Quadratic growth

In

, ${\displaystyle f(x)=\Theta (x^{2})}$.^[1] This can be defined both continuously (for a real-valued function of a real variable) or discretely (for a sequence of real numbers, i.e., real-valued function of an integer or natural number variable).

Examples

Examples of quadratic growth include:

• Any
  quadratic polynomial
  .
• Certain integer sequences such as the triangular numbers. The ${\displaystyle n}$th triangular number has value ${\displaystyle n(n+1)/2}$, approximately ${\displaystyle n^{2}/2}$.

For a real function of a real variable, quadratic growth is equivalent to the

of the third derivative operator ${\displaystyle D^{3}}$. Similarly, for a sequence (a real function of an integer or natural number variable), quadratic growth is equivalent to the second (if discrete).

Algorithmic examples include:

• The amount of time taken in the worst case by certain algorithms, such as insertion sort, as a function of the input length.^[3]
• The numbers of live cells in space-filling cellular automaton patterns such as the breeder, as a function of the number of time steps for which the pattern is simulated.^[4]
• Metcalfe's law stating that the value of a communications network grows quadratically as a function of its number of users.^[5]

See also

• Exponential growth

References

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