Portability	portable
Stability	experimental
Maintainer	mik@konecny.aow.cz

Data.Number.ER.RnToRm.UnitDom.ChebyshevBase.Polynom.Bounds

Description

Internal module for Data.Number.ER.RnToRm.UnitDom.ChebyshevBase.Polynom.

Implementation of various functions related to the bounds of polynomials.

Synopsis

Documentation

chplUpperBound Source

Arguments

:: (ERRealBase b, DomainBox box varid Int, Ord box, Show varid, DomainIntBox boxra varid (ERInterval b), DomainBoxMappable boxra boxras varid (ERInterval b) [ERInterval b])
=> EffortIndex	how hard to try
-> ERChebPoly box b
-> b

Find an upper bound on a polynomial over the unit domain [-1,1]^n.

Quick method that does not converge to exact result with increasing effort index.

chplLowerBound Source

Arguments

:: (ERRealBase b, DomainBox box varid Int, Ord box, Show varid, DomainIntBox boxra varid (ERInterval b), DomainBoxMappable boxra boxras varid (ERInterval b) [ERInterval b])
=> EffortIndex	how hard to try
-> ERChebPoly box b
-> b

Find a lower bound on a polynomial over the unit domain [-1,1]^n.

Quick method that does not converge to exact result with increasing effort index.

chplBounds Source

Arguments

:: (ERRealBase b, DomainBox box varid Int, Ord box, Show varid, DomainIntBox boxra varid (ERInterval b), DomainBoxMappable boxra boxras varid (ERInterval b) [ERInterval b])
=> EffortIndex	how hard to try
-> ERChebPoly box b
-> (b, b)

Find both lower and upper bounds on a polynomial over the unit domain [-1,1]^n.

Quick method that does not converge to exact result with increasing effort index.

chplUpperBoundExpensive Source

Arguments

:: (ERRealBase b, DomainBox box varid Int, Ord box, Show varid, DomainIntBox boxra varid (ERInterval b), DomainBoxMappable boxra boxras varid (ERInterval b) [ERInterval b])
=> EffortIndex	how hard to try
-> ERChebPoly box b
-> b

Find an upper bound on a polynomial over the unit domain [-1,1]^n.

chplLowerBoundExpensive Source

Arguments

:: (ERRealBase b, DomainBox box varid Int, Ord box, Show varid, DomainIntBox boxra varid (ERInterval b), DomainBoxMappable boxra boxras varid (ERInterval b) [ERInterval b])
=> EffortIndex	how hard to try
-> ERChebPoly box b
-> b

Find a lower bound on a polynomial over the unit domain [-1,1]^n.

Quick method that does not converge to exact result with increasing effort index.

chplBoundsExpensive Source

Arguments

:: (ERRealBase b, DomainBox box varid Int, Ord box, Show varid, DomainIntBox boxra varid (ERInterval b), DomainBoxMappable boxra boxras varid (ERInterval b) [ERInterval b])
=> EffortIndex	how hard to try
-> ERChebPoly box b
-> (b, b)

Find both lower and upper bounds on a polynomial over the unit domain [-1,1]^n.

Quick method that does not converge to exact result with increasing effort index.

chplBoundsAffine Source

Arguments

:: (ERRealBase b, DomainBox box varid Int, Ord box)
=> EffortIndex	how hard to try
-> ERChebPoly box b
-> (b, b)

Find bounds on a polynomial over the unit domain [-1,1]^n.

Fast but inaccurate method, in essence taking the maximum of the upper affine reduction.

chplBoundsByDerivative Source

Arguments

:: (ERRealBase b, DomainBox box varid Int, Ord box, Show varid, DomainIntBox boxra varid (ERInterval b), DomainBoxMappable boxra boxras varid (ERInterval b) [ERInterval b])
=> EffortIndex	how hard to try looking for peaks
-> ERChebPoly box b
-> (b, b)

Find a close upper bound of a polynomial over the unit domain [-1,1]^n.

Approximates all local extrema and computes their maximum.

chplMax Source

Arguments

:: (ERRealBase b, DomainBox box varid Int, Ord box, Show varid, DomainIntBox boxra varid (ERInterval b), DomainBoxMappable boxra boxras varid (ERInterval b) [ERInterval b])
=> Int	maximum polynomial degree
-> Int	maximum term count
-> ERChebPoly box b
-> ERChebPoly box b
-> (ERChebPoly box b, ERChebPoly box b)

Approximate from below and from above the pointwise maximum of two polynomials

chplMin Source

Arguments

:: (ERRealBase b, DomainBox box varid Int, Ord box, Show varid, DomainIntBox boxra varid (ERInterval b), DomainBoxMappable boxra boxras varid (ERInterval b) [ERInterval b])
=> Int	maximum polynomial degree
-> Int	maximum term count
-> ERChebPoly box b
-> ERChebPoly box b
-> (ERChebPoly box b, ERChebPoly box b)

Approximate from below and from above the pointwise minimum of two polynomials

chplNonneg Source

Arguments

:: (ERRealBase b, DomainBox box varid Int, Ord box, Show varid, DomainIntBox boxra varid (ERInterval b), DomainBoxMappable boxra boxras varid (ERInterval b) [ERInterval b])
=> Int	maximum polynomial degree
-> Int	maximum term count
-> ERChebPoly box b
-> (ERChebPoly box b, ERChebPoly box b)

Approximate the function max(0,p(x)) by a polynomial from below and from above.

chplNonnegCubic Source

Arguments

:: (ERRealBase b, DomainBox box varid Int, Ord box, Show varid, DomainIntBox boxra varid (ERInterval b), DomainBoxMappable boxra boxras varid (ERInterval b) [ERInterval b])
=> Int	maximum polynomial degree
-> Int	maximum term count
-> ERChebPoly box b
-> (ERChebPoly box b, ERChebPoly box b)

A version of chplNonneg using a cubic approximation.

chplTimesLoHi :: (ERRealBase b, DomainBox box varid Int, Ord box, Show varid, DomainIntBox boxra varid (ERInterval b), DomainBoxMappable boxra boxras varid (ERInterval b) [ERInterval b]) => ERChebPoly box b -> (ERChebPoly box b, ERChebPoly box b, b) -> (ERChebPoly box b, ERChebPoly box b, b)Source

Multiply a polynomial by an enclosure (with non-negated lower bound).