lsqcurvefit

Syntax

x = lsqcurvefit(@func,x0,xdata,ydata)

x = lsqcurvefit(@func,x0,xdata,ydata,lb,ub)

x = lsqcurvefit(@func,x0,xdata,ydata,lb,ub,options)

[x,resnorm,residual,exitflag,output] = lsqcurvefit(...)

Inputs

func

The function of system residuals. See the optimset option Jacobian for details.

x0

An estimate of the best fit parameters.

xdata

The domain values for which the best fit is performed. If the domain is multivariate then each variable is stored in xdata by column.

ydata

The range values for which the best fit is performed.

lb

The fitting parameter lower bounds.

Use [ ] or -inf values if needed to specify unbounded conditions.

ub

The fitting parameter upper bounds.

Use [ ] or inf values if needed to specify unbounded conditions.

options

A struct containing option settings.

See optimset for details.

Outputs

x

The best fit parameters.

resnorm

The squared length of the residuals vector.

residual

The residuals vector.

info

The convergence status flag.

info = 4

Relative step size converged to within tolX.

info = 3

Relative function value converged to within tolFun.

info = 2

Step size converged to within tolX.

info = 1

Function value converged to within tolFun.

info = 0

Reached maximum number of iterations or function calls, or the algorithm aborted because it was not converging.

info = -3

Trust region became too small to continue.

output

A struct containing iteration details. The members are as follows:

iterations

The number of iterations.

nfev

The number of function evaluations.

xiter

The candidate solution at each iteration.

resnormiter

The objective function value at each iteration.

Examples

function y = FittingFunc(p, x)
    y = p(1) * exp(-p(2)*x);
end

x = [1; 2; 3; 4];
y = [8.025, 3.975, 2.025, 0.975];
p0 = [15; 1];
[p,res] = lsqcurvefit(@FittingFunc,p0,x,y)

p = [Matrix] 2 x 1
16.09850
 0.69669
res = 0.00190995097

function y = FittingFunc(p, x, offset)
    y = p(1) * exp(-p(2)*x) + offset;
end

handle = @(x, p) FittingFunc(x, p, 2);
[p,res] = lsqcurvefit(handle,p0,x,y+2)

p = [Matrix] 2 x 1
16.09850
 0.69669
res = 0.00190995097

[p,res] = lsqcurvefit(@FittingFunc,p0,x,y,[10,0.7],[16.2,2])

p = [Matrix] 2 x 1
16.16859
 0.70000
resnorm = 0.00226086103

Comments

lsqcurvefit uses a modified Gauss-Netwon algorithm with a trust region method. Bounds are supported with the addition of an affine scaling method drawn from the following sources.

• Francisco, J.B., N. Krejic, M. Martí­nez. 2005. "An interior-point method for solving box-constrained underdetermined nonlinear systems." Journal of Computational and Applied Mathematics 177, no. 1: 67-88. https://doi.org/10.1016/j.cam.2004.08.013
• Bellavia, Stefania, Maria Macconi, Sandra Pieraccini. 2012. "Constrained Dogleg Methods for nonlinear systems with simple bounds." Computational Optimization and Applications 53, 771-794 https://doi.org/10.1007/s10589-012-9469-8
• Klug, Andreas. 2006. "Affine-Scaling Methods for Nonlinear Minimization Problems and Nonlinear Systems of Equations with Bound Constraints" PhD Dissertation, Bavarian Julius Maximilians University, Wurzburg https://opus.bibliothek.uni-wuerzburg.de/opus4-wuerzburg/frontdoor/deliver/index/docId/1628/file/thesis.pdf

Options for convergence tolerance controls and analytical derivatives are specified with optimset.

To pass additional parameters to a function argument, use an anonymous function.

When using lb or ub with unbounded variables, use or +/-inf rather than arbitrarily large +/- numeric values. Large numeric bound ranges will degrade the performance of the function.