Math.ExpPairs

Package implements an algorithm to minimize the maximum of a list of rational objective functions over the set of exponent pairs. See full description in A. V. Lelechenko, Linear programming over exponent pairs. Acta Univ. Sapientiae, Inform. 5, No. 2, 271-287 (2013). http://www.acta.sapientia.ro/acta-info/C5-2/info52-7.pdf

A set of useful applications can be found in Math.ExpPairs.Ivic, Math.ExpPairs.Kratzel and Math.ExpPairs.MenzerNowak.

Documentation

optimize :: [RationalForm Rational] -> [Constraint Rational] -> OptimizeResult

This function takes a list of rational forms and a list of constraints and returns an exponent pair, which satisfies all constraints and minimizes the maximum of all rational forms.

data OptimizeResult

Container for the result of optimization.

Instances

Eq OptimizeResult

Ord OptimizeResult

Show OptimizeResult

optimalValue :: OptimizeResult -> RationalInf

The minimal value of objective function.

optimalPair :: OptimizeResult -> InitPair

The initial exponent pair, on which minimal value was achieved.

optimalPath :: OptimizeResult -> Path

The sequence of processes, after which minimal value was achieved.

simulateOptimize :: Rational -> OptimizeResult

Wrap Rational into OptimizeResult.

simulateOptimize' :: RationalInf -> OptimizeResult

Wrap RationalInf into OptimizeResult.

data LinearForm t

Define an affine linear form of two variables: a*k + b*l + c*m. First argument of LinearForm stands for a, second for b and third for c. Linear forms form a monoid by addition.

Constructors

LinearForm t t t

Instances

Eq t => Eq (LinearForm t)

Num t => Num (LinearForm t)

(Num t, Eq t, Show t) => Show (LinearForm t)

Num t => Monoid (LinearForm t)

data RationalForm t

Define a rational form of two variables, equal to the ratio of two LinearForm.

Constructors

RationalForm (LinearForm t) (LinearForm t)

Instances

Num t => Fractional (RationalForm t)

Num t => Num (RationalForm t)

(Eq t, Num t, Show t) => Show (RationalForm t)

data IneqType

Constants to specify the strictness of Constraint.

Constructors

Strict

Strict inequality (>0).

NonStrict

Non-strict inequality (≥0).

Instances

Eq IneqType

Show IneqType

data Constraint t

A linear constraint of two variables.

Constructors

Constraint (LinearForm t) IneqType

Instances

(Eq t, Num t, Show t) => Show (Constraint t)

type InitPair = InitPair' Rational

Exponent pair built from rational fractions of Corput16, HuxW87b1 and Hux05

data Path

Holds a list of Process and a matrix of projective transformation, which they define. It also provides a fancy Show instance. E. g.,

show (mconcat $ replicate 10 aPath) == "A^10"

Instances

Eq Path

Ord Path

Read Path

Show Path

Monoid Path

data RatioInf t

Extends a rational type with positive and negative infinities.

Constructors

InfMinus

Negative infinity

Finite (Ratio t)

Finite value

InfPlus

Positive infinity

Instances

Eq t => Eq (RatioInf t)

Integral t => Fractional (RatioInf t)

Integral t => Num (RatioInf t)

Integral t => Ord (RatioInf t)

Integral t => Real (RatioInf t)

(Integral t, Show t) => Show (RatioInf t)

type RationalInf = RatioInf Integer

Arbitrary-precision rational numbers with positive and negative infinities.