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 = Θ. This can be defined both continuously or discretely.

Examples

Examples of quadratic growth include:
For a real function of a real variable, quadratic growth is equivalent to the second derivative being constant, 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, quadratic growth is equivalent to the second finite difference being constant, 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 or Newton polynomial.
Algorithmic examples include: