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Double exponential function
 
 
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A double exponential function is a constant raised to the power of an exponential function
Exponential function

The exponential function is a function in mathematics. The application of this function to a value x is written as exp. Equivalently, this can be written in the form ex, where e is the mathematical constant that is the base of the natural logarithm and that is also known as Euler's number....
. The general formula is , which grows even faster than an exponential function. For example, if a = b = 10:

Factorial
Factorial

In mathematics, the factorial of a negative and non-negative numbers integer n, denoted by n!, is the Product of all positive integers less than or equal to n....
s grow faster than exponential functions, but much slower than double-exponential functions. The hyper-exponential function
Tetration

In mathematics, tetration is an iterated function exponential function, the first hyper operator after exponentiation. The portmanteau tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration....
 and Ackermann function
Ackermann function

In computability theory, the Ackermann function or Ackermann?P?ter function is a simple example of a computable function that is not Primitive recursive function....
 grow even faster. See Big O notation
Big O notation

In mathematics, big O notation describes the asymptotic analysis of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions....
 for a comparison of the rate of growth of various functions.

The inverse of a double exponential function is a double logarithm
Logarithm

In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number....
.

and Sloane observed that in several important integer sequence
Integer sequence

In mathematics, an integer sequence is a sequence of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms....
s, each term is a constant plus the square of the previous term, and show that such sequences can be formed by rounding to the nearest integer the values of a doubly exponential function in which the middle exponent is two.






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A double exponential function is a constant raised to the power of an exponential function
Exponential function

The exponential function is a function in mathematics. The application of this function to a value x is written as exp. Equivalently, this can be written in the form ex, where e is the mathematical constant that is the base of the natural logarithm and that is also known as Euler's number....
. The general formula is , which grows even faster than an exponential function. For example, if a = b = 10:
  • f(-1) ˜ 1.26
  • f(0) = 10
  • f(1) = 1010
  • f(2) = 10100 = googol
    Googol

    A googol is the large number 10100, that is, the numerical digit 1 followed by one hundred 0 .The term was coined in 1938 by Milton Sirotta , nephew of American mathematician Edward Kasner....
  • f(3) = 101000
  • f(100) = 1010100 = googolplex
    Googolplex

    A googolplex is the number 10googol, which can also be written as the number 1 followed by a googol of 0 ....
    .


Factorial
Factorial

In mathematics, the factorial of a negative and non-negative numbers integer n, denoted by n!, is the Product of all positive integers less than or equal to n....
s grow faster than exponential functions, but much slower than double-exponential functions. The hyper-exponential function
Tetration

In mathematics, tetration is an iterated function exponential function, the first hyper operator after exponentiation. The portmanteau tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration....
 and Ackermann function
Ackermann function

In computability theory, the Ackermann function or Ackermann?P?ter function is a simple example of a computable function that is not Primitive recursive function....
 grow even faster. See Big O notation
Big O notation

In mathematics, big O notation describes the asymptotic analysis of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions....
 for a comparison of the rate of growth of various functions.

The inverse of a double exponential function is a double logarithm
Logarithm

In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number....
.

Doubly exponential sequences

Aho and Sloane observed that in several important integer sequence
Integer sequence

In mathematics, an integer sequence is a sequence of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms....
s, each term is a constant plus the square of the previous term, and show that such sequences can be formed by rounding to the nearest integer the values of a doubly exponential function in which the middle exponent is two. Integer sequences with this squaring behavior include

  • The Fermat number
    Fermat number

    In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a natural number of the formwhere n is a nonnegative integer....
    s
  • The Double Mersenne number
    Double Mersenne number

    In mathematics, a double Mersenne number is a Mersenne prime of the formwhere p is a Mersenne prime exponent....
    s
  • The elements of Sylvester's sequence
    Sylvester's sequence

    In number theory, Sylvester's sequence is a sequence of integers in which each member of the sequence is the product of the previous members, plus one....
     
where E ˜ 1.264084735305 is Vardi's constant.
  • The number of k-ary
    Arity

    In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the number of domains in the corresponding Cartesian product....
     operators
    Logical connective

    In logic, two sentences may be joined by means of a logical connective to form a compound sentence. The truth-value of the compound is uniquely determined by the truth-values of the simpler sentences....
    :


More generally, if the nth value of an integer sequences is proportional to a double exponential function of n, Ionascu and Stanica call the sequence "almost doubly-exponential" and describe conditions under which it can be defined as the floor of a doubly-exponential sequence plus a constant. Additional sequences of this type include

  • The prime numbers 2, 11, 1361, ...
where A ˜ 1.306377883863 is Mills' constant
Mills' constant

In number theory, Mills' constant is defined as the smallest positive real number A; such that the floor function of the double exponential function...
.


Applications


Algorithmic complexity

In computational complexity theory
Computational complexity theory

Computational complexity theory, as a branch of the theory of computation in computer science, investigates the problems related to the Computational resource required for the execution of algorithms , and the inherent difficulty in providing efficient algorithms for specific computational problems....
, some algorithms take double-exponential time:
  • Decision procedures for Presburger arithmetic
    Presburger arithmetic

    Presburger arithmetic is the first-order predicate calculus of the natural numbers with addition, named in honor of Mojzesz Presburger, who introduced it in 1929....
  • Computing a Gröbner basis
    Gröbner basis

    In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gr?bner basis is a particular kind of generating subset of an ring ideal I in a polynomial ring R....
     (in the worst case)
  • Finding a complete set of associative-commutative unifiers
  • Satisfying CTL+ (which is, in fact, 2-EXPTIME
    2-EXPTIME

    In computational complexity theory, the complexity class 2-EXPTIME is the Set of all decision problems solvable by a deterministic Turing machine in big O notation time, where p is a polynomial function of n....
    -complete)
  • Quantifier elimination
    Quantifier elimination

    Quantifier elimination is a concept in mathematical logic, model theory, and theoretical computer science. One way of classifying Well-formed formula is by the amount of quantifiers....
     on real closed field
    Real closed field

    In mathematics, a real closed field is a Field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers....
    s takes at least doubly-exponential time (but is not even known to be computable in ELEMENTARY
    ELEMENTARY

    In computational complexity theory, the complexity class ELEMENTARY is the union of the classes in the exponential hierarchy.The name was coined by Laszlo Kalmar, in the context of recursive functions and undecidability; most problems in it are far from elementary....
    )


Number theory

Some number theoretical
Number theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
 bounds are double exponential. Odd perfect numbers with n distinct prime factors are known to be at most a result of Nielsen (2003).

The Largest known prime number in the electronic era has grown roughly as a double exponential function of the year since Miller
J. C. P. Miller

Jeffrey Charles Percy Miller was an England mathematician and computing pioneer. He worked in number theory and on geometry, particularly polyhedra, where Miller's monster refers to the Great dirhombicosidodecahedron....
 and Wheeler found a 79-digit prime on EDSAC
EDSAC

Electronic Discrete Storage Automatic Calculator was an early United Kingdom computer. The machine, having been inspired by John von Neumann's seminal First Draft of a Report on the EDVAC, was constructed by Maurice Wilkes and his team at the University of Cambridge University of Cambridge Mathematical Laboratory in England....
1 in 1951.

Theoretical biology

In population dynamics
Population dynamics

Population dynamics is the branch of life sciences that studies short- and long-term changes in the size and age composition of populations, and the biology and environment processes influencing those changes....
 the growth of human population is sometimes supposed to be double exponential. Gurevich and Varfolomeyev experimentally fit where N(y) is the population in year y in millions.

Physics

In the Oscillator Toda
Oscillator Toda

Oscillator Toda is special kind of nonlinear oscillator; it is vulgarization of the Toda field theory, which refers to a continuous limit of Toda's chain, of chain of particles, with exponential potential of interaction between neighbors...
 model of self-pulsation
Self-pulsation

Self-pulsation takes place at the beginning of laser action.As the pump is switched on, the gainin the active medium rises and exceeds the steady-state value....
, the logarithm of amplitude varies exponentially with time (for large amplitudes), thus the amplitude varies as double-exponential function of time.