Orbital library

## Uses of Interfaceorbital.math.functional.Function

Packages that use Function
orbital.algorithm.evolutionary Genetic algorithms simulate nature on a very abstract level to get solutions for sophisticated problems.
orbital.algorithm.template A framework for general algorithmic evaluation schemes including search and planning algorithms.
orbital.math Defines arithmetic objects and provides mathematical algorithms.
orbital.math.functional Contains mathematical functors and extended functional operations.

Uses of Function in orbital.algorithm.evolutionary

Methods in orbital.algorithm.evolutionary that return Function
` Function` `GeneticAlgorithm.complexity()`

` Function` `GeneticAlgorithm.spaceComplexity()`

Uses of Function in orbital.algorithm.template

Methods in orbital.algorithm.template that return Function
` Function` `ThresholdAccepting.complexity()`
O(∞).
` Function` `SimulatedAnnealing.complexity()`
O(∞).
` Function` `RealTimeDynamicProgramming.complexity()`

` Function` `ParallelBranchAndBound.complexity()`
O(d) on parallel machines where d the solution depth.
` Function` `IterativeExpansion.complexity()`
O(bd) where b is the branching factor and d the solution depth.
` Function` `IterativeDeepeningAStar.complexity()`
O(bd) where b is the branching factor and d the solution depth.
` Function` `IterativeDeepening.complexity()`
O(bd) where b is the branching factor and d the solution depth.
` Function` `IterativeBroadening.complexity()`

` Function` `HillClimbing.complexity()`
O(∞).
` Function` `Greedy.complexity()`
O(n*log n + n*f(n)) for n=|C| candidates.
` Function` `GaussSeidelDynamicProgramming.complexity()`

` Function` `DynamicProgramming.complexity()`
O(n2)
` Function` `DivideAndConquer.complexity()`
≈O(n*㏒ n).
` Function` `DepthFirstSearch.complexity()`
O(∞).
` Function` `BreadthFirstSearch.complexity()`
O(bd) where b is the branching factor and d the solution depth.
` Function` `BranchAndBound.complexity()`
O(bd) where b is the branching factor and d the solution depth.
` Function` `Backtracking.complexity()`
O(nn) in the worst case.
` Function` `AStar.complexity()`
O(bd) where b is the branching factor and d the solution depth.
` Function` `AlgorithmicTemplate.complexity()`
Measure for the asymptotic time complexity of the central solution operation in O-notation.
` Function` `ThresholdAccepting.spaceComplexity()`
O(b) where b is the branching factor and d the solution depth.
` Function` `SimulatedAnnealing.spaceComplexity()`
O(b) where b is the branching factor and d the solution depth.
` Function` `RealTimeDynamicProgramming.spaceComplexity()`

` Function` `ParallelBranchAndBound.spaceComplexity()`
O(bd) where b is the branching factor and d the solution depth.
` Function` `IterativeExpansion.spaceComplexity()`
O(b*d) where b is the branching factor and d the solution depth.
` Function` `HillClimbing.spaceComplexity()`
O(b) where b is the branching factor and d the solution depth.
` Function` `Greedy.spaceComplexity()`

` Function` `GaussSeidelDynamicProgramming.spaceComplexity()`

` Function` `DynamicProgramming.spaceComplexity()`

` Function` `DivideAndConquer.spaceComplexity()`

` Function` `DepthFirstSearch.spaceComplexity()`
O(b*d) where b is the branching factor and d the solution depth.
` Function` `BreadthFirstSearch.spaceComplexity()`
O(bd) where b is the branching factor and d the solution depth.
` Function` `Backtracking.spaceComplexity()`

` Function` `AStar.spaceComplexity()`
O(bd) where b is the branching factor and d the solution depth.
` Function` `AlgorithmicTemplate.spaceComplexity()`
Measure for the asymptotic space complexity of the central solution operation in O-notation.

Uses of Function in orbital.math

Subinterfaces of Function in orbital.math
` interface` `Polynomial`
Polynomial p∈R[S] := R(S).
` interface` `UnivariatePolynomial`
(Univariate) polynomial p∈R[X].

Methods in orbital.math that return Function
`static Function` ```NumericalAlgorithms.bezierCurve(double t0, double tz, Matrix bezierNodes)```
Bezier curve.
`static Function` ```NumericalAlgorithms.dSolve(BinaryFunction f, Real tau, Real eta, Real a, Real b, int steps, int order)```
Returns a numerical solution x of the one-dimensional differential equation x'(t) = f(t,x(t)), x(τ)=η on [a,b].
`static Function` ```NumericalAlgorithms.dSolve(BinaryFunction f, Real tau, Real eta, Real min, Real max, int steps, Matrix butcher)```
Returns a numerical solution x of the one-dimensional differential equation x'(t) = f(t,x(t)), x(τ)=η on [a,b].
`static Function` ```NumericalAlgorithms.dSolve(BinaryFunction f, Real tau, Vector eta, Real a, Real b, int steps, int order)```
Returns a numerical solution x of the differential equation x'(t) = f(t,x(t)), x(τ)=η on [a,b].
`static Function` ```NumericalAlgorithms.dSolve(BinaryFunction f, Real tau, Vector eta, Real min, Real max, int steps, Matrix butcher)```
Returns a numerical solution x of the differential equation x'(t) = f(t,x(t)), x(τ)=η on [a,b].
`static Function` ```AlgebraicAlgorithms.dSolve(Matrix A, Vector b, Real tau, Vector eta)```
Symbolically solves ordinary differential equation system.
`static Function` ```Stat.functionalRegression(Function composedFunc, Matrix experiment)```
Performs linear regression to estimate a composed function with least squares.
`static Function` `NumericalAlgorithms.polynomialInterpolation(Matrix A)`
Polynomial interpolation.
`static Function` ```NumericalAlgorithms.splineInterpolation(int k, Matrix A, int interpolationType)```
Spline interpolation.
`static Function` ```NumericalAlgorithms.splineInterpolation(int k, Matrix A, int interpolationType, java.lang.Object[] config)```
Deprecated. Use `NumericalAlgorithms.splineInterpolation(int,Matrix,int,Real,Real)`, or `NumericalAlgorithms.splineInterpolation(int,Matrix,int)` instead since they have a more reasonable argument list.
`static Function` ```NumericalAlgorithms.splineInterpolation(int k, Matrix A, int interpolationType, Real fp_a, Real fp_b)```
Spline interpolation.

Methods in orbital.math with parameters of type Function
`static Function` ```Stat.functionalRegression(Function composedFunc, Matrix experiment)```
Performs linear regression to estimate a composed function with least squares.
`static Arithmetic` ```NumericalAlgorithms.integrate(Function f, Arithmetic a, Arithmetic b)```
Returns ≈ ∫ab f dx.
`static Arithmetic` ```MathUtilities.integrate(Function f, Arithmetic a, Arithmetic b)```
Returns ∫ab f dx.
`static Vector` ```Stat.regression(Function[] funcs, Matrix experiment)```
Performs linear regression to estimate the statistical mean according to least squares.

Uses of Function in orbital.math.functional

Subinterfaces of Function in orbital.math.functional
`static interface` `Function.Composite`
A composite function.

Fields in orbital.math.functional declared as Function
`static Function` `Functions.arccos`
arccos: [-1,1]→[0,π]; x ↦ arccos x = cos-1 x.
`static Function` `Functions.arccot`
arccot: R→(0,π); x ↦ arccot x = cot-1 x.
`static Function` `Functions.arcosh`
arcosh: [1,∞)→[0,∞); x ↦ arcosh x = (cosh|[0,∞))-1 x = ㏒(x ± √x2-1).
`static Function` `Functions.arcoth`
arcoth: R\[-1,1]→R\{0}; x ↦ arcoth x = coth-1 x = ㏒((x+1) / (x-1)) / 2.
`static Function` `Functions.arcsin`
arcsin: [-1,1]→[-π/2,π/2]; x ↦ arcsin x = sin-1 x.
`static Function` `Functions.arctan`
arctan: R→(-π/2,π/2); x ↦ arctan x = tan-1 x.
`static Function` `Functions.arsinh`
arsinh: R→R; x ↦ arsinh x = sinh-1 x = ㏒(x + √x2+1).
`static Function` `Functions.artanh`
artanh: (-1,1)→R; x ↦ artanh x = tanh-1 x = ㏒((1+x) / (1-x)) / 2.
`static Function` `Functions.cos`
cos: CC; x ↦ cos x = ∑n=0 (-1)n * x2n / (2n)!.
`static Function` `Functions.cosh`
cosh: CC; x ↦ cosh x = (ex+e-x) / 2 = ∑n=0 x2n / (2n)!.
`static Function` `Functions.cot`
cot: CZC; x ↦ cot x = cos x / sin x = 1 / tan x.
`static Function` `Functions.coth`
coth: R\{0}→R; x ↦ coth x = cosh x / sinh x = 1 / tanh x.
`static Function` `Functions.csc`
csc: R\{0}→R; x ↦ csc x = 1 / sin(x).
`static Function` `Functions.csch`
csch: R\{0}→R; x ↦ csch x = 1 / sinh(x).
`static Function` `Functions.diracDelta`
diracDelta δ: M\{0}→{0}; x ↦ 0 if x≠0.
`static Function` `Functions.exp`
exp: CC\{0}; x ↦ ex = ∑n=0 xn / n!.
`static Function` `Functions.id`
id: R→R; x ↦ x .
`static Function` `Operations.inf`
inf ⊓: An→A; (xi) ↦ ⊓i {xi} = (|∞,min|) (xi).
`static Function` `Operations.inverse`
inverse -1: A→A; x ↦ x-1.
`static Function` `Functions.log`
㏒: C\{0}→C; x ↦ ㏒e x.
`static Function` `Functions.logistic`
logistic: A→(0,1); x ↦ 1 / (1 + e-x).
`static Function` `Operations.minus`
minus −: A→A; x ↦ −x.
`static Function` `Functions.nondet`
Represents a nondeterministic function.
`static Function` `Functions.norm`
norm: A→[0,∞); x ↦ ||x||.
`static Function` `Functions.one`
one: R→R; x ↦ 1 .
`static Function` `Operations.product`
product ∏: An→A; (xi) ↦ ∏i xi = (|1,⋅|) (xi).
`static Function` `Functions.reciprocal`
reciprocal: C\{0}→C; x ↦ x-1 = 1 / x.
`static Function` `Functions.sec`
sec: R→R; x ↦ sec x = 1 / cos(x).
`static Function` `Functions.sech`
sech: R→R; x ↦ sech x = 1/cosh(x).
`static Function` `Functions.sign`
sign: A→{-1,0,1}; x ↦ -1 if x<0, x ↦ 0 if x=0, x ↦ 1 if x>0.
`static Function` `Functions.sin`
sin: CC; x ↦ sin x = ∑n=0 (-1)n * x2n+1 / (2n+1)!.
`static Function` `Functions.sinh`
sinh: CC; x ↦ sinh x = (ex-e-x) / 2 = ∑n=0 x2n+1 / (2n+1)!.
`static Function` `Functions.sqrt`
sqrt √ : CC; x ↦ √x = x1/2.
`static Function` `Functions.square`
square: R→R; x ↦ x2 .
`static Function` `Operations.sum`
sum ∑: An→A; (xi) ↦ ∑i xi = (|0,+|) (xi).
`static Function` `Operations.sup`
sup ⊔: An→A; (xi) ↦ ⊔i {xi} = (|-∞,max|) (xi).
`static Function` `Functions.tan`
tan: C\(π/2+πZ)→C; x ↦ tan x = sin x / cos x.
`static Function` `Functions.tanh`
tanh: C\(πi/2*Z)→C; x ↦ tanh x = sinh x / cosh x.
`static Function` `Functions.zero`
zero: R→R; x ↦ 0 .

Methods in orbital.math.functional that return Function
`static Function` `Functionals.bind(BinaryFunction f)`
Binds both arguments of a BinaryFunction together.
`static Function` ```Functionals.bindFirst(BinaryFunction f, java.lang.Object x)```
Binds the first argument of a BinaryFunction to a fixed value.
`static Function` ```Functionals.bindSecond(BinaryFunction f, java.lang.Object y)```
Binds the second argument of a BinaryFunction to a fixed value.
`static Function` ```Functionals.compose(BinaryFunction f, Function g, Function h)```
compose: (f,g,h) ↦ f ∘ (g × h) := f(g,h) .
`static Function` ```Functionals.compose(Function f, Function g)```
compose: (f,g) ↦ f ∘ g := f(g).
`static Function` `Functions.constant(java.lang.Object a)`
constant â: R→R; x ↦ a .
` Function` `Function.derive()`
Derives this function and returns the resulting Function df/dx.
`static Function` `Functions.exp(Arithmetic b)`
expb: CC\{0}; x ↦ bx .
` Function` `Function.integrate()`
Integrates this function and returns the resulting indefinite integral ∫ f dx.
`static Function` `Functions.linear(Arithmetic a)`
linear: A→B; x ↦ a*x.
`static Function` ```Functionals.nest(Function f, int n)```
Nests a function n times within itself.
`static Function` ```Functions.piecewise(Predicate[] cond, Function[] value)```
Get a function defined piecewise.
`static Function` `Functionals.pointwise(Function elemental)`
A function that performs an operation pointwise.
`static Function` `Functions.pow(Arithmetic p)`
powp: R→R; x ↦ xp .
`static Function` `Functions.pow(double p)`

`static Function` `Functions.projection(int component)`
projection πc: An→A; (x1,...xn)T ↦ xc.
`static Function` ```Functions.projection(int i, int j)```
projection πi,j: An×m→A; (xi,j) ↦ xi,j.
`static Function` `Functions.step(Real t)`
step ht: A→{0,1}; x ↦ 1 if x≥t, x ↦ 0 if x<t.
`static Function` `Functions.symbolic(java.lang.String name)`
symbolic f:R→R; x ↦ f(x).

Methods in orbital.math.functional with parameters of type Function
`static Function` ```Functionals.compose(BinaryFunction f, Function g, Function h)```
compose: (f,g,h) ↦ f ∘ (g × h) := f(g,h) .
`static Function` ```Functionals.compose(Function f, Function g)```
compose: (f,g) ↦ f ∘ g := f(g).
`static MathFunctor` ```Functionals.genericCompose(Function f, java.lang.Object g)```
generic compose calls the compose function appropriate for the type of g.
`static Matrix` ```Functionals.map(Function f, Matrix a)```

`static Tensor` ```Functionals.map(Function f, Tensor a)```
Maps a list of arguments with a function.
`static Vector` ```Functionals.map(Function f, Vector a)```

`static Function` ```Functionals.nest(Function f, int n)```
Nests a function n times within itself.
`static BinaryFunction` `Functionals.onFirst(Function f)`
Applies a function on the first argument, ignoring the second.
`static BinaryFunction` `Functionals.onSecond(Function f)`
Applies a function on the second argument, ignoring the first.
`static Function` ```Functions.piecewise(Predicate[] cond, Function[] value)```
Get a function defined piecewise.
`static Function` `Functionals.pointwise(Function elemental)`
A function that performs an operation pointwise.

Orbital library
1.3.0: 11 Apr 2009

Copyright © 1996-2009 André Platzer
All Rights Reserved.