Friday, October 19, 2007

Math stuf is tuff to remember..

How many of you know what is a closed form expression? Here's what wikipedia tells me ( i'd have wasted a lot of time finding out math books if not for wikipedia..)..

"In mathematics, an equation or system of equations is said to have a closed-form solution if, and only if, at least one solution can be expressed analytically in terms of a bounded number of certain "well-known" functions. Typically, these well-known functions are defined to be elementary functions; so infinite series, limits, and continued fractions are not permitted.

For example, the roots of any quadratic equation with complex coefficients can be expressed in closed form in terms of addition, subtraction, multiplication, division, and square root extraction, all elementary functions. However, there are quintic equations without closed-form solutions using elementary functions — see Galois theory."

So, I'm reading up the steerable filter design for feature detection using canny-like criteria which claims to have a closed form solution to its filter design and not a fully computational approach . Got stumped by the first sentence itself and decided to look up on whats the closed form solution. Ok. now lemme proceed.

Here's another one. Whats the Taylor series expansion of an expression? ( deja vu???).. If you wanna approximate a function with the sum of the values of its derivatives at a point, you need the Taylor series . And if that point is x=0, then its called a McLauren series. Here's wiki to the rescue , yet again...

"The Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It may be regarded as the limit of the Taylor polynomials. Taylor series are named in honour of English mathematician Brook Taylor. If the series uses the derivatives at zero, the series is also called a Maclaurin series, named after Scottish mathematician Colin Maclaurin."

As the degree of the Taylor series rises, it approaches the correct function. This image shows sinx and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.

As the degree of the Taylor series rises, it approaches the correct function. This image shows sinx and Taylor approximations,polynomialsof degree 1, 3, 5, 7, 9, 11 and 13.

complex function f that is infinitely differentiable in a neighbourhood of a real or complex number a, is the power series
f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+\cdots

which in a more compact form can be written

\sum_{n=0}^{\infin} \frac{f^{(n)}(a)}{n!} (x-a)^{n}\,,

where n! is the factorial of n and f (n)(a) denotes the nth derivative of f at the point a; the zeroth derivative of f is defined to be f itself and (xa)0 and 0! are both defined to be 1.


THIS COULD BE USEFUL

The Maclaurin series for the exponential function ex at a = 0 is

1 + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!}+ \cdots  \qquad = \qquad  1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120} + \cdots\ .



No comments: