lsqcurvefit with a 'zero' in denominator

Hello Everyone,

I have a data set [xdata,ydata], which are in fact calculated by a Fortran code. All I want to do is to fit these data into a specific nonlinear function. I have a clear definition of how this specific nonlinear function should be, and therefore I am trying to fit into that.

However, in order to try lsqcurvefit for a easier case, I used;

xdata=linspace(0,3,1000)
ydata=trip(xdata)

where trip is

tripl=1+xdata+xdata.^2+2*xdata.^3+153*xdata.^4-100*xdata.^5+(5-xdata).^-1;

Nonlinear function to be fitted is

function ftripl=ftrip(b,x)
ftripl=b(1)+b(2)*x+b(3)*x.^2+b(4)*x.^3+b(5)*x.^4+b(6)*x.^5+(b(7)-x).^-1;
end

I create initial parameters via

guess=4*rand(1,7)-2;

and run

fit=lsqcurvefit(@ftrip,guess,xdata,ydata);

I get decent values for all coefficients save b(7), which should attain the value 5. I recieve b(7)=1.02 with warning 'local minimum is possible'. How can I overcome this problem and find the value I require?

PS. In reality I will fit my data into a function of 5th order polynomial plus arctangent(a/(x-b)) where a,b and five coefficients of the polynomial will be found and x-b becomes very close to zero frequently.

[the cyclist]

@Batuhan,

I edited your question to format the code, but I am not 100% certain that I got your definition of trip exactly right, because you had written

trip = trip1 = ...

and I was not quite sure what you meant there. Please check to see if I interpreted it correctly. Sorry if I messed up your intention.

Maybe even better would be to express trip as an anonymous function:

trip = @(x) 1+x+x.^2+2*x.^3+153*x.^4-100*x.^5+(5-x).^-1;

and then

y = trip(xdata);

[Batuhan]

@the cyclist

No, you are right, I made a mistake there. trip = @(x) 1+x+x.^2+2*x.^3+153*x.^4-100*x.^5+(5-x).^-1;

would be the correct one. Thank you.

Answer 1

[Matt J]

In reality I will fit my data into a function of 5th order polynomial plus arctangent(a/(x-b)) where a,b and five coefficients of the polynomial will be found and x-b becomes very close to zero frequently.

If x-b is near zero at most data points x, you should probably just replace the arctangent term by pi/2*sign(a) and fit the remaining polynomial terms based on those data points only (using POLYFIT, of course, not LSQCURVEFIT). You can try each of the 3 possibilities for sign(a) and see which gives you the best fit.

If you want a good fit to the arctangent term, you'll need more data points away from the singularity.

[Matt J]

If you want a good fit to the arctangent term, you'll need more data points away from the singularity.

As a compromise, after you've fit the polynomial terms b1...b6, you could also try obtaining a and b from the arctan term by doing

p=@(x) b(1)+b(2)*x+b(3)*x.^2+b(4)*x.^3+b(5)*x.^4+b(6)*x.^5 ;
zdata= tan(ydata - p(xdata));

A=[ones(length(zdata),1), zdata(:)];
B=zdata(:).*xdata(:);
ab=A\B;

a=ab(1);
b=ab(2);