Force polynomial fit through multiple points

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Bernoulli Lizard 2013 年 4 月 29 日
I have a set of x, y data that I want to fit to a quadratic polynomial. Is it possible to force the fit through BOTH zero points?
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Kye Taylor 2013 年 4 月 29 日

Remember that a third order polynomial has the form
g(x) = a*x^3 + b*x^2 + c*x + d
A second order polynomial has the form
f(x) = a*x^2 + b*x + c
This same second order polynomial can be written
f(x) = a*(x-x1)*(x-x2)
where x1 and x2 are the roots of the polynomial and a is the coefficient on x^2. So the model y(x) = constant*(x-x1)*(x-x2) is exactly what you want.

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回答 (1 件)

Kye Taylor 2013 年 4 月 29 日

I assume your data is given by two row vectors xData and yData, given for example by
xData = linspace(-2,2);
yData = 2.3*(xData-1).*(xData+1) + 0.2*rand(size(xData));
Then, since you know the roots, try this
% the roots you know
x1 = 1;
x2 = -1;
% the coefficient that makes the model
% y(x) = a*(x-x1)*x-x2) fit the data with
% smallest squared-error In other words
% a minimizes l2-error in a*designMatrix - yData'
designMatrix = ((xData-x1).*(xData-x2))';
a = designMatrix\yData'
plot(xData, yData, 'ko', xData, a*designMatrix, 'r-')
legend('Data','Model')

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