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LORENTZFIT fits a single- or multi-parameter Lorentzian function to data



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% LORENTZFIT fits a single- or multi-parameter Lorentzian function to data
% LORENTZFIT(X,Y) returns YPRIME(X), a Lorentzian fit to the data
% found using LSQCURVEFIT. The function Y(X) is fit by the model:
% YPRIME(X) = P1./((X - P2).^2 + P3) + C.
% values in addition to fit-parameters PARAMS = [P1 P2 P3 C]. The RESNORM
% and RESIDUAL outputs from LSQCURVEFIT are also returned.
% [...] = LORENTZFIT(X,Y,P0) can be used to provide starting
% values (P0 = [P01 P02 P03 C0]) for the parameters in PARAMS.
% [...] = LORENTZFIT(X,Y,P0,BOUNDS) may be used to define lower
% and upper bounds for the possbile values for each parameter in PARAMS.
% BOUNDS = [LB1 LB2 LB3 LB4;
% UB1 UB2 UB3 UB4].
% If the user does not wish to manually define values for P0, it may be
% enetered as an empty matrix P0 = []. In this case, default values will
% be used. The default bounds for all parameters are (-Inf,Inf).
% [...] = LORENTZFIT(X,Y,P0,BOUNDS,NPARAMS) may be used to specify the
% number of parameters used in the Lorentzian fitting function. The
% number of parameters defined in P0 and BOUNDS must match the function
% specified by NPARAMS. If the user does not wish to manually define
% values for P0 or BOUNDS, both may be enetered as empty matricies:
% P0 = []; BOUNDS = [].
% -NPARAMS options
% '1' - Single parameter Lorentzian (no constant term)
% L1(X) = 1./(P1(X.^2 + 1))
% '1c' - Single parameter Lorentzian (with constant term)
% L1C(X) = 1./(P1(X.^2 + 1)) + C
% '2' - Two parameter Lorentzian (no constant term)
% L2(X) = P1./(X.^2 + P2)
% '2c' - Two parameter Lorentzian (with constant term)
% L2C(X) = P1./(X.^2 + P2) + C
% '3' - Three parameter Lorentzian (no constant term)
% L3(X) = P1./((X - P2).^2 + P3)
% [DEFAULT] '3c' - Three parameter Lorentzian (with constant term)
% L3C(X) = P1./((X - P2).^2 + P3) + C
% array for the MATLAB function LSQCURVEFIT. OPTIONS can be set using the
% following command:
% OPTIONS = optimset('PARAM1',VALUE1,'PARAM2',VALUE2,...);
% See the help documentation for OPTIMSET for more details.
% X and Y must be the same size, numeric, and non-complex. P0 and BOUNDS
% must also be numeric and non-complex. NPARAMS is a character array.
% Examples:
% x = -16:0.1:35;
% y = 19.4./((x - 7).^2 + 15.8) + randn(size(x))./10;
% [yprime1 params1 resnorm1 residual1] = lorentzfit(x,y,[20 10 15 0]);
% figure; plot(x,y,'b.','LineWidth',2)
% hold on; plot(x,yprime1,'r-','LineWidth',2)
% [yprime2 params2 resnorm2 residual2] = lorentzfit(x,y,[],[],'3');
% figure; plot(x,y,'b.','LineWidth',2)
% hold on; plot(x,yprime2,'r-','LineWidth',2)
% See also: lsqcurvefit.

Comments and Ratings (23)

Leo YU

Leo YU (view profile)

Hi Jared, I met some problems using this function. Could you please check this? Thank you.
' Local minimum possible.

lsqcurvefit stopped because the size of the current step is less than
the selected value of the step size tolerance.

<stopping criteria details>'

'Local minimum found.

Optimization completed because the size of the gradient is less than
the selected value of the optimality tolerance.

<stopping criteria details>

Optimization completed: The first-order optimality measure, 7.891109e-11,
is less than options.OptimalityTolerance = 1.366818e-09.

Optimization Metric Options
relative first-order optimality = 7.89e-11 OptimalityTolerance = 1e-09 (selected)'

which settings I should change? Cheers.

Jered Wells

Joseph. To get the covariance matrix associated with the fit variables, you can follow the instructions here: Using the LORENTZFIT code, you will need to output the Jacobian from the call to LSQCURVEFIT. This will require a little editing on your part, but should be fairly straightforward. Hope this helps.

Joseph Horne

Jared, I'm using a Lorentz fit to find the position of a single peak to quote it as a resonance value. If it is possible to find the uncertainty in the value, or some uncertainty maybe from lsqcurvefit that would be useful.
Cheers for the quick reply.

Jered Wells

Joseph, I'm not sure what you mean by calculating the error in P2. Without a ground truth, it's hard to say. If you expect a shift of say x0 based on knowledge of the original data, maybe you could compute the difference P2-x0? Otherwise, maybe you are referring to the uncertainty in P2? Let me know.

Joseph Horne

Hi Jared,
Probably really basic but I want to find the error in the peak position, P2. Can you tell me how?

Jered Wells

Alexandra, you need to define additional output variables in order to capture the fit parameters. For example, [YPRIME, PARAMS] = lorentzfit(x,y); should give you the parameters you are looking for. Also see the help documentation by calling >> help lotentzfit from the command line.

How can I access the fitted distribution parameters? Thank you.

Hello Jered,
thanks for the file. It is very useful!
I just didn't understand how to extract the FWHM.
Thanks in advance

Jered Wells

Heather, thank you for your comment. If you need to fit a "double-Lorentzian function," you may consider modifying the code to do this. I suspect that fitting will be more unstable due to the increase in unknowns and the need to have more accurate initial seeds. I leave this exercise to you :-)


Thanks Jered for sharing the excellent file! Could there be a double-peak fitting function added? Thanks again.


Jered Wells

Sangeeta, please defer to the help documentation provided in the code. X and Y are the independent and dependent variables, respectively. Varargin is indicated because the user may choose to define a variable number of input parameters depending on his or her needs and applications.

what is varargin input?
I suppose and x and y are Wavelength and intensity data

ihsan AMIN

ihsan AMIN

Galya h

Tianhan Liu

This file is great and Jered is very helpful. It helped me save a lot of time analyzing data.

Jered Wells

Hamed, if you could provide me with some sample data, I could be more helpful. My contact information is stored in the MATLAB function just past the help documentation. Thanks


Hamed (view profile)

Dear Jered,

Thanks for sharing the code. It is really helpful.

In the application that I am interested in, I want to measure the quality factor of resonances. Meaning that the fit should perfectly follow the fast transition at the peak. But, when I use the code, it sort of smooths the response.
Could you please give me an idea how to resolve this problem?


Ramu (view profile)

Yea, Thanks for the script. It was very helpful

Jered Wells

Thank you, Ramu, for your comments. I trust that all of your questions have been answered.


Ramu (view profile)

I changed the 3 parameter fitting fucntion to
F= p(4) + (2*p(1)./pi)*(p(3)./(4*(x-p(2)).^2 + p(3).^2))

now p1 gives Area, p2 gives the center, p3 gives the FWHM and p4 gives the y offset.


Ramu (view profile)

I tried this function on my data and found that parameter calculated are very different from the one I get from Orgin.

I recalculated the width by taking the sq. root of P3. Is that the correct way ?



Added stopping thresholds for fitting based on magnitude of input data.


Added OPTIONS to the LSQCURVEFIT routine.


Added INPUTCHECK and OPTIONS. Rearranged SWITCH loop.


Better identified default functionality in help file


Included option to select one, two, or three parameter Lorentzian model with or without constant parameter.
Improved documentation
Provided two additional outputs from LSQCURVEFIT: RESNORM and RESIDUAL.


Documentation updated to match MATLAB standard for help file
H1 line has been included
Error checking included
Outputs modified for easier use


Cleaned up description

MATLAB Release
MATLAB 8.5 (R2015a)

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