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Fitting Curve With an Inverse Which Fits a Polynomial

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Hi. I have 8 data points with x and y values. I would like to input new y values and interpolate x values.
I am able to input new x values and interpolate y values. I can fit the points with a sixth order polynomial for y vs. x which is valid in the range. However, I cannot fit a polynomial for x vs. y
One approach is to solve the polynomial for each y value; however, I have thousands of y values I want to interpolate for, so it would be extremely computationally intensive.
Does anyone know a faster approach? Thanks!

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Accepted Answer

Ameer Hamza
Ameer Hamza on 5 Oct 2020
Edited: Ameer Hamza on 5 Oct 2020
The inverse of a polynomial is not a polynomial, so you cannot simply interpolate the inverse function. Following shows two approaches
1) fzero()
x = linspace(0, 2, 8);
y = 5*x.^6 + 3*x.^5; % y varies from 0 to 416.
pf = polyfit(x, y, 6);
y_pred = @(x) polyval(pf, x);
% find x, when y = 100;
y_val = 100;
x_val = fzero(@(x) y_pred(x)-y_val, rand);
2) Polynomial root finding. This method gives all possible solutions
x = linspace(0, 2, 8);
y = 5*x.^6 + 3*x.^5; % y varies from 0 to 416.
pf = polyfit(x, y, 6);
% find x, when y = 100;
y_val = 100;
pf(end) = pf(end)-y_val;
x_vals = roots(pf);
x_vals = x_vals(imag(x_vals)==0); % if you only want real roots.

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Ephraim Bryski
Ephraim Bryski on 5 Oct 2020
I used the fzero() approach; it works perfectly and is much faster than solve(). Thank you!
Ameer Hamza
Ameer Hamza on 5 Oct 2020
I am glad to be of help!
Yes, symbolic mathematics is much slower as compared to numerical equivalent.

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