Algebraic Function

An algebraic function is a scalar-output function with a formal specification composed of algebraic expressions (polynomial equation with integer coefficients) that specifies a set of scalar-output function instances).

• Context:
  • It can range from being a Scalar-Input Scalar-Output Algebraic Function to being a Vector-Input Scalar-Output Algebraic Function.
  • It can range from being a Continuous Algebraic Function to being a Discontinuous Algebraic Function.
  • It can range from being a Function with a Finite Algebraic Expression to being a Function with an Infinite Algebraic Expression.
  • It can be associated to an Algebraic Function Family.
  • It can be represented by an Algebraic Function Structure.
  • …
• Example(s):
  • a Linear Function, such as [math]\displaystyle{ f(x) = 2.3 x + 6 }[/math].
  • a Polynomial Function.
  • a Vector-Input Polynomial Function of [math]\displaystyle{ f(x,y) = \frac{1}{3} y^2 − 2.3 x + 6 }[/math].
  • a Probability Function Family.
  • …
• Counter-Example(s):
  • a Vector-Output Function.
  • an Algebraic Equation.
  • a Transcendental Function.
  • a Class-Output Function, such as a Boolean function.
  • a Software Function.
• See: Bound Parameter, Measure Function, Metric Space, Algebraic Sentence, Algebraic Operations.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/list_of_mathematical_functions#Algebraic_functions Retrieved:2015-6-14.
  • Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.
    • Polynomials: Can be generated by addition, multiplication, and exponentiation alone.
      • Constant function: polynomial of degree zero, graph is a horizontal straight line
      • Linear function: First degree polynomial, graph is a straight line.
      • Quadratic function: Second degree polynomial, graph is a parabola.
      • Cubic function: Third degree polynomial.
      • Quartic function: Fourth degree polynomial.
      • Quintic function: Fifth degree polynomial.
      • Sextic function: Sixth degree polynomial.
    • Rational functions: A ratio of two polynomials.
    • Nth root.
      • Square root: Yields a number whose square is the given one [math]\displaystyle{ x^{\frac{1}{2}} \!\ }[/math] .
      • Cube root: Yields a number whose cube is the given one [math]\displaystyle{ x^{\frac{1}{3}} \!\ }[/math] .

2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/algebraic_function Retrieved:2014-11-23.
  • In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions can be expressed using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power:  :[math]\displaystyle{ f(x)=1/x, f(x)=\sqrt{x}, f(x)=\frac{ \sqrt{1+x^3}}{x^{3/7}-\sqrt{7} x^{1/3}} }[/math] are typical examples.

    However, some algebraic functions cannot be expressed by such finite expressions (as proven by Galois and Niels Abel), as it is for example the case of the function defined by  : [math]\displaystyle{ f(x)^5+f(x)^4+x=0 }[/math].

    In more precise terms, an algebraic function of degree n in one variable x is a function [math]\displaystyle{ y = f(x) }[/math] that satisfies a polynomial equation  : [math]\displaystyle{ a_n(x)y^n+a_{n-1}(x)y^{n-1}+\cdots+a_0(x)=0 }[/math] where the coefficients ai(x) are polynomial functions of x, with coefficients belonging to a set S.

    Quite often, [math]\displaystyle{ S=\mathbb Q }[/math], and one then talks about "function algebraic over [math]\displaystyle{ \mathbb Q }[/math]", and the evaluation at a given rational value of such an algebraic function gives an algebraic number.

    A function which is not algebraic is called a transcendental function, as it is for example the case of [math]\displaystyle{ \exp(x), \tan(x), \ln(x), \Gamma(x) }[/math]. A composition of transcendental functions can give an algebraic function: [math]\displaystyle{ f(x)=\cos (\arcsin(x)) = \sqrt{1-x^2} }[/math].

    As an equation of degree n has n roots, a polynomial equation does not implicitly define a single function, but n

    functions, sometimes also called branches. Consider for example the equation of the unit circle:

    [math]\displaystyle{ y^2+x^2=1.\, }[/math]

    This determines y, except only up to an overall sign; accordingly, it has two branches:

    [math]\displaystyle{ y=\pm \sqrt{1-x^2}.\, }[/math]

    An algebraic function in m variables is similarly defined as a function y which solves a polynomial equation in m + 1 variables:  :[math]\displaystyle{ p(y,x_1,x_2,\dots,x_m)=0.\, }[/math]

    It is normally assumed that p should be an irreducible polynomial. The existence of an algebraic function is then guaranteed by the implicit function theorem.

    Formally, an algebraic function in m variables over the field K is an element of the algebraic closure of the field of rational functions K(x₁,...,x_m).

2013

• http://www.mathworks.com/help/optim/ug/writing-objective-functions.html
  • Many Optimization Toolbox™ solvers minimize a scalar function of a multidimensional vector. The objective function is the function the solvers attempt to minimize. Several solvers accept vector-valued objective functions, and some solvers use objective functions you specify by vectors or matrices.

2009

• http://en.wiktionary.org/wiki/algebraic_function
  • (Algebra) Any function that only uses the operations of addition, subtraction, multiplication, division and raising to a rational power.

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