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ConvexCurve

This class should be used to represent convex curves with f(0)=0f(0) = 0 (see IsRegularConvex), and exploit these properties to optimize computations.

note

f(0)=0f(0) = 0 is required for the curve to be IsRegularSuperAdditive, hence derive from SuperAdditiveCurve, but is not required for convexity to be stable on maximum, addition and convolution. To keep the type system simple for the common cases of DNC, we opted not to support non-regular convex functions with their own type.

public class ConvexCurve : SuperAdditiveCurve, Unipi.Nancy.MinPlusAlgebra.IToCodeString, Unipi.Nancy.Utility.IStableHashCode

Inheritance Object → Curve → SuperAdditiveCurve → ConvexCurve
Implements IToCodeString, IStableHashCode

Properties​

IsSuperAdditive​

True if the curve is super-additive.

note

For a SuperAdditiveCurve this will always return true, without performing any checks.

public bool IsSuperAdditive { get; }

Property Value​

Boolean

PseudoPeriodStart​

Point in time after which the curve has a pseudo-periodic behavior.

note

Referred to as TT or Rank in [BT08] Section 4.1

public Rational PseudoPeriodStart { get; set; }

Property Value​

Rational

PseudoPeriodLength​

Time length of each pseudo-period.

note

Referred to as dd in [BT08] Section 4.1

public Rational PseudoPeriodLength { get; set; }

Property Value​

Rational

PseudoPeriodHeight​

Static value gain applied after each pseudo-period. If it's 0, the curve is truly periodic.

note

Referred to as cc in [BT08] Section 4.1

public Rational PseudoPeriodHeight { get; set; }

Property Value​

Rational

PseudoPeriodSlope​

Average slope of curve in pseudo-periodic behavior. If it's 0, the curve is truly periodic.

public Rational PseudoPeriodSlope { get; }

Property Value​

Rational

FirstPseudoPeriodEnd​

End time of the first pseudo period.

public Rational FirstPseudoPeriodEnd { get; }

Property Value​

Rational

SecondPseudoPeriodEnd​

End time of the second pseudo period.

public Rational SecondPseudoPeriodEnd { get; }

Property Value​

Rational

BaseSequence​

Sequence describing behavior of the curve in [0,T+d[[0, T + d[. Combined with the UPP property, this is also allows to derive f(t)f(t) for any t≥T+dt \ge T + d.

note

Referred to as [t1,...,tk][t_1, ..., t_k] in [BT08] Section 4.1

public Sequence BaseSequence { get; set; }

Property Value​

Sequence

IsFinite​

True if the curve has finite value for any tt.

public bool IsFinite { get; }

Property Value​

Boolean

FirstFiniteTime​

The first instant around which the curve is not infinite. Does not specify whether it's inclusive or not, i.e. if f(t)f(t) is finite.

public Rational FirstFiniteTime { get; }

Property Value​

Rational

FirstFiniteTimeExceptOrigin​

The first instant around which the curve is not infinite, excluding the origin point. Does not specify whether it's inclusive or not, i.e. if f(t)f(t) is finite.

public Rational FirstFiniteTimeExceptOrigin { get; }

Property Value​

Rational

FirstNonZeroTime​

The first instant around which the curve is not 0. Does not specify whether it's inclusive or not, i.e. if f(t‾)=0f(\overline{t}) = 0.

public Rational FirstNonZeroTime { get; }

Property Value​

Rational

PseudoPeriodStartInfimum​

Returns the minimum TLT_L such that f(t+d)=f(t)+cf(t + d) = f(t) + c for all t>TLt > T_L. It is the infimum of all valid PseudoPeriodStart, i.e. TL=inf⁡{T∣f(t+d)=f(t)+c∀t≥T}T_L = \inf\{ T | f(t + d) = f(t) + c \forall t \ge T \}.

public Rational PseudoPeriodStartInfimum { get; }

Property Value​

Rational

IsZero​

True if the curve is 0 for all tt.

public bool IsZero { get; }

Property Value​

Boolean

IsPlusInfinite​

True if the curve has +∞+\infty value for any tt.

public bool IsPlusInfinite { get; }

Property Value​

Boolean

IsMinusInfinite​

True if the curve has −∞-\infty value for any tt.

public bool IsMinusInfinite { get; }

Property Value​

Boolean

IsContinuous​

True if there is no infinite value or discontinuity within the curve.

public bool IsContinuous { get; }

Property Value​

Boolean

IsContinuousExceptOrigin​

True if there is no discontinuity within the curve, except at most in origin.

public bool IsContinuousExceptOrigin { get; }

Property Value​

Boolean

IsLeftContinuous​

True if there is no left-discontinuity within the curve.

public bool IsLeftContinuous { get; }

Property Value​

Boolean

IsRightContinuous​

True if there is no right-discontinuity within the curve.

public bool IsRightContinuous { get; }

Property Value​

Boolean

IsNonNegative​

True if the curve is non-negative, i.e. f(t)≥0f(t) \ge 0 for any tt.

public bool IsNonNegative { get; }

Property Value​

Boolean

FirstNonNegativeTime​

The first instant around which the curve is non-negative. Does not specify whether it's inclusive or not, i.e. if f(t‾)>=0f(\overline{t}) >= 0.

public Rational FirstNonNegativeTime { get; }

Property Value​

Rational

IsNonDecreasing​

True if for any t>st > s, f(t)≥f(s)f(t) \ge f(s).

public bool IsNonDecreasing { get; }

Property Value​

Boolean

IsUltimatelyFinite​

True if for all t≥t \ge PseudoPeriodStart the curve is finite.

note

This property does not check if f(t),t<f(t), t < PseudoPeriodStart is either infinite, finite or both.

public bool IsUltimatelyFinite { get; }

Property Value​

Boolean

IsUltimatelyInfinite​

True if, for b∈{−∞,+∞}b \in \{-\infty, +\infty\} and some TT, f(t)=bf(t) = b for all t≥Tt \ge T.

public bool IsUltimatelyInfinite { get; }

Property Value​

Boolean

IsUltimatelyPlain​

True if for all t>t > PseudoPeriodStart the curve is either always finite or always infinite.

note

Defined in [BT08], Definition 1.

public bool IsUltimatelyPlain { get; }

Property Value​

Boolean

IsPlain​

True if ff is plain, i.e., it is either a) always finite, b) always plus or minus infinite (without changing sign), or c) finite up to a TT, then always plus or minus infinite (without changing sign)

note

Formally defined in [BT08], Definition 1.

public bool IsPlain { get; }

Property Value​

Boolean

IsUltimatelyAffine​

True if for all t≥t \ge PseudoPeriodStart the curve is affine.

public bool IsUltimatelyAffine { get; }

Property Value​

Boolean

IsUltimatelyConstant​

True if for t≥t \ge PseudoPeriodStart the curve is constant.

public bool IsUltimatelyConstant { get; }

Property Value​

Boolean

IsSubAdditive​

True if the curve is sub-additive, i.e. f(t+s)≤f(t)+f(s)f(t+s) \le f(t) + f(s).

note

Based on [Zippo23] Lemma 9.3: f(0)≥0,ff(0) \ge 0, f is sub-additive   ⟺  f∘=f∘⊗f∘\iff f^\circ = f^\circ \otimes f^\circ, where f∘f^\circ is defined in WithZeroOrigin(). Can be computationally expensive the first time it is invoked, the result is cached afterwards.

public bool IsSubAdditive { get; }

Property Value​

Boolean

IsRegularSubAdditive​

True if the curve is sub-additive with f(0)=0f(0) = 0.

public bool IsRegularSubAdditive { get; }

Property Value​

Boolean

IsRegularSuperAdditive​

True if the curve is super-additive with f(0)=0f(0) = 0.

public bool IsRegularSuperAdditive { get; }

Property Value​

Boolean

IsConcave​

Tests if the curve is concave, i.e. for any two points (t,f(t))(t, f(t)) the straight line joining them is below ff.

note

The property is checked via the following property: ff is concave   ⟺  \iff a) ff is continuous, or it is continuous for t>0t > 0 and f(0)≤f(0+)f(0) \le f(0^+), and b) ff is composed of segments with decreasing slopes.

public bool IsConcave { get; }

Property Value​

Boolean

IsRegularConcave​

Tests if the curve is concave with f(0)=0f(0) = 0.

public bool IsRegularConcave { get; }

Property Value​

Boolean

IsConvex​

Tests if the curve is convex, i.e. for any two points (t,f(t))(t, f(t)) the straight line joining them is above ff.

note

The property is checked via the following property: ff is convex   ⟺  \iff a) ff is continuous, or it is continuous for t>0t > 0 and f(0)≥f(0+)f(0) \ge f(0^+), and b) ff is composed of segments with increasing slopes.

public bool IsConvex { get; }

Property Value​

Boolean

IsRegularConvex​

Tests if the curve is convex with f(0)=0f(0) = 0.

public bool IsRegularConvex { get; }

Property Value​

Boolean

HasTransient​

True if pseudo-periodic behavior starts at T>0T > 0.

public bool HasTransient { get; }

Property Value​

Boolean

TransientSequence​

Sequence describing the curve in [0,T[[0, T[, before pseudo-periodic behavior.

public Sequence TransientSequence { get; }

Property Value​

Sequence

TransientElements​

Elements describing the curve from [0,T[[0, T[, before pseudo-periodic behavior.

note

Referred to as [t1,...,ti0−1][t_1, ..., t_{i_0 - 1}] in [BT08] Section 4.1

public IEnumerable<Element> TransientElements { get; }

Property Value​

IEnumerable<Element>

PseudoPeriodicSequence​

Sequence describing the pseudo-periodic behavior of the curve in [T,T+d[[T, T + d[.

public Sequence PseudoPeriodicSequence { get; }

Property Value​

Sequence

PseudoPeriodicElements​

Elements describing the pseudo-periodic behavior of the curve in [T,T+d[[T, T + d[.

note

Referred to as [ti0,...,tk][t_{i_0}, ..., t_k] in [BT08] Section 4.1

public IEnumerable<Element> PseudoPeriodicElements { get; }

Property Value​

IEnumerable<Element>

Constructors​

ConvexCurve(Sequence, Rational, Rational, Rational)​

Constructor.

public ConvexCurve(Sequence baseSequence, Rational pseudoPeriodStart, Rational pseudoPeriodLength, Rational pseudoPeriodHeight)

Parameters​

baseSequence Sequence

pseudoPeriodStart Rational

pseudoPeriodLength Rational

pseudoPeriodHeight Rational

ConvexCurve(Curve)​

Copy constructor.

public ConvexCurve(Curve other)

Parameters​

other Curve

Methods​

Addition(Curve)​

public Curve Addition(Curve b)

Parameters​

b Curve

Returns​

Curve

Addition(ConvexCurve)​

public ConvexCurve Addition(ConvexCurve b)

Parameters​

b ConvexCurve

Returns​

ConvexCurve

Maximum(Curve, ComputationSettings)​

public Curve Maximum(Curve curve, ComputationSettings settings)

Parameters​

curve Curve

settings ComputationSettings

Returns​

Curve

Maximum(ConvexCurve, ComputationSettings)​

public ConvexCurve Maximum(ConvexCurve curve, ComputationSettings settings)

Parameters​

curve ConvexCurve

settings ComputationSettings

Returns​

ConvexCurve

Convolution(Curve, ComputationSettings)​

public Curve Convolution(Curve curve, ComputationSettings settings)

Parameters​

curve Curve

settings ComputationSettings

Returns​

Curve

Convolution(ConvexCurve, ComputationSettings)​

public ConvexCurve Convolution(ConvexCurve curve, ComputationSettings settings)

Parameters​

curve ConvexCurve

settings ComputationSettings

Returns​

ConvexCurve