math-functions-0.3.4.2: Collection of tools for numeric computations
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 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 #
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Defined in Numeric.RootFinding

Functor Root Source #
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Applicative Root Source #
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Foldable Root Source #
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Traversable Root Source #
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Alternative Root Source #
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MonadPlus Root Source #
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Eq a => Eq ( Root a) Source #
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Data a => Data ( Root a) Source #
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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 #
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Show a => Show ( Root a) Source #
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Generic ( Root a) Source #
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Associated Types

type Rep ( Root a) :: Type -> Type Source #

NFData a => NFData ( Root a) Source #
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Methods

rnf :: Root a -> () Source #

type Rep ( Root a) Source #
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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))))

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 #
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Data Tolerance Source #
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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 #
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Show Tolerance Source #
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Generic Tolerance Source #
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type Rep Tolerance Source #
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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.

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

Instances

Instances details
Eq RiddersParam Source #
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Defined in Numeric.RootFinding

Data RiddersParam Source #
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Methods

gfoldl :: ( forall d b. Data d => c (d -> b) -> d -> c b) -> ( forall g. g -> c g) -> RiddersParam -> c RiddersParam Source #

gunfold :: ( forall b r. Data b => c (b -> r) -> c r) -> ( forall r. r -> c r) -> Constr -> c RiddersParam Source #

toConstr :: RiddersParam -> Constr Source #

dataTypeOf :: RiddersParam -> DataType Source #

dataCast1 :: Typeable t => ( forall d. Data d => c (t d)) -> Maybe (c RiddersParam ) Source #

dataCast2 :: Typeable t => ( forall d e. ( Data d, Data e) => c (t d e)) -> Maybe (c RiddersParam ) Source #

gmapT :: ( forall b. Data b => b -> b) -> RiddersParam -> RiddersParam Source #

gmapQl :: (r -> r' -> r) -> r -> ( forall d. Data d => d -> r') -> RiddersParam -> r Source #

gmapQr :: forall r r'. (r' -> r -> r) -> r -> ( forall d. Data d => d -> r') -> RiddersParam -> r Source #

gmapQ :: ( forall d. Data d => d -> u) -> RiddersParam -> [u] Source #

gmapQi :: Int -> ( forall d. Data d => d -> u) -> RiddersParam -> u Source #

gmapM :: Monad m => ( forall d. Data d => d -> m d) -> RiddersParam -> m RiddersParam Source #

gmapMp :: MonadPlus m => ( forall d. Data d => d -> m d) -> RiddersParam -> m RiddersParam Source #

gmapMo :: MonadPlus m => ( forall d. Data d => d -> m d) -> RiddersParam -> m RiddersParam Source #

Read RiddersParam Source #
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Defined in Numeric.RootFinding

Show RiddersParam Source #
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Generic RiddersParam Source #
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Default RiddersParam Source #
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type Rep RiddersParam Source #
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type Rep RiddersParam = D1 (' MetaData "RiddersParam" "Numeric.RootFinding" "math-functions-0.3.4.2-8cxCYxAIZeU8rpmBedV7Yk" ' 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 #
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Defined in Numeric.RootFinding

Data RiddersStep Source #
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Methods

gfoldl :: ( forall d b. Data d => c (d -> b) -> d -> c b) -> ( forall g. g -> c g) -> RiddersStep -> c RiddersStep Source #

gunfold :: ( forall b r. Data b => c (b -> r) -> c r) -> ( forall r. r -> c r) -> Constr -> c RiddersStep Source #

toConstr :: RiddersStep -> Constr Source #

dataTypeOf :: RiddersStep -> DataType Source #

dataCast1 :: Typeable t => ( forall d. Data d => c (t d)) -> Maybe (c RiddersStep ) Source #

dataCast2 :: Typeable t => ( forall d e. ( Data d, Data e) => c (t d e)) -> Maybe (c RiddersStep ) Source #

gmapT :: ( forall b. Data b => b -> b) -> RiddersStep -> RiddersStep Source #

gmapQl :: (r -> r' -> r) -> r -> ( forall d. Data d => d -> r') -> RiddersStep -> r Source #

gmapQr :: forall r r'. (r' -> r -> r) -> r -> ( forall d. Data d => d -> r') -> RiddersStep -> r Source #

gmapQ :: ( forall d. Data d => d -> u) -> RiddersStep -> [u] Source #

gmapQi :: Int -> ( forall d. Data d => d -> u) -> RiddersStep -> u Source #

gmapM :: Monad m => ( forall d. Data d => d -> m d) -> RiddersStep -> m RiddersStep Source #

gmapMp :: MonadPlus m => ( forall d. Data d => d -> m d) -> RiddersStep -> m RiddersStep Source #

gmapMo :: MonadPlus m => ( forall d. Data d => d -> m d) -> RiddersStep -> m RiddersStep Source #

Read RiddersStep Source #
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Defined in Numeric.RootFinding

Show RiddersStep Source #
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Generic RiddersStep Source #
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NFData RiddersStep Source #
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IterationStep RiddersStep Source #
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type Rep RiddersStep Source #
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Newton-Raphson algorithm

data NewtonParam Source #

Parameters for ridders root finding

Constructors

NewtonParam

Fields

Instances

Instances details
Eq NewtonParam Source #
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Defined in Numeric.RootFinding

Data NewtonParam Source #
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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 Source #

gunfold :: ( forall b r. Data b => c (b -> r) -> c r) -> ( forall r. r -> c r) -> Constr -> c NewtonParam Source #

toConstr :: NewtonParam -> Constr Source #

dataTypeOf :: NewtonParam -> DataType Source #

dataCast1 :: Typeable t => ( forall d. Data d => c (t d)) -> Maybe (c NewtonParam ) Source #

dataCast2 :: Typeable t => ( forall d e. ( Data d, Data e) => c (t d e)) -> Maybe (c NewtonParam ) Source #

gmapT :: ( forall b. Data b => b -> b) -> NewtonParam -> NewtonParam Source #

gmapQl :: (r -> r' -> r) -> r -> ( forall d. Data d => d -> r') -> NewtonParam -> r Source #

gmapQr :: forall r r'. (r' -> r -> r) -> r -> ( forall d. Data d => d -> r') -> NewtonParam -> r Source #

gmapQ :: ( forall d. Data d => d -> u) -> NewtonParam -> [u] Source #

gmapQi :: Int -> ( forall d. Data d => d -> u) -> NewtonParam -> u Source #

gmapM :: Monad m => ( forall d. Data d => d -> m d) -> NewtonParam -> m NewtonParam Source #

gmapMp :: MonadPlus m => ( forall d. Data d => d -> m d) -> NewtonParam -> m NewtonParam Source #

gmapMo :: MonadPlus m => ( forall d. Data d => d -> m d) -> NewtonParam -> m NewtonParam Source #

Read NewtonParam Source #
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Defined in Numeric.RootFinding

Show NewtonParam Source #
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Generic NewtonParam Source #
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Default NewtonParam Source #
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type Rep NewtonParam Source #
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type Rep NewtonParam = D1 (' MetaData "NewtonParam" "Numeric.RootFinding" "math-functions-0.3.4.2-8cxCYxAIZeU8rpmBedV7Yk" ' 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 #
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Defined in Numeric.RootFinding

Data NewtonStep Source #
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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 Source #

gunfold :: ( forall b r. Data b => c (b -> r) -> c r) -> ( forall r. r -> c r) -> Constr -> c NewtonStep Source #

toConstr :: NewtonStep -> Constr Source #

dataTypeOf :: NewtonStep -> DataType Source #

dataCast1 :: Typeable t => ( forall d. Data d => c (t d)) -> Maybe (c NewtonStep ) Source #

dataCast2 :: Typeable t => ( forall d e. ( Data d, Data e) => c (t d e)) -> Maybe (c NewtonStep ) Source #

gmapT :: ( forall b. Data b => b -> b) -> NewtonStep -> NewtonStep Source #

gmapQl :: (r -> r' -> r) -> r -> ( forall d. Data d => d -> r') -> NewtonStep -> r Source #

gmapQr :: forall r r'. (r' -> r -> r) -> r -> ( forall d. Data d => d -> r') -> NewtonStep -> r Source #

gmapQ :: ( forall d. Data d => d -> u) -> NewtonStep -> [u] Source #

gmapQi :: Int -> ( forall d. Data d => d -> u) -> NewtonStep -> u Source #

gmapM :: Monad m => ( forall d. Data d => d -> m d) -> NewtonStep -> m NewtonStep Source #

gmapMp :: MonadPlus m => ( forall d. Data d => d -> m d) -> NewtonStep -> m NewtonStep Source #

gmapMo :: MonadPlus m => ( forall d. Data d => d -> m d) -> NewtonStep -> m NewtonStep Source #

Read NewtonStep Source #
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Show NewtonStep Source #
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Generic NewtonStep Source #
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NFData NewtonStep Source #
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IterationStep NewtonStep Source #
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type Rep NewtonStep Source #
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Defined in Numeric.RootFinding

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.