataCast2 :: Typeable t => ( forall d e. ( Data d, Data e) => c (t d e)) -> Maybe (c Tolerance ) Source # gmapT :: ( forall b. Data b => b -> b) -> Tolerance -> Tolerance Source # gmapQl :: (r -> r' -> r) -> r -> ( forall d. Data d => d -> r') -> Tolerance -> r Source # gmapQr :: forall r r'. (r' -> r -> r) -> r -> ( forall d. Data d => d -> r') -> Tolerance -> r Source # gmapQ :: ( forall d. Data d => d -> u) -> Tolerance -> [u] Source # gmapQi :: Int -> ( forall d. Data d => d -> u) -> Tolerance -> u Source # gmapM :: Monad m => ( forall d. Data d => d -> m d) -> Tolerance -> m Tolerance Source # gmapMp :: MonadPlus m => ( forall d. Data d => d -> m d) -> Tolerance -> m Tolerance Source # gmapMo :: MonadPlus m => ( forall d. Data d => d -> m d) -> Tolerance -> m Tolerance Source # |
|
| Read Tolerance Source # | |
| Show Tolerance Source # | |
| Generic Tolerance Source # | |
| type Rep Tolerance Source # | |
|
Defined in Numeric.RootFinding
type
Rep
Tolerance
=
D1
('
MetaData
"Tolerance" "Numeric.RootFinding" "math-functions-0.3.4.2-8cxCYxAIZeU8rpmBedV7Yk" '
False
) (
C1
('
MetaCons
"RelTol" '
PrefixI
'
False
) (
S1
('
MetaSel
('
Nothing
::
Maybe
Symbol
) '
NoSourceUnpackedness
'
SourceStrict
'
DecidedStrict
) (
Rec0
Double
))
:+:
C1
('
MetaCons
"AbsTol" '
PrefixI
'
False
) (
S1
('
MetaSel
('
Nothing
::
Maybe
Symbol
) '
NoSourceUnpackedness
'
SourceStrict
'
DecidedStrict
) (
Rec0
Double
)))
|
|
withinTolerance :: Tolerance -> Double -> Double -> Bool Source #
Check that two values are approximately equal. In addition to specification values are considered equal if they're within 1ulp of precision. No further improvement could be done anyway.
class IterationStep a where Source #
Type class for checking whether iteration converged already.
Methods
matchRoot :: Tolerance -> a -> Maybe ( Root Double ) Source #
Return
Just root
is current iteration converged within
required error tolerance. Returns
Nothing
otherwise.
Instances
| IterationStep NewtonStep Source # | |
|
Defined in Numeric.RootFinding |
|
| IterationStep RiddersStep Source # | |
|
Defined in Numeric.RootFinding |
|
Arguments
| :: IterationStep a | |
| => Int |
Maximum |
| -> Tolerance |
Error tolerance |
| -> [a] | |
| -> Root Double |
Find root in lazy list of iterations.
Ridders algorithm
data RiddersParam Source #
Parameters for
ridders
root finding
Constructors
| RiddersParam | |
|
Fields
|
|
Instances
Arguments
| :: RiddersParam |
Parameters for algorithms.
|
| -> ( Double , Double ) |
Bracket for root |
| -> ( Double -> Double ) |
Function to find roots |
| -> Root Double |
Use the method of Ridders[Ridders1979] to compute a root of a function. It doesn't require derivative and provide quadratic convergence (number of significant digits grows quadratically with number of iterations).
The function must have opposite signs when evaluated at the lower and upper bounds of the search (i.e. the root must be bracketed). If there's more that one root in the bracket iteration will converge to some root in the bracket.
riddersIterations :: ( Double , Double ) -> ( Double -> Double ) -> [ RiddersStep ] Source #
List of iterations for Ridders methods. See
ridders
for
documentation of parameters
data RiddersStep Source #
Single Ridders step. It's a bracket of root
Constructors
| RiddersStep ! Double ! Double |
Ridders step. Parameters are bracket for the root |
| RiddersBisect ! Double ! Double |
Bisection step. It's fallback which is taken when Ridders update takes us out of bracket |
| RiddersRoot ! Double |
Root found |
| RiddersNoBracket |
Root is not bracketed |
Instances
Newton-Raphson algorithm
data NewtonParam Source #
Parameters for
ridders
root finding
Constructors
| NewtonParam | |
|
Fields
|
|
Instances
Arguments
| :: NewtonParam |
Parameters for algorithm.
|
| -> ( Double , Double , Double ) |
Triple of
|
| -> ( Double -> ( Double , Double )) |
Function to find root of. It returns pair of function value and its first derivative |
| -> Root Double |
Solve equation using Newton-Raphson iterations.
This method require both initial guess and bounds for root. If Newton step takes us out of bounds on root function reverts to bisection.
newtonRaphsonIterations :: ( Double , Double , Double ) -> ( Double -> ( Double , Double )) -> [ NewtonStep ] Source #
List of iteration for Newton-Raphson algorithm. See documentation
for
newtonRaphson
for meaning of parameters.
data NewtonStep Source #
Steps for Newton iterations
Constructors
| NewtonStep ! Double ! Double |
Normal Newton-Raphson update. Parameters are: old guess, new guess |
| NewtonBisection ! Double ! Double |
Bisection fallback when Newton-Raphson iteration doesn't work. Parameters are bracket on root |
| NewtonRoot ! Double |
Root is found |
| NewtonNoBracket |
Root is not bracketed |
Instances
References
- Ridders, C.F.J. (1979) A new algorithm for computing a single root of a real continuous function. IEEE Transactions on Circuits and Systems 26:979–980.
- Press W.H.; Teukolsky S.A.; Vetterling W.T.; Flannery B.P. (2007). "Section 9.2.1. Ridders' Method". /Numerical Recipes: The Art of Scientific Computing (3rd ed.)./ New York: Cambridge University Press. ISBN 978-0-521-88068-8.