Recursive Exponential Function

Recursion in recursive exponential function is the method of programming whereby a function directly or indirectly calls itself. Recursion is frequently presented as an alternative to iteration. There are different types of recursion, they are:Linear, Tail, Binary, Exponential, Nested, Mutual recursions.

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Recursive exponential function or Exponential recursion

In Exponential  recursive function where more than one call (normally we take this function for more than two calls)is made to the function  within itself. This function leads to exponential growth in the number of recursive calls.An exponential recursive function is one that, if we want  to draw out a representation of all the function calls, we must have an exponential number of calls in relation to the size of the data set (exponential meaning if there were n elements, there would be O(an) function calls where a is a positive number).

A simple and good example for an exponentially recursive function is a function to compute all the permutations of a given data set. typically, complicated exponential recursive functions are carried out by computers, with specially written programs. However, there is primitive(fundamental) method of exponential recursive functions , that we can restrict to pen and paper.

primitive recursive exponential function

The  recursive exponential function in primitive method follows the following rules.

1. Take n natural numbers as inputs and produce a natural number

2. Basic primitive recursive  functions are given by three axioms, they are

a)The “Constant Function” zero. Z = λx.0

b)The “Successor Function”. S = λx.(x+1)

c)The “Projection Function” Pin.  Pin= λx1⋅⋅⋅xn.xi

Primitive recursion exponential function method allows us to reduce complicated mathematical exponential function to very simple operations

Computing Exponential recursive function

•The exponential function computes xy

•xyis simply x multiplied by itself y times

•x⋅xis simply x added to itself x times

•x+x is simply x+1 or S(x) x times


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Mathematically, Recursive definition in exponential function is defined as-

• Exp function f = λx,y.(x^ y)

f(x,0)= 1

f(x,y+1) = f(x,y).x

• In recursive notation

f(x,0) = g(x)

f(x,y+1) = h(x,y,f(x,y))