Math.Root.Finder

Documentation

class RootFinder r a b where

General interface for numerical root finders.

Methods

initRootFinder :: (a -> b) -> a -> a -> r a b

initRootFinder f x0 x1: Initialize a root finder for the given function with the initial bracketing interval (x0,x1).

stepRootFinder :: (a -> b) -> r a b -> r a b

Step a root finder for the given function (which should generally be the same one passed to initRootFinder), refining the finder's estimate of the location of a root.

estimateRoot :: r a b -> a

Extract the finder's current estimate of the position of a root.

estimateError :: r a b -> a

Extract the finder's current estimate of the upper bound of the distance from estimateRoot to an actual root in the function.

Generally, estimateRoot r +- estimateError r should bracket a root of the function.

converged :: (Num a, Ord a) => a -> r a b -> Bool

Test whether a root finding algorithm has converged to a given relative accuracy.

defaultNSteps :: Tagged (r a b) Int

Default number of steps after which root finding will be deemed to have failed. Purely a convenience used to control the behavior of built-in functions such as findRoot and traceRoot. The default value is 250.

Instances

(Fractional a, Ord b, Num b) => RootFinder Bisect a b

(Fractional a, Ord a, Real b, Fractional b, Ord b) => RootFinder Dekker a b

(Fractional a, Ord a) => RootFinder FalsePosition a a

(Fractional a, Ord a, Real b, Fractional b) => RootFinder InverseQuadratic a b

(Floating a, Ord a) => RootFinder RiddersMethod a a

(Fractional a, Ord a) => RootFinder SecantMethod a a

(RealFloat a, Real b, Fractional b) => RootFinder Brent a b

Fractional a => RootFinder Newton a (a, a)

traceRoot :: (Eq (r a b), RootFinder r a b, Num a, Ord a) => (a -> b) -> a -> a -> Maybe a -> [r a b]

traceRoot f x0 x1 mbEps initializes a root finder and repeatedly steps it, returning each step of the process in a list. When the algorithm terminates or the defaultNSteps limit is exceeded, the list ends. Termination criteria depends on mbEps; if it is of the form Just eps then convergence to eps is used (using the converged method of the root finder). Otherwise, the trace is not terminated until subsequent states are equal (according to ==). This is a stricter condition than convergence to 0; subsequent states may have converged to zero but as long as any internal state changes the trace will continue.

findRoot :: (RootFinder r a b, Num a, Ord a) => (a -> b) -> a -> a -> a -> Either (r a b) (r a b)

findRoot f x0 x1 eps initializes a root finder and repeatedly steps it. When the algorithm converges to eps or the defaultNSteps limit is exceeded, the current best guess is returned, with the Right constructor indicating successful convergence or the Left constructor indicating failure to converge.

eps :: RealFloat a => a

A useful constant: eps is (for most RealFloat types) the smallest positive number such that 1 + eps /= 1.