Factory.Math.Precision

AUTHOR

Dr. Alistair Ward

DESCRIPTION

Defines the unit with which precision is measured, and operations on it.

Types

Type-synonyms

type ConvergenceOrder = Int

The order of convergence; http://en.wikipedia.org/wiki/Rate_of_convergence.

type ConvergenceRate = Double

The rate of convergence; http://en.wikipedia.org/wiki/Rate_of_convergence.

type DecimalDigits = Int

A number of decimal digits.

Constants

linearConvergence :: ConvergenceOrder

Linear convergence-rate; which may be qualified by the rate of convergence.

quadraticConvergence :: ConvergenceOrder

Quadratic convergence-rate.

cubicConvergence :: ConvergenceOrder

Cubic convergence-rate.

quarticConvergence :: ConvergenceOrder

Quartic convergence-rate.

Functions

getIterationsRequired

Arguments

:: Integral i

=> ConvergenceOrder

-> DecimalDigits

The precision of the initial estimate.

-> DecimalDigits

The required precision.

-> i

The predicted number of iterations, required to achieve a specific accuracy, at a given order of convergence.

getTermsRequired

Arguments

:: Integral i

=> ConvergenceRate

-> DecimalDigits

The additional number of correct decimal digits.

-> i

• The predicted number of terms which must be extracted from a series, if it is to converge to the required accuracy, at the specified linear convergence-rate.
• The convergence-rate of a series, is the error in the series after summation of (n+1)th terms, divided by the error after only n terms, as the latter tends to infinity. As such, for a convergent series (in which the error get smaller with successive terms), it's value lies in the range 0 .. 1.
• http://en.wikipedia.org/wiki/Rate_of_convergence.

promote :: Num n => n -> DecimalDigits -> n

Promotes the specified number, by a number of DecimalDigits.

simplify

Arguments

:: RealFrac operand

=> DecimalDigits

The number of places after the decimal point, which are required.

-> operand

-> Rational

• Reduces a Rational to the minimal form required for the specified number of fractional decimal places; irrespective of the number of integral decimal places.
• A Rational approximation to an irrational number, may be very long, and provide an unknown excess precision. Whilst this doesn't sound harmful, it costs in performance and memory-requirement, and being unpredictable isn't actually useful.