Quadratic growth

In mathematics, a function or sequence is said to exhibit quadratic growth when its values are proportional to the square of the function argument or sequence position. "Quadratic growth" often means more generally "quadratic growth in the limit", as the argument or sequence position goes to infinity – in big Theta notation, f(x) = Θ(x²).^[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 nth triangular number has value n(n+1)/2, approximately n²/2.

For a real function of a real variable, quadratic growth is equivalent to the second derivative being constant (i.e., the third derivative being zero), and thus functions with quadratic growth are exactly the quadratic polynomials, as these are the kernel of the third derivative operator D³. Similarly, for a sequence (a real function of an integer or natural number variable), quadratic growth is equivalent to the second finite difference being constant (the third finite difference being zero),^[2] and thus a sequence with quadratic growth is also a quadratic polynomial. Indeed, an integer-valued sequence with quadratic growth is a polynomial in the zeroth, first, and second binomial coefficient with integer values. The coefficients can be determined by taking the Taylor polynomial (if continuous) or Newton polynomial (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

1. Moore, Cristopher; Mertens, Stephan (2011), The Nature of Computation, Oxford University Press, p. 22, ISBN 9780191620805.
2. Kalman, Dan (1997), Elementary Mathematical Models: Order Aplenty and a Glimpse of Chaos, Cambridge University Press, p. 81, ISBN 9780883857076.
3. Estivill-Castro, Vladimir (1999), "Sorting and order statistics", in Atallah, Mikhail J., Algorithms and Theory of Computation Handbook, Boca Raton, FL: CRC, pp. 3-1–3-25, MR 1797171.
4. Griffeath, David; Hickerson, Dean (2003), "A two-dimensional cellular automaton crystal with irrational density", New constructions in cellular automata, St. Fe Inst. Stud. Sci. Complex., New York: Oxford Univ. Press, pp. 79–91, MR 2079729. See in particular p. 81: "A breeder is any pattern which grows quadratically by creating a steady stream of copies of a second object, each of which creates a stream of a third."
5. Rohlfs, Jeffrey H. (2003), "3.3 Metcalfe's law", Bandwagon Effects in High-technology Industries, MIT Press, pp. 29–30, ISBN 9780262681384.