Polynomial Curve Fitting

A procedure in which the basic problem is to pass a curve through a set of points, representing experimental data, in such a way that the curve shows as well as possible the relationship between the two quantities plotted. It is always possible to pass some smooth curve through all the points plotted, but since there is assumed to be some experimental error present, such a procedure would ordinarily not be desirable.

Algebraic fit versus geometric fit for curves

For algebraic analysis of data, "fitting" usually means trying to find the curve that minimizes the vertical (y-axis) displacement of a point from the curve (e.g., ordinary least squares). However for graphical and image applications geometric fitting seeks to provide the best visual fit; which usually means trying to minimize the orthogonal distance to the curve (e.g., total least squares), or to otherwise include both axes of displacement of a point from the curve. Geometric fits are not popular because they usually require non-linear and/or iterative calculations, although they have the advantage of a more aesthetic and geometrically accurate result.

Suppose a collection of date is represented by n points in the xy-plane

 

(X1,Y1), ( X2, Y2 ) ,..., (Xn, Yn)  

 

and you are asked to find a polynomial function of degree n-1 

 

 

whose graph passes through the specified points. This procedure is called polynomial curve fitting. If all x- coordinates of the points are distinct,  then there is precisely one polynomial function of degree n-1 (or less) that fits the n points. 

 

To solve for the n coefficients of p(x) , substitute each pf the n points into the polynomial function and obtain n linear equations in n variables a0, a1, 12 ,... an-1

 

 

Sources : http://www.answers.com/topic/curve-fitting#ixzz379zcn2pV