AERN-Real-0.9.2: datatypes and abstractions for approximating exact real numbers

Portabilityportable
Stabilityalpha
Maintainermik@konecny.aow.cz

Data.Number.ER.Real.Arithmetic.Newton

Description

Interval Newton's method for root finding.

To be used for obtaining functions out of their inverse(s) over various intervals.

Synopsis

Documentation

erNewton_FullArgsSource

Arguments

:: ERIntApprox ira 
=> (EffortIndex -> ira -> ira, EffortIndex -> ira -> ira)

a function and its derivative

-> ira

a starting point

-> ira

a lower bound of the absolute value of the derivative over the working interval

-> Int

number of iterations

-> EffortIndex

the initial index to use for argument function and its derivative

-> ira

the result

erNewton_mdfd_FullArgsSource

Arguments

:: ERIntApprox ira 
=> (EffortIndex -> ira -> ira, EffortIndex -> ira -> ira)

a function and its derivative

-> ira

a starting point

-> ira

The minimum of absolute value of derivative over the working interval

-> Int

number of iterations

-> EffortIndex

It triggers the initial index to be called by the argument function and its derivative.

-> ira

the result

Modified Newton Method Notes:

  1. It has a cubic convergence speed, as opposed to the original Newton's square convergence speed.
  2. It does not deal with multiple roots.
  3. Per iteration, it makes two queries on the derivative, so it best suits the cases where computation of the derivative is at most as expensive as the function itself.