orbital.math.functional
Class Functions

java.lang.Object
  orbital.math.functional.Functions

public final class Functions
extends java.lang.Object

Common function implementations.

Author:

André Platzer

Stereotype:

Module

Field Summary

static Function	arccos arccos: [-1,1]->[0,π]; x |-> arccos x = cos-1 x.
static Function	arccot arccot: R->(0,π); x |-> arccot x = cot-1 x.
static Function	arcosh arcosh: [1, inf )->[0, inf ); x |-> arcosh x = (cosh|[0, inf ))-1 x = log(x ± sqrt x2-1).
static Function	arcoth arcoth: R\[-1,1]->R\{0}; x |-> arcoth x = coth-1 x = log((x+1) / (x-1)) / 2.
static Function	arcsin arcsin: [-1,1]->[-π/2,π/2]; x |-> arcsin x = sin-1 x.
static Function	arctan arctan: R->(-π/2,π/2); x |-> arctan x = tan-1 x.
static Function	arsinh arsinh: R->R; x |-> arsinh x = sinh-1 x = log(x + sqrt x2+1).
static Function	artanh artanh: (-1,1)->R; x |-> artanh x = tanh-1 x = log((1+x) / (1-x)) / 2.
static Function	cos cos: C->C; x |-> cos x = Σn=0 inf (-1)n * x2n / (2n)!.
static Function	cosh cosh: C->C; x |-> cosh x = (ex+e-x) / 2 = Σn=0 inf x2n / (2n)!.
static Function	cot cot: C\πZ->C; x |-> cot x = cos x / sin x = 1 / tan x.
static Function	coth coth: R\{0}->R; x |-> coth x = cosh x / sinh x = 1 / tanh x.
static Function	csc csc: R\{0}->R; x |-> csc x = 1 / sin(x).
static Function	csch csch: R\{0}->R; x |-> csch x = 1 / sinh(x).
static BinaryFunction	delta delta: R×R->R; (x,x) |-> 1, (x,y) |-> 0 for x!=y.
static Function	diracDelta diracDelta δ: M\{0}->{0}; x |-> 0 if x!=0.
static Function	exp exp: C->C\{0}; x |-> ex = Σn=0 inf xn / n!.
static Functions	functions Class alias object.
static Function	id id: R->R; x |-> x .
static Function	log log: C\{0}->C; x |-> loge x.
static Function	logistic logistic: A->(0,1); x |-> 1 / (1 + e-x).
static Function	nondet Represents a nondeterministic function.
static Function	norm norm: A->[0, inf ); x |-> ||x||.
static Function	one one: R->R; x |-> 1 .
static BinaryFunction	projectFirst Projects to the first argument, ignoring the second.
static BinaryFunction	projectSecond Projects to the second argument, ignoring the first.
static Function	reciprocal reciprocal: C\{0}->C; x |-> x-1 = 1 / x.
static Function	sec sec: R->R; x |-> sec x = 1 / cos(x).
static Function	sech sech: R->R; x |-> sech x = 1/cosh(x).
static Function	sign sign: A->{-1,0,1}; x |-> -1 if x<0, x |-> 0 if x=0, x |-> 1 if x>0.
static Function	sin sin: C->C; x |-> sin x = Σn=0 inf (-1)n * x2n+1 / (2n+1)!.
static Function	sinh sinh: C->C; x |-> sinh x = (ex-e-x) / 2 = Σn=0 inf x2n+1 / (2n+1)!.
static Function	sqrt sqrt sqrt : C->C; x |-> sqrt x = x1/2.
static Function	square square: R->R; x |-> x2 .
static Function	tan tan: C\(π/2+πZ)->C; x |-> tan x = sin x / cos x.
static Function	tanh tanh: C\(πi/2*Z)->C; x |-> tanh x = sinh x / cosh x.
static Function	zero zero: R->R; x |-> 0 .

Method Summary

static BinaryFunction	binaryConstant(Arithmetic a) constant â: R×R->R; (x,y) |-> a.
static BinaryFunction	binarySymbolic(java.lang.String name) symbolic f:R×R->R; (x,y) |-> f(x, y).
static Function	constant(java.lang.Object a) constant â: R->R; x |-> a .
static int	delta(int i, int j)
static Function	exp(Arithmetic b) expb: C->C\{0}; x |-> bx .
static Function	linear(Arithmetic a) linear: A->B; x |-> a*x.
static Function	piecewise(Predicate[] cond, Function[] value) Get a function defined piecewise.
static UnivariatePolynomial	polynom(int degree) polynom: R'->R'; X|->Σi=0d Xi.
static UnivariatePolynomial	polynom(Vector coeff) polynom: R'->R'; X|->Σi=0d aiXi.
static Function	pow(Arithmetic p) powp: R->R; x |-> xp .
static Function	pow(double p)
static Function	projection(int component) projection πc: An->A; (x1,...xn)T |-> xc.
static Function	projection(int i, int j) projection πi,j: An×m->A; (xi,j) |-> xi,j.
static Function	step(Real t) step ht: A->{0,1}; x |-> 1 if x>=t, x |-> 0 if x<t.
static Function	symbolic(java.lang.String name) symbolic f:R->R; x |-> f(x).

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

functions

public static final Functions functions

Class alias object.

To alias the methods in this class, use an idiom like

 // alias object
 Functions F = Functions.functions;
 Operations op = Operations.operations;
 // use alias
 Function f = (Function) op.times.apply(F.sin, op.plus.apply(F.square, F.cos));
 // instead of the long form
 Function f = (Function) Operations.times.apply(Functions.sin, Operations.plus.apply(Functions.square, Functions.cos));
 

zero

public static final Function zero

zero: R->R; x |-> 0 .

one

public static final Function one

one: R->R; x |-> 1 .

id

public static final Function id

id: R->R; x |-> x .

derive: id' = 1
integrate: integral id dx = x²/2

reciprocal

public static final Function reciprocal

reciprocal: C\{0}->C; x |-> x^-1 = 1 / x.

derive: reciprocal' = -x^-2
integrate: integral x^-1 dx = log x

It is generally preferred to use Operations.inverse instead!

See Also:

Operations.inverse

square

public static final Function square

square: R->R; x |-> x² .

derive: square' = 2*id
integrate: integral x² dx = x³/3

Implementation uses faster x*x for Values.

sqrt

public static final Function sqrt

sqrt sqrt : C->C; x |-> sqrt x = x^1/2.

derive: sqrt x' = -1/2/ sqrt x
integrate: integral sqrt xdx = 2/3*x^3/2

Implementation uses faster Math.sqrt(double) for scalars in [0, inf ).

For complex numbers z=r*e^i*φ!=0 this function returns

sqrt |z|*e^i*φ/2 = sqrt r * (cos(φ/2) + i*sin(φ/2))

But just like real numbers, the negative of this is a square root as well.

exp

public static final Function exp

exp: C->C\{0}; x |-> e^x = Σ_n=0 ^inf xⁿ / n!.

derive: (e^x)' = e^x
integrate: integral e^x dx = e^x

For complex numbers x = a + i*b in C; a,b in R this function returns e^{a + i*b} = e^ae^i*b = e^a * (cos(b) + i*sin(b)).

exp-function is exactly 2kπi-periodic. This is due to the relation for all z in C e^z+ω = e^z <=> e^ω = 1 <=> ω = 2kπi; k in Z.

See Also:

log, ValueFactory.polar(Real,Real)

log

public static final Function log

log: C\{0}->C; x |-> log_e x.

derive: log' = 1 / x
integrate: integral log x dx = x*log |x| - x

For complex numbers x=r*e^iφ in C this function returns the principal logarithm

log(z) = log(r) + i*φ = log |z| + i*arg(z)

But adding 2kπi will lead to all other (complex) logarithms. These multiple logarithms of complex numbers result from exp-function being 2kπi periodic.

See Also:

exp

sin

public static final Function sin

sin: C->C; x |-> sin x = Σ_n=0 ^inf (-1)ⁿ * x²ⁿ⁺¹ / (2n+1)!. Sine function.

derive: sin' = cos
integrate: integral sin x dx = -cos x

For complex numbers z this function returns sin z = sinh(i*z)/i = (e^i*z-e^-i*z) / (2i).

functional equations: sin(x+y) = sin x * cos y + cos x * sin y, cos(x+y) = cos x * cos y - sin x * sin y.
sin(x)² + cos(x)² = 1

This sinus looks like ∿.

See Also:

arcsin, sinh

cos

public static final Function cos

cos: C->C; x |-> cos x = Σ_n=0 ^inf (-1)ⁿ * x²ⁿ / (2n)!. Cosine function

derive: cos' = -sin
integrate: integral cos x dx = sin x

For complex numbers z this function returns cos z = cosh(i*z) = (e^i*z+e^-i*z) / 2.

reductions: cos x = sin(π/2+x)

See Also:

arccos, cosh

tan

public static final Function tan

tan: C\(π/2+πZ)->C; x |-> tan x = sin x / cos x. Tangent function.

derive: tan' = sec² = 1/cos² = 1 + tan²
integrate: integral tan x dx = - log(cos x)

For complex numbers z this function is tan z = tanh(i*z)/i.

See Also:

arctan, cot

cot

public static final Function cot

cot: C\πZ->C; x |-> cot x = cos x / sin x = 1 / tan x. Cotangent function.

derive: cot' = -csc² = -(1 + cot²).

integrate: integral cot x dx = log(sin x)

For complex numbers z this function is cot z = coth(i*z)*i.

See Also:

tan

csc

public static final Function csc

csc: R\{0}->R; x |-> csc x = 1 / sin(x). Cosecant function.

derive: csc' = -cot*csc

sec

public static final Function sec

sec: R->R; x |-> sec x = 1 / cos(x). Secant function.

derive: sec' = sec*tan.

arcsin

public static final Function arcsin

arcsin: [-1,1]->[-π/2,π/2]; x |-> arcsin x = sin^-1 x. Arc sine function.

derive: arcsin' = 1 / sqrt 1 - x²
integrate: integral arcsin x dx = x * arcsin x + sqrt 1 - x²

arcsin x = Σ_n=0 ^inf (-1)ⁿ * nCr(-1/2, n) * x²ⁿ⁺¹ / (2n+1) on (-1,1).

See Also:

sin

arccos

public static final Function arccos

arccos: [-1,1]->[0,π]; x |-> arccos x = cos^-1 x. Arc cosine function.

derive: arccos' = - 1 / sqrt 1 - x²
integrate: integral arccos x dx = x * arccos x - sqrt 1 - x²

arccos x = π/2 - arcsin x.

See Also:

cos

arctan

public static final Function arctan

arctan: R->(-π/2,π/2); x |-> arctan x = tan^-1 x. Arc tangent function.

derive: arctan' = 1 / (1 + x²)
integrate: integral arctan x dx = x * arctan x - log(x² + 1)/2

arctan x = Σ_n=0 ^inf (-1)ⁿ * x²ⁿ⁺¹ / (2n+1) on [-1,1].

See Also:

tan

arccot

public static final Function arccot

arccot: R->(0,π); x |-> arccot x = cot^-1 x. Arc cotangent function.

derive: arccot' = - 1 / (1 + x²)
integrate: integral arccot x dx = x * arccot x + log(x² + 1)/2

arccot x = π/2 - arctan x.

See Also:

cot

sinh

public static final Function sinh

sinh: C->C; x |-> sinh x = (e^x-e^-x) / 2 = Σ_n=0 ^inf x²ⁿ⁺¹ / (2n+1)!. Hyperbolic sine function.

derive: sinh' = cosh
integrate: integral sinh x dx = cosh x

functional equations: sinh(x+y) = sinh x * cosh y + cosh x * sinh y, cosh(x+y) = cosh x * cosh y + sinh x * sinh y.
cosh(x)² - sinh(x)² = 1
(cosh x+ sinh x)ⁿ = cosh(n*x) + sinh(n*x)

See Also:

arsinh

cosh

public static final Function cosh

cosh: C->C; x |-> cosh x = (e^x+e^-x) / 2 = Σ_n=0 ^inf x²ⁿ / (2n)!. Hyperbolic cosine function.

derive: cosh' = sinh
integrate: integral cosh x dx = sinh x

See Also:

arcosh

tanh

public static final Function tanh

tanh: C\(πi/2*Z)->C; x |-> tanh x = sinh x / cosh x. Hyperbolic tangent function.

derive: tanh' = sech² = 1/cosh(x)² = (1 + tanh x) * (1 - tanh x) = 1 - (tanh x)²
integrate: integral tanh x dx = log(cosh x)

The hyperbolic tangent is a sigmoid function.

See Also:

coth

csch

public static final Function csch

csch: R\{0}->R; x |-> csch x = 1 / sinh(x). Hyperbolic cosecant function.

derive: csch' = -coth*csch.

sech

public static final Function sech

sech: R->R; x |-> sech x = 1/cosh(x). Hyperbolic secant function.

derive: sech' = -sech*tanh.

coth

public static final Function coth

coth: R\{0}->R; x |-> coth x = cosh x / sinh x = 1 / tanh x. Hyperbolic cotangent function.

derive: coth' = -csch².

integrate: integral coth x dx = log(sinh x)

|coth 0| = inf is a singularity.

See Also:

tanh

arsinh

public static final Function arsinh

arsinh: R->R; x |-> arsinh x = sinh^-1 x = log(x + sqrt x²+1). Area hyperbolic sine function.

derive: arsinh' = 1 / sqrt x²+1)
integrate: integral arsinh x dx = x * arsinh x - sqrt x²+1)

See Also:

sinh

arcosh

public static final Function arcosh

arcosh: [1, inf )->[0, inf ); x |-> arcosh x = (cosh|_{[0, inf )})^-1 x = log(x ± sqrt x²-1). Area hyperbolic cosine function.

derive: arcosh' = 1 / sqrt x²-1)
integrate: integral arcosh x dx = x*arcosh x - sqrt x²-1)

See Also:

sinh

artanh

public static final Function artanh

artanh: (-1,1)->R; x |-> artanh x = tanh^-1 x = log((1+x) / (1-x)) / 2. Area hyperbolic tangent function.

derive: artanh' = 1 / (1 - x²)
integrate: integral artanh x dx = x*artanh x + log(x²-1) / 2

See Also:

tanh

arcoth

public static final Function arcoth

arcoth: R\[-1,1]->R\{0}; x |-> arcoth x = coth^-1 x = log((x+1) / (x-1)) / 2. Area hyperbolic cotangent function.

derive: arcoth' = 1 / (1 - x²)
integrate: integral arcoth x dx = x*arcoth x + log(x²-1) / 2

See Also:

coth

norm

public static final Function norm

norm: A->[0, inf ); x |-> ||x||.

derive: norm'|_Rⁿ\{0} = x / ||x||. which is true for the euclidian 2-norm ||x||₂ = sqrt x.x, only.

derive: abs'|_{(- inf ,0)} = -1, abs'|_{(0, inf )} = 1.

See Also:

Vector.multiply(orbital.math.Vector)

nondet

public static final Function nondet

Represents a nondeterministic function.

This nondeterministic function returns results randomly. It is provided for theoretical reasons and cannot be used as a random generator.

logistic

public static final Function logistic

logistic: A->(0,1); x |-> 1 / (1 + e^-x).

derive: logistic' = e^-x / (1 + e^-x)² = logistic(x) * (1 - logistic(x)).
integrate: integral logistic(x)dx =

The logistic function is a sigmoid function, and resembles the continuous logistic distribution.

See Also:

sign

sign

public static final Function sign

sign: A->{-1,0,1}; x |-> -1 if x<0, x |-> 0 if x=0, x |-> 1 if x>0.

derive: sign' = diracDelta = 0 on R\{0}.

See Also:

step(Real), logistic

diracDelta

public static final Function diracDelta

diracDelta δ: M\{0}->{0}; x |-> 0 if x!=0.

derive: δ' = δ ??.
integrate: integral δ(x)dx = step₀(x)

See Also:

delta, step(Real), piecewise(Predicate[], Function[])

projectFirst

public static final BinaryFunction projectFirst

Projects to the first argument, ignoring the second.

projectFirst: (x,y) |-> x.

Equals Functionals.onFirst(id)

Evolves: might be renamed.

projectSecond

public static BinaryFunction projectSecond

Projects to the second argument, ignoring the first.

projectSecond: (x,y) |-> y.

Equals Functionals.onSecond(id)

Evolves: might be renamed.

delta

public static final BinaryFunction delta

delta: R×R->R; (x,x) |-> 1, (x,y) |-> 0 for x!=y. Kronecker-delta function.

See Also:

diracDelta

Method Detail

constant

public static final Function constant(java.lang.Object a)

constant â: R->R; x |-> a .

derive: â' = 0
integrate: integral adx = a*x

symbolic

public static final Function symbolic(java.lang.String name)

symbolic f:R->R; x |-> f(x).

derive: (f)' = f'
integrate: integral f(x)dx = integral f(x)dx

Parameters:

name - the name of the symbolic function.

Returns:

a pure symbolic function with a specified name.

linear

public static final Function linear(Arithmetic a)

linear: A->B; x |-> a*x.

derive: linear' = a.
integrate: integral a*x dx = a*x²/2

linear functions are Lipschitz-continuous.

The concrete sets A and B depend on the exact type of a. For instance if a is a Matrix in K^m×n this function is the linear homomorphism Kⁿ->K^m; x |-> a*x.

pow

public static final Function pow(Arithmetic p)

pow_p: R->R; x |-> x^p .

derive: (x^p)' = p*x^p-1
integrate: integral x^p dx = x^p+1/(p+1)

pow

public static final Function pow(double p)

exp

public static final Function exp(Arithmetic b)

exp_b: C->C\{0}; x |-> b^x .

derive: (b^x)' = log b * b^x
integrate: integral b^x dx = b^x / log b

projection

public static final Function projection(int component)

projection π_c: Aⁿ->A; (x₁,...x_n)^T |-> x_c.

projection

public static final Function projection(int i,
                                        int j)

projection π_i,j: A^n×m->A; (x_i,j) |-> x_i,j.

polynom

public static final UnivariatePolynomial polynom(int degree)

polynom: R'->R'; X|->Σ_i=0^d Xⁱ.

This method will return the polynom in R[X]_d of the given degree that corresponds to the Vandermonde Matrix so all coefficients are 1. dim R[X]_d = d+1. The ring R' is "compatible" with R.

See Also:

polynom(Vector), UnivariatePolynomial

polynom

public static final UnivariatePolynomial polynom(Vector coeff)

polynom: R'->R'; X|->Σ_i=0^d a_iXⁱ.

This method will return the polynom in R[X] with the specified coefficients vector in R^d. Although dim R[X] = inf this method will return a polynomial in R[X]_d of degree =<d:=coeff.dimension(). The ring R' is "compatible" with R.

See Also:

polynom(int), ValueFactory.asPolynomial(Vector), UnivariatePolynomial

step

public static final Function step(Real t)

step h_t: A->{0,1}; x |-> 1 if x>=t, x |-> 0 if x<t.

derive: step_t' = diracDelta(t-x).
integrate: integral step_t(x)dx = step_t(x) * (x - t)

Step function alias Heaviside function.

See Also:

sign, diracDelta

piecewise

public static final Function piecewise(Predicate[] cond,
                                       Function[] value)

Get a function defined piecewise. piecewise: A->B; x|->f_{min {i ¦ ci(x) and 1=<i=<m}}(x).

deriving this function requires it to be in C¹(A, B). Similarly, integrating requires it to be integrable, at all. Unless you make sure that these requirements are met, the implementation will return values that are completely meaningless.

Note that piecewise functions can be defined in terms of appropriate compositions with heaviside functions which in turn can be defined as translations of the unit step function.

Parameters:

cond - the condition predicates c_i.

value - the functions of whom the first one is applied whose associated predicate yields true.

Throws:

java.lang.IllegalArgumentException - if no condition predicate matches for an argument x.

See Also:

step(orbital.math.Real)

Preconditions:

cond.length == value.length

binaryConstant

public static final BinaryFunction binaryConstant(Arithmetic a)

constant â: R×R->R; (x,y) |-> a.

derive: â' = 0.
integrate: integral adx_i = a*x_i

binarySymbolic

public static final BinaryFunction binarySymbolic(java.lang.String name)

symbolic f:R×R->R; (x,y) |-> f(x, y).

derive: (f)' = f'
integrate: integral f(x,y)dx_i = integral f(x,y)dx_i

Parameters:

name - the name of the symbolic function.

Returns:

a pure symbolic function with a specified name.

delta

public static final int delta(int i,
                              int j)