Adjusting the recto/curvlinearity of a polynomial fit

Neil3030

I was wondering if a procedure exists (ideally one that can be programmed in MATLAB) to adjust how curvlinear or rectolinear a polynomial fit is.

For example:

Computing 10 points around 0:

[HTML]apexes = randn(1,11);
x = [-5 -4 -3 -2 -1 0 1 2 3 4 5];
plot(x,apexes,'xk');hold on;
axis([-6 6 -6 6]);[/HTML]



and fitting a 5th order polynomial (for the purpose of demonstration) across a range of x values between -5 and 5 in steps of .01.

[HTML]x2 = -5:.01:5;
order = polyfit(x,apexes,5);
curve = polyval(order,x2);
plot(x2,curve);[/HTML]



changing the rate of change in x2 to whole units, the fitted curve becomes rectolinear; at least, this is the term I have been using to describe peaks and troughs that have sudden angular change, as opposed to gradual curves.

[HTML]x2 = -5:1:5;
order = polyfit(x,apexes,5);
curve = polyval(order,x2);
plot(x2,curve);[/HTML]



What I'd now like to do is vary the degree to which the curves are curvelinear and rectolinear in a graded fashion. Is this possible? Fitting higher order polynomials kind of achieves this, but (a) they go a bit wonky once the order excedes the number of datapoints and (b) I'd ideally like a procedure that won't alter the overall curvelength. I would also like something that maintains the number of kinks in the curve, as opposed to, say, simply adjusting the rate of change of x2 for a value between 0 and 1, in which case you get something like this:

[HTML]x2 = -5:.5:5;
order = polyfit(x,apexes,5);
curve = polyval(order,x2);
plot(x2,curve);[/HTML]



Any help greatly appreciated, I hope I've explained that well!

Michael Collins

I don't have an answer to your question, but I do have a question about your question!

It's almost like you're suggesting that the fit itself is changing? I'm guessing you're not because you seem fairly familiar with MATLAB...

All that's happening is that MATLAB is evaluating the polynomial at fewer points, and hence the curve appears to be (but is not actually) developing corners - which of course a polynomial can never have.

If you want the actual fit itself to change, you'll need to look at a different interpolation method (splines, for example).