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

Contents

Data types
Ridders algorithm
Newton-Raphson algorithm
References

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
- = RelTol !Double
- | AbsTol !Double
withinTolerance :: Tolerance -> Double -> Double -> Bool
class IterationStep a where
- matchRoot :: Tolerance -> a -> Maybe (Root Double)
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

data Root a Source #

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

Instances details

Monad Root Source #
Instance details Defined in Numeric.RootFinding Methods (>>=) :: Root a -> (a -> Root b) -> Root b # (>>) :: Root a -> Root b -> Root b # return :: a -> Root a #
Functor Root Source #
Instance details Defined in Numeric.RootFinding Methods fmap :: (a -> b) -> Root a -> Root b # (<$) :: a -> Root b -> Root a #
Applicative Root Source #
Instance details Defined in Numeric.RootFinding Methods pure :: a -> Root a # (<>) :: Root (a -> b) -> Root a -> Root b # liftA2 :: (a -> b -> c) -> Root a -> Root b -> Root c # (>) :: Root a -> Root b -> Root b # (<*) :: Root a -> Root b -> Root a #
Foldable Root Source #
Instance details Defined in Numeric.RootFinding Methods fold :: Monoid m => Root m -> m # foldMap :: Monoid m => (a -> m) -> Root a -> m # foldMap' :: Monoid m => (a -> m) -> Root a -> m # foldr :: (a -> b -> b) -> b -> Root a -> b # foldr' :: (a -> b -> b) -> b -> Root a -> b # foldl :: (b -> a -> b) -> b -> Root a -> b # foldl' :: (b -> a -> b) -> b -> Root a -> b # foldr1 :: (a -> a -> a) -> Root a -> a # foldl1 :: (a -> a -> a) -> Root a -> a # toList :: Root a -> [a] # null :: Root a -> Bool # length :: Root a -> Int # elem :: Eq a => a -> Root a -> Bool # maximum :: Ord a => Root a -> a # minimum :: Ord a => Root a -> a # sum :: Num a => Root a -> a # product :: Num a => Root a -> a #
Traversable Root Source #
Instance details Defined in Numeric.RootFinding Methods traverse :: Applicative f => (a -> f b) -> Root a -> f (Root b) # sequenceA :: Applicative f => Root (f a) -> f (Root a) # mapM :: Monad m => (a -> m b) -> Root a -> m (Root b) # sequence :: Monad m => Root (m a) -> m (Root a) #
Alternative Root Source #
Instance details Defined in Numeric.RootFinding Methods empty :: Root a # (<\|>) :: Root a -> Root a -> Root a # some :: Root a -> Root [a] # many :: Root a -> Root [a] #
MonadPlus Root Source #
Instance details Defined in Numeric.RootFinding Methods mzero :: Root a # mplus :: Root a -> Root a -> Root a #
Eq a => Eq (Root a) Source #
Instance details Defined in Numeric.RootFinding Methods (==) :: Root a -> Root a -> Bool # (/=) :: Root a -> Root a -> Bool #
Data a => Data (Root a) Source #
Instance details 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) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Root a) # toConstr :: Root a -> Constr # dataTypeOf :: Root a -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Root a)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Root a)) # gmapT :: (forall b. Data b => b -> b) -> Root a -> Root a # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Root a -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Root a -> r # gmapQ :: (forall d. Data d => d -> u) -> Root a -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Root a -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Root a -> m (Root a) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Root a -> m (Root a) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Root a -> m (Root a) #
Read a => Read (Root a) Source #
Instance details Defined in Numeric.RootFinding Methods readsPrec :: Int -> ReadS (Root a) # readList :: ReadS [Root a] # readPrec :: ReadPrec (Root a) # readListPrec :: ReadPrec [Root a] #
Show a => Show (Root a) Source #
Instance details Defined in Numeric.RootFinding Methods showsPrec :: Int -> Root a -> ShowS # show :: Root a -> String # showList :: [Root a] -> ShowS #
Generic (Root a) Source #
Instance details Defined in Numeric.RootFinding Associated Types type Rep (Root a) :: Type -> Type # Methods from :: Root a -> Rep (Root a) x # to :: Rep (Root a) x -> Root a #
NFData a => NFData (Root a) Source #
Instance details Defined in Numeric.RootFinding Methods rnf :: Root a -> () #
type Rep (Root a) Source #
Instance details Defined in Numeric.RootFinding type Rep (Root a) = D1 ('MetaData "Root" "Numeric.RootFinding" "math-functions-0.3.4.2-58VLxRxhvDgL5rM8ecKJZI" '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))))

fromRoot Source #

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.

data Tolerance Source #

Error tolerance for finding root. It describes when root finding algorithm should stop trying to improve approximation.

Constructors

RelTol !Double	Relative error tolerance. Given `RelTol ε` two values are considered approximately equal if \[ \frac{\|a - b\|}{\|\operatorname{max}(a,b)} < \varepsilon \]
AbsTol !Double	Absolute error tolerance. Given `AbsTol δ` two values are considered approximately equal if \[ \|a - b\| < \delta \]. Note that `AbsTol 0` could be used to require to find approximation within machine precision.

Instances

Instances details

Eq Tolerance Source #
Instance details Defined in Numeric.RootFinding Methods (==) :: Tolerance -> Tolerance -> Bool # (/=) :: Tolerance -> Tolerance -> Bool #
Data Tolerance Source #
Instance details 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 # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c Tolerance # toConstr :: Tolerance -> Constr # dataTypeOf :: Tolerance -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c Tolerance) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Tolerance) # gmapT :: (forall b. Data b => b -> b) -> Tolerance -> Tolerance # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Tolerance -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Tolerance -> r # gmapQ :: (forall d. Data d => d -> u) -> Tolerance -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Tolerance -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Tolerance -> m Tolerance # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Tolerance -> m Tolerance # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Tolerance -> m Tolerance #
Read Tolerance Source #
Instance details Defined in Numeric.RootFinding Methods readsPrec :: Int -> ReadS Tolerance # readList :: ReadS [Tolerance] # readPrec :: ReadPrec Tolerance # readListPrec :: ReadPrec [Tolerance] #
Show Tolerance Source #
Instance details Defined in Numeric.RootFinding Methods showsPrec :: Int -> Tolerance -> ShowS # show :: Tolerance -> String # showList :: [Tolerance] -> ShowS #
Generic Tolerance Source #
Instance details Defined in Numeric.RootFinding Associated Types type Rep Tolerance :: Type -> Type # Methods from :: Tolerance -> Rep Tolerance x # to :: Rep Tolerance x -> Tolerance #
type Rep Tolerance Source #
Instance details Defined in Numeric.RootFinding type Rep Tolerance = D1 ('MetaData "Tolerance" "Numeric.RootFinding" "math-functions-0.3.4.2-58VLxRxhvDgL5rM8ecKJZI" '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

Instances details

IterationStep NewtonStep Source #
Instance details Defined in Numeric.RootFinding Methods matchRoot :: Tolerance -> NewtonStep -> Maybe (Root Double) Source #
IterationStep RiddersStep Source #
Instance details Defined in Numeric.RootFinding Methods matchRoot :: Tolerance -> RiddersStep -> Maybe (Root Double) Source #

findRoot Source #

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 riddersMaxIter :: !Int Maximum number of iterations. Default = 100 riddersTol :: !Tolerance Error tolerance for root approximation. Default is relative error 4·ε, where ε is machine precision.

Instances

Instances details

Eq RiddersParam Source #
Instance details Defined in Numeric.RootFinding Methods (==) :: RiddersParam -> RiddersParam -> Bool # (/=) :: RiddersParam -> RiddersParam -> Bool #
Data RiddersParam Source #
Instance details Defined in Numeric.RootFinding Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> RiddersParam -> c RiddersParam # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c RiddersParam # toConstr :: RiddersParam -> Constr # dataTypeOf :: RiddersParam -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c RiddersParam) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c RiddersParam) # gmapT :: (forall b. Data b => b -> b) -> RiddersParam -> RiddersParam # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> RiddersParam -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> RiddersParam -> r # gmapQ :: (forall d. Data d => d -> u) -> RiddersParam -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> RiddersParam -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> RiddersParam -> m RiddersParam # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> RiddersParam -> m RiddersParam # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> RiddersParam -> m RiddersParam #
Read RiddersParam Source #
Instance details Defined in Numeric.RootFinding Methods readsPrec :: Int -> ReadS RiddersParam # readList :: ReadS [RiddersParam] # readPrec :: ReadPrec RiddersParam # readListPrec :: ReadPrec [RiddersParam] #
Show RiddersParam Source #
Instance details Defined in Numeric.RootFinding Methods showsPrec :: Int -> RiddersParam -> ShowS # show :: RiddersParam -> String # showList :: [RiddersParam] -> ShowS #
Generic RiddersParam Source #
Instance details Defined in Numeric.RootFinding Associated Types type Rep RiddersParam :: Type -> Type # Methods from :: RiddersParam -> Rep RiddersParam x # to :: Rep RiddersParam x -> RiddersParam #
Default RiddersParam Source #
Instance details Defined in Numeric.RootFinding Methods def :: RiddersParam #
type Rep RiddersParam Source #
Instance details Defined in Numeric.RootFinding type Rep RiddersParam = D1 ('MetaData "RiddersParam" "Numeric.RootFinding" "math-functions-0.3.4.2-58VLxRxhvDgL5rM8ecKJZI" 'False) (C1 ('MetaCons "RiddersParam" 'PrefixI 'True) (S1 ('MetaSel ('Just "riddersMaxIter") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Int) :*: S1 ('MetaSel ('Just "riddersTol") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Tolerance)))

ridders Source #

Arguments

:: RiddersParam	Parameters for algorithms. `def` provides reasonable defaults
-> (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

Instances details

Eq RiddersStep Source #
Instance details Defined in Numeric.RootFinding Methods (==) :: RiddersStep -> RiddersStep -> Bool # (/=) :: RiddersStep -> RiddersStep -> Bool #
Data RiddersStep Source #
Instance details Defined in Numeric.RootFinding Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> RiddersStep -> c RiddersStep # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c RiddersStep # toConstr :: RiddersStep -> Constr # dataTypeOf :: RiddersStep -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c RiddersStep) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c RiddersStep) # gmapT :: (forall b. Data b => b -> b) -> RiddersStep -> RiddersStep # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> RiddersStep -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> RiddersStep -> r # gmapQ :: (forall d. Data d => d -> u) -> RiddersStep -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> RiddersStep -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> RiddersStep -> m RiddersStep # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> RiddersStep -> m RiddersStep # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> RiddersStep -> m RiddersStep #
Read RiddersStep Source #
Instance details Defined in Numeric.RootFinding Methods readsPrec :: Int -> ReadS RiddersStep # readList :: ReadS [RiddersStep] # readPrec :: ReadPrec RiddersStep # readListPrec :: ReadPrec [RiddersStep] #
Show RiddersStep Source #
Instance details Defined in Numeric.RootFinding Methods showsPrec :: Int -> RiddersStep -> ShowS # show :: RiddersStep -> String # showList :: [RiddersStep] -> ShowS #
Generic RiddersStep Source #
Instance details Defined in Numeric.RootFinding Associated Types type Rep RiddersStep :: Type -> Type # Methods from :: RiddersStep -> Rep RiddersStep x # to :: Rep RiddersStep x -> RiddersStep #
NFData RiddersStep Source #
Instance details Defined in Numeric.RootFinding Methods rnf :: RiddersStep -> () #
IterationStep RiddersStep Source #
Instance details Defined in Numeric.RootFinding Methods matchRoot :: Tolerance -> RiddersStep -> Maybe (Root Double) Source #
type Rep RiddersStep Source #
Instance details Defined in Numeric.RootFinding type Rep RiddersStep = D1 ('MetaData "RiddersStep" "Numeric.RootFinding" "math-functions-0.3.4.2-58VLxRxhvDgL5rM8ecKJZI" 'False) ((C1 ('MetaCons "RiddersStep" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)) :+: C1 ('MetaCons "RiddersBisect" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double))) :+: (C1 ('MetaCons "RiddersRoot" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)) :+: C1 ('MetaCons "RiddersNoBracket" 'PrefixI 'False) (U1 :: Type -> Type)))

Newton-Raphson algorithm

data NewtonParam Source #

Parameters for ridders root finding

Constructors

NewtonParam
Fields newtonMaxIter :: !Int Maximum number of iterations. Default = 50 newtonTol :: !Tolerance Error tolerance for root approximation. Default is relative error 4·ε, where ε is machine precision

Instances

Instances details

Eq NewtonParam Source #
Instance details Defined in Numeric.RootFinding Methods (==) :: NewtonParam -> NewtonParam -> Bool # (/=) :: NewtonParam -> NewtonParam -> Bool #
Data NewtonParam Source #
Instance details Defined in Numeric.RootFinding Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> NewtonParam -> c NewtonParam # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c NewtonParam # toConstr :: NewtonParam -> Constr # dataTypeOf :: NewtonParam -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c NewtonParam) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c NewtonParam) # gmapT :: (forall b. Data b => b -> b) -> NewtonParam -> NewtonParam # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> NewtonParam -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> NewtonParam -> r # gmapQ :: (forall d. Data d => d -> u) -> NewtonParam -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> NewtonParam -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> NewtonParam -> m NewtonParam # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> NewtonParam -> m NewtonParam # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> NewtonParam -> m NewtonParam #
Read NewtonParam Source #
Instance details Defined in Numeric.RootFinding Methods readsPrec :: Int -> ReadS NewtonParam # readList :: ReadS [NewtonParam] # readPrec :: ReadPrec NewtonParam # readListPrec :: ReadPrec [NewtonParam] #
Show NewtonParam Source #
Instance details Defined in Numeric.RootFinding Methods showsPrec :: Int -> NewtonParam -> ShowS # show :: NewtonParam -> String # showList :: [NewtonParam] -> ShowS #
Generic NewtonParam Source #
Instance details Defined in Numeric.RootFinding Associated Types type Rep NewtonParam :: Type -> Type # Methods from :: NewtonParam -> Rep NewtonParam x # to :: Rep NewtonParam x -> NewtonParam #
Default NewtonParam Source #
Instance details Defined in Numeric.RootFinding Methods def :: NewtonParam #
type Rep NewtonParam Source #
Instance details Defined in Numeric.RootFinding type Rep NewtonParam = D1 ('MetaData "NewtonParam" "Numeric.RootFinding" "math-functions-0.3.4.2-58VLxRxhvDgL5rM8ecKJZI" 'False) (C1 ('MetaCons "NewtonParam" 'PrefixI 'True) (S1 ('MetaSel ('Just "newtonMaxIter") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Int) :*: S1 ('MetaSel ('Just "newtonTol") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Tolerance)))

newtonRaphson Source #

Arguments

:: NewtonParam	Parameters for algorithm. `def` provide reasonable defaults.
-> (Double, Double, Double)	Triple of `(low bound, initial guess, upper bound)`. If initial guess if out of bracket middle of bracket is taken as approximation
-> (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

Instances details

Eq NewtonStep Source #
Instance details Defined in Numeric.RootFinding Methods (==) :: NewtonStep -> NewtonStep -> Bool # (/=) :: NewtonStep -> NewtonStep -> Bool #
Data NewtonStep Source #
Instance details Defined in Numeric.RootFinding Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> NewtonStep -> c NewtonStep # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c NewtonStep # toConstr :: NewtonStep -> Constr # dataTypeOf :: NewtonStep -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c NewtonStep) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c NewtonStep) # gmapT :: (forall b. Data b => b -> b) -> NewtonStep -> NewtonStep # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> NewtonStep -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> NewtonStep -> r # gmapQ :: (forall d. Data d => d -> u) -> NewtonStep -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> NewtonStep -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> NewtonStep -> m NewtonStep # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> NewtonStep -> m NewtonStep # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> NewtonStep -> m NewtonStep #
Read NewtonStep Source #
Instance details Defined in Numeric.RootFinding Methods readsPrec :: Int -> ReadS NewtonStep # readList :: ReadS [NewtonStep] # readPrec :: ReadPrec NewtonStep # readListPrec :: ReadPrec [NewtonStep] #
Show NewtonStep Source #
Instance details Defined in Numeric.RootFinding Methods showsPrec :: Int -> NewtonStep -> ShowS # show :: NewtonStep -> String # showList :: [NewtonStep] -> ShowS #
Generic NewtonStep Source #
Instance details Defined in Numeric.RootFinding Associated Types type Rep NewtonStep :: Type -> Type # Methods from :: NewtonStep -> Rep NewtonStep x # to :: Rep NewtonStep x -> NewtonStep #
NFData NewtonStep Source #
Instance details Defined in Numeric.RootFinding Methods rnf :: NewtonStep -> () #
IterationStep NewtonStep Source #
Instance details Defined in Numeric.RootFinding Methods matchRoot :: Tolerance -> NewtonStep -> Maybe (Root Double) Source #
type Rep NewtonStep Source #
Instance details Defined in Numeric.RootFinding type Rep NewtonStep = D1 ('MetaData "NewtonStep" "Numeric.RootFinding" "math-functions-0.3.4.2-58VLxRxhvDgL5rM8ecKJZI" 'False) ((C1 ('MetaCons "NewtonStep" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)) :+: C1 ('MetaCons "NewtonBisection" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double))) :+: (C1 ('MetaCons "NewtonRoot" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)) :+: C1 ('MetaCons "NewtonNoBracket" 'PrefixI 'False) (U1 :: Type -> Type)))

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.