lsqcurvefit

Solve nonlinear curve-fitting (data-fitting) problems in least-squares sense

Syntax

x = lsqcurvefit(fun,x0,xdata,ydata)

x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub)

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

x = lsqcurvefit(problem)

[x,resnorm] = lsqcurvefit(___)

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

[x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqcurvefit(___)

Description

Nonlinear least-squares solver

Find coefficients x that solve the problem

$$\underset{x}{\min }{\Vert F(x,xdata)-ydata\Vert }_2^2=\underset{x}{\min }{\displaystyle \sum _i{\left(F\left(x,xdat{a}_i\right)-ydat{a}_i\right)}^2},$$

given input data xdata, and the observed output ydata, where xdata and ydata are matrices or vectors, and F (x, xdata) is a matrix-valued or vector-valued function of the same size as ydata.

Optionally, the components of x can have lower and upper bounds lb, and ub. The arguments x, lb, and ub can be vectors or matrices; see Matrix Arguments.

The lsqcurvefit function uses the same algorithm as lsqnonlin. lsqcurvefit simply provides a convenient interface for data-fitting problems.

Rather than compute the sum of squares, lsqcurvefit requires the user-defined function to compute the vector-valued function

$$F(x,xdata)=\left[\begin{array}{c}F\left(x,xdata(1)\right)\\ F\left(x,xdata(2)\right)\\ ⋮\\ F\left(x,xdata(k)\right)\end{array}\right].$$

x = lsqcurvefit(fun,x0,xdata,ydata) starts at x0 and finds coefficients x to best fit the nonlinear function fun(x,xdata) to the data ydata (in the least-squares sense). ydata must be the same size as the vector (or matrix) F returned by fun.

Note

Passing Extra Parameters explains how to pass extra parameters to the vector function fun(x), if necessary.

x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub) defines a set of lower and upper bounds on the design variables in x, so that the solution is always in the range lb ≤ x ≤ ub. You can fix the solution component x(i) by specifying lb(i) = ub(i).

Note

If the specified input bounds for a problem are inconsistent, the output x is x0 and the outputs resnorm and residual are [].

Components of x0 that violate the bounds lb ≤ x ≤ ub are reset to the interior of the box defined by the bounds. Components that respect the bounds are not changed.

x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,options) minimizes with the optimization options specified in options. Use optimoptions to set these options. Pass empty matrices for lb and ub if no bounds exist.

x = lsqcurvefit(problem) finds the minimum for problem, a structure described in problem.

[x,resnorm] = lsqcurvefit(___), for any input arguments, returns the value of the squared 2-norm of the residual at x: sum((fun(x,xdata)-ydata).^2).

[x,resnorm,residual,exitflag,output] = lsqcurvefit(___) additionally returns the value of the residual fun(x,xdata)-ydata at the solution x, a value exitflag that describes the exit condition, and a structure output that contains information about the optimization process.

[x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqcurvefit(___) additionally returns a structure lambda whose fields contain the Lagrange multipliers at the solution x, and the Jacobian of fun at the solution x.

Examples

Simple Exponential Fit

Suppose that you have observation time data xdata and observed response data ydata, and you want to find parameters $$x(1)$$ and $$x(2)$$ to fit a model of the form

$$\text{ydata}=x(1)\exp (x(2)\text{xdata}).$$

Input the observation times and responses.

xdata = ...
 [0.9 1.5 13.8 19.8 24.1 28.2 35.2 60.3 74.6 81.3];
ydata = ...
 [455.2 428.6 124.1 67.3 43.2 28.1 13.1 -0.4 -1.3 -1.5];

Create a simple exponential decay model.

fun = @(x,xdata)x(1)*exp(x(2)*xdata);

Fit the model using the starting point x0 = [100,-1].

x0 = [100,-1];
x = lsqcurvefit(fun,x0,xdata,ydata)

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

x = 1×2

  498.8309   -0.1013

Plot the data and the fitted curve.

times = linspace(xdata(1),xdata(end));
plot(xdata,ydata,'ko',times,fun(x,times),'b-')
legend('Data','Fitted exponential')
title('Data and Fitted Curve')

Best Fit with Bound Constraints

Find the best exponential fit to data where the fitting parameters are constrained.

Generate data from an exponential decay model plus noise. The model is

$$y=\exp (-1.3t)+\epsilon ,$$

with $$t$$ ranging from 0 through 3, and $$\epsilon$$ normally distributed noise with mean 0 and standard deviation 0.05.

rng default % for reproducibility
xdata = linspace(0,3);
ydata = exp(-1.3*xdata) + 0.05*randn(size(xdata));

The problem is: given the data (xdata, ydata), find the exponential decay model $$y=x(1)\exp (x(2)\text{xdata})$$ that best fits the data, with the parameters bounded as follows:

$$0\le x(1)\le 3/4$$

$$-2\le x(2)\le -1.$$

lb = [0,-2];
ub = [3/4,-1];

Create the model.

fun = @(x,xdata)x(1)*exp(x(2)*xdata);

Create an initial guess.

x0 = [1/2,-2];

Solve the bounded fitting problem.

x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub)

Local minimum found.

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

x = 1×2

    0.7500   -1.0000

Examine how well the resulting curve fits the data. Because the bounds keep the solution away from the true values, the fit is mediocre.

plot(xdata,ydata,'ko',xdata,fun(x,xdata),'b-')
legend('Data','Fitted exponential')
title('Data and Fitted Curve')

Compare Algorithms

Compare the results of fitting with the default 'trust-region-reflective' algorithm and the 'levenberg-marquardt' algorithm.

Suppose that you have observation time data xdata and observed response data ydata, and you want to find parameters $$x(1)$$ and $$x(2)$$ to fit a model of the form

$$\text{ydata}=x(1)\exp (x(2)\text{xdata}).$$

Input the observation times and responses.

xdata = ...
 [0.9 1.5 13.8 19.8 24.1 28.2 35.2 60.3 74.6 81.3];
ydata = ...
 [455.2 428.6 124.1 67.3 43.2 28.1 13.1 -0.4 -1.3 -1.5];

Create a simple exponential decay model.

fun = @(x,xdata)x(1)*exp(x(2)*xdata);

Fit the model using the starting point x0 = [100,-1].

x0 = [100,-1];
x = lsqcurvefit(fun,x0,xdata,ydata)

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

x = 1×2

  498.8309   -0.1013

Compare the solution with that of a 'levenberg-marquardt' fit.

options = optimoptions('lsqcurvefit','Algorithm','levenberg-marquardt');
lb = [];
ub = [];
x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,options)

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

x = 1×2

  498.8309   -0.1013

The two algorithms converged to the same solution. Plot the data and the fitted exponential model.

times = linspace(xdata(1),xdata(end));
plot(xdata,ydata,'ko',times,fun(x,times),'b-')
legend('Data','Fitted exponential')
title('Data and Fitted Curve')

Compare Algorithms and Examine Solution Process

Compare the results of fitting with the default 'trust-region-reflective' algorithm and the 'levenberg-marquardt' algorithm. Examine the solution process to see which is more efficient in this case.

Suppose that you have observation time data xdata and observed response data ydata, and you want to find parameters $$x(1)$$ and $$x(2)$$ to fit a model of the form

$$\text{ydata}=x(1)\exp (x(2)\text{xdata}).$$

Input the observation times and responses.

xdata = ...
 [0.9 1.5 13.8 19.8 24.1 28.2 35.2 60.3 74.6 81.3];
ydata = ...
 [455.2 428.6 124.1 67.3 43.2 28.1 13.1 -0.4 -1.3 -1.5];

Create a simple exponential decay model.

fun = @(x,xdata)x(1)*exp(x(2)*xdata);

Fit the model using the starting point x0 = [100,-1].

x0 = [100,-1];
[x,resnorm,residual,exitflag,output] = lsqcurvefit(fun,x0,xdata,ydata);

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

Compare the solution with that of a 'levenberg-marquardt' fit.

options = optimoptions('lsqcurvefit','Algorithm','levenberg-marquardt');
lb = [];
ub = [];
[x2,resnorm2,residual2,exitflag2,output2] = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,options);

Local minimum possible.
lsqcurvefit stopped because the relative size of the current step is less than
the value of the step size tolerance.

Are the solutions equivalent?

norm(x-x2)

ans = 2.0626e-06

Yes, the solutions are equivalent.

Which algorithm took fewer function evaluations to arrive at the solution?

fprintf(['The ''trust-region-reflective'' algorithm took %d function evaluations,\n',...
   'and the ''levenberg-marquardt'' algorithm took %d function evaluations.\n'],...
   output.funcCount,output2.funcCount)

The 'trust-region-reflective' algorithm took 87 function evaluations,
and the 'levenberg-marquardt' algorithm took 72 function evaluations.

Plot the data and the fitted exponential model.

times = linspace(xdata(1),xdata(end));
plot(xdata,ydata,'ko',times,fun(x,times),'b-')
legend('Data','Fitted exponential')
title('Data and Fitted Curve')

The fit looks good. How large are the residuals?

fprintf(['The ''trust-region-reflective'' algorithm has residual norm %f,\n',...
   'and the ''levenberg-marquardt'' algorithm has residual norm %f.\n'],...
   resnorm,resnorm2)

The 'trust-region-reflective' algorithm has residual norm 9.504887,
and the 'levenberg-marquardt' algorithm has residual norm 9.504887.

Input Arguments

fun — Function you want to fit

function handle | name of function

Function you want to fit, specified as a function handle or the name of a function. fun is a function that takes two inputs: a vector or matrix x, and a vector or matrix xdata. fun returns a vector or matrix F, the objective function evaluated at x and xdata. The function fun can be specified as a function handle for a function file:

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

where myfun is a MATLAB^® function such as

function F = myfun(x,xdata)
F = ...     % Compute function values at x, xdata

fun can also be a function handle for an anonymous function.

f = @(x,xdata)x(1)*xdata.^2+x(2)*sin(xdata);
x = lsqcurvefit(f,x0,xdata,ydata);

lsqcurvefit passes x to your objective function in the shape of the x0 argument. For example, if x0 is a 5-by-3 array, then lsqcurvefit passes x to fun as a 5-by-3 array.

Note

fun should return fun(x,xdata), and not the sum-of-squares sum((fun(x,xdata)-ydata).^2). lsqcurvefit implicitly computes the sum of squares of the components of fun(x,xdata)-ydata. See Examples.

If the Jacobian can also be computed and the 'SpecifyObjectiveGradient' option is true, set by

options = optimoptions('lsqcurvefit','SpecifyObjectiveGradient',true)

then the function fun must return a second output argument with the Jacobian value J (a matrix) at x. By checking the value of nargout, the function can avoid computing J when fun is called with only one output argument (in the case where the optimization algorithm only needs the value of F but not J).

function [F,J] = myfun(x,xdata)
F = ...          % objective function values at x
if nargout > 1   % two output arguments
   J = ...   % Jacobian of the function evaluated at x
end

If fun returns a vector (matrix) of m components and x has n elements, where n is the number of elements of x0, the Jacobian J is an m-by-n matrix where J(i,j) is the partial derivative of F(i) with respect to x(j). (The Jacobian J is the transpose of the gradient of F.) For more information, see Writing Vector and Matrix Objective Functions.

Example: @(x,xdata)x(1)*exp(-x(2)*xdata)

Data Types: char | function_handle | string

x0 — Initial point

real vector | real array

Initial point, specified as a real vector or real array. Solvers use the number of elements in x0 and the size of x0 to determine the number and size of variables that fun accepts.

Example: x0 = [1,2,3,4]

Data Types: double

xdata — Input data for model

real vector | real array

Input data for model, specified as a real vector or real array. The model is

ydata = fun(x,xdata),

where xdata and ydata are fixed arrays, and x is the array of parameters that lsqcurvefit changes to search for a minimum sum of squares.

Example: xdata = [1,2,3,4]

Data Types: double

ydata — Response data for model

real vector | real array

Response data for model, specified as a real vector or real array. The model is

ydata = fun(x,xdata),

where xdata and ydata are fixed arrays, and x is the array of parameters that lsqcurvefit changes to search for a minimum sum of squares.

The ydata array must be the same size and shape as the array fun(x0,xdata).

Example: ydata = [1,2,3,4]

Data Types: double

lb — Lower bounds

real vector | real array

Lower bounds, specified as a real vector or real array. If the number of elements in x0 is equal to the number of elements in lb, then lb specifies that

x(i) >= lb(i) for all i.

If numel(lb) < numel(x0), then lb specifies that

x(i) >= lb(i) for 1 <= i <= numel(lb).

If lb has fewer elements than x0, solvers issue a warning.

Example: To specify that all x components are positive, use lb = zeros(size(x0)).

Data Types: double

ub — Upper bounds

real vector | real array

Upper bounds, specified as a real vector or real array. If the number of elements in x0 is equal to the number of elements in ub, then ub specifies that

x(i) <= ub(i) for all i.

If numel(ub) < numel(x0), then ub specifies that

x(i) <= ub(i) for 1 <= i <= numel(ub).

If ub has fewer elements than x0, solvers issue a warning.

Example: To specify that all x components are less than 1, use ub = ones(size(x0)).

Data Types: double

options — Optimization options

output of optimoptions | structure such as optimset returns

Optimization options, specified as the output of optimoptions or a structure such as optimset returns.

Some options apply to all algorithms, and others are relevant for particular algorithms. See Optimization Options Reference for detailed information.

Some options are absent from the optimoptions display. These options appear in italics in the following table. For details, see View Options.

All Algorithms
Algorithm	Choose between 'trust-region-reflective' (default) and 'levenberg-marquardt'. The Algorithm option specifies a preference for which algorithm to use. It is only a preference, because certain conditions must be met to use each algorithm. For the trust-region-reflective algorithm, the nonlinear system of equations cannot be underdetermined; that is, the number of equations (the number of elements of F returned by fun) must be at least as many as the length of x. For more information on choosing the algorithm, see Choosing the Algorithm.
CheckGradients	Compare user-supplied derivatives (gradients of objective or constraints) to finite-differencing derivatives. Choices are false (default) or true. For optimset, the name is DerivativeCheck and the values are 'on' or 'off'. See Current and Legacy Option Names.
Diagnostics	Display diagnostic information about the function to be minimized or solved. Choices are 'off' (default) or 'on'.
DiffMaxChange	Maximum change in variables for finite-difference gradients (a positive scalar). The default is Inf.
DiffMinChange	Minimum change in variables for finite-difference gradients (a positive scalar). The default is 0.
Display	Level of display (see Iterative Display): 'off' or 'none' displays no output. 'iter' displays output at each iteration, and gives the default exit message. 'iter-detailed' displays output at each iteration, and gives the technical exit message. 'final' (default) displays just the final output, and gives the default exit message. 'final-detailed' displays just the final output, and gives the technical exit message.
FiniteDifferenceStepSize	Scalar or vector step size factor for finite differences. When you set FiniteDifferenceStepSize to a vector v, the forward finite differences delta are delta = v.*sign′(x).*max(abs(x),TypicalX); where sign′(x) = sign(x) except sign′(0) = 1. Central finite differences are delta = v.*max(abs(x),TypicalX); Scalar FiniteDifferenceStepSize expands to a vector. The default is sqrt(eps) for forward finite differences, and eps^(1/3) for central finite differences. For optimset, the name is FinDiffRelStep. See Current and Legacy Option Names.
FiniteDifferenceType	Finite differences, used to estimate gradients, are either 'forward' (default), or 'central' (centered). 'central' takes twice as many function evaluations, but should be more accurate. The algorithm is careful to obey bounds when estimating both types of finite differences. So, for example, it could take a backward, rather than a forward, difference to avoid evaluating at a point outside bounds. For optimset, the name is FinDiffType. See Current and Legacy Option Names.
FunctionTolerance	Termination tolerance on the function value, a positive scalar. The default is 1e-6. See Tolerances and Stopping Criteria. For optimset, the name is TolFun. See Current and Legacy Option Names.
FunValCheck	Check whether function values are valid. 'on' displays an error when the function returns a value that is complex, Inf, or NaN. The default 'off' displays no error.
MaxFunctionEvaluations	Maximum number of function evaluations allowed, a positive integer. The default is 100*numberOfVariables for the 'trust-region-reflective' algorithm, and 200*numberOfVariables for the 'levenberg-marquardt' algorithm. See Tolerances and Stopping Criteria and Iterations and Function Counts. For optimset, the name is MaxFunEvals. See Current and Legacy Option Names.
MaxIterations	Maximum number of iterations allowed, a positive integer. The default is 400. See Tolerances and Stopping Criteria and Iterations and Function Counts. For optimset, the name is MaxIter. See Current and Legacy Option Names.
OptimalityTolerance	Termination tolerance on the first-order optimality (a positive scalar). The default is 1e-6. See First-Order Optimality Measure. Internally, the 'levenberg-marquardt' algorithm uses an optimality tolerance (stopping criterion) of 1e-4 times FunctionTolerance and does not use OptimalityTolerance. For optimset, the name is TolFun. See Current and Legacy Option Names.
OutputFcn	Specify one or more user-defined functions that an optimization function calls at each iteration. Pass a function handle or a cell array of function handles. The default is none ([]). See Output Function and Plot Function Syntax.
PlotFcn	Plots various measures of progress while the algorithm executes; select from predefined plots or write your own. Pass a name, a function handle, or a cell array of names or function handles. For custom plot functions, pass function handles. The default is none ([]): 'optimplotx' plots the current point. 'optimplotfunccount' plots the function count. 'optimplotfval' plots the function value. 'optimplotresnorm' plots the norm of the residuals. 'optimplotstepsize' plots the step size. 'optimplotfirstorderopt' plots the first-order optimality measure. Custom plot functions use the same syntax as output functions. See Output Functions for Optimization Toolbox and Output Function and Plot Function Syntax. For optimset, the name is PlotFcns. See Current and Legacy Option Names.
SpecifyObjectiveGradient	If false (default), the solver approximates the Jacobian using finite differences. If true, the solver uses a user-defined Jacobian (defined in fun), or Jacobian information (when using JacobMult), for the objective function. For optimset, the name is Jacobian, and the values are 'on' or 'off'. See Current and Legacy Option Names.
StepTolerance	Termination tolerance on x, a positive scalar. The default is 1e-6. See Tolerances and Stopping Criteria. For optimset, the name is TolX. See Current and Legacy Option Names.
TypicalX	Typical x values. The number of elements in TypicalX is equal to the number of elements in x0, the starting point. The default value is ones(numberofvariables,1). The solver uses TypicalX for scaling finite differences for gradient estimation.
UseParallel	When true, the solver estimates gradients in parallel. Disable by setting to the default, false. See Parallel Computing.
Trust-Region-Reflective Algorithm
JacobianMultiplyFcn	Jacobian multiply function, specified as a function handle. For large-scale structured problems, this function computes the Jacobian matrix product J*Y, J'*Y, or J'*(J*Y) without actually forming J. The function is of the form W = jmfun(Jinfo,Y,flag) where Jinfo contains the matrix used to compute J*Y (or J'*Y, or J'*(J*Y)). The first argument Jinfo must be the same as the second argument returned by the objective function fun, for example, by [F,Jinfo] = fun(x) Y is a matrix that has the same number of rows as there are dimensions in the problem. flag determines which product to compute: If flag == 0 then W = J'*(J*Y). If flag > 0 then W = J*Y. If flag < 0 then W = J'*Y. In each case, J is not formed explicitly. The solver uses Jinfo to compute the preconditioner. See Passing Extra Parameters for information on how to supply values for any additional parameters jmfun needs. Note 'SpecifyObjectiveGradient' must be set to true for the solver to pass Jinfo from fun to jmfun. See Minimization with Dense Structured Hessian, Linear Equalities and Jacobian Multiply Function with Linear Least Squares for similar examples. For optimset, the name is JacobMult. See Current and Legacy Option Names.
JacobPattern	Sparsity pattern of the Jacobian for finite differencing. Set JacobPattern(i,j) = 1 when fun(i) depends on x(j). Otherwise, set JacobPattern(i,j) = 0. In other words, JacobPattern(i,j) = 1 when you can have ∂fun(i)/∂x(j) ≠ 0. Use JacobPattern when it is inconvenient to compute the Jacobian matrix J in fun, though you can determine (say, by inspection) when fun(i) depends on x(j). The solver can approximate J via sparse finite differences when you give JacobPattern. If the structure is unknown, do not set JacobPattern. The default behavior is as if JacobPattern is a dense matrix of ones. Then the solver computes a full finite-difference approximation in each iteration. This can be expensive for large problems, so it is usually better to determine the sparsity structure.
MaxPCGIter	Maximum number of PCG (preconditioned conjugate gradient) iterations, a positive scalar. The default is max(1,numberOfVariables/2). For more information, see Large Scale Nonlinear Least Squares.
PrecondBandWidth	Upper bandwidth of preconditioner for PCG, a nonnegative integer. The default PrecondBandWidth is Inf, which means a direct factorization (Cholesky) is used rather than the conjugate gradients (CG). The direct factorization is computationally more expensive than CG, but produces a better quality step towards the solution. Set PrecondBandWidth to 0 for diagonal preconditioning (upper bandwidth of 0). For some problems, an intermediate bandwidth reduces the number of PCG iterations.
SubproblemAlgorithm	Determines how the iteration step is calculated. The default, 'factorization', takes a slower but more accurate step than 'cg'. See Trust-Region-Reflective Least Squares.
TolPCG	Termination tolerance on the PCG iteration, a positive scalar. The default is 0.1.
Levenberg-Marquardt Algorithm
InitDamping	Initial value of the Levenberg-Marquardt parameter, a positive scalar. Default is 1e-2. For details, see Levenberg-Marquardt Method.
ScaleProblem	'jacobian' can sometimes improve the convergence of a poorly scaled problem; the default is 'none'.

Example: options = optimoptions('lsqcurvefit','FiniteDifferenceType','central')

problem — Problem structure

structure

Problem structure, specified as a structure with the following fields:

Field Name	Entry
objective	Objective function of x and xdata
x0	Initial point for x, active set algorithm only
xdata	Input data for objective function
ydata	Output data to be matched by objective function
lb	Vector of lower bounds
ub	Vector of upper bounds
solver	'lsqcurvefit'
options	Options created with optimoptions

You must supply at least the objective, x0, solver, xdata, ydata, and options fields in the problem structure.

Data Types: struct

Output Arguments

x — Solution

real vector | real array

Solution, returned as a real vector or real array. The size of x is the same as the size of x0. Typically, x is a local solution to the problem when exitflag is positive. For information on the quality of the solution, see When the Solver Succeeds.

resnorm — Squared norm of the residual

nonnegative real

Squared norm of the residual, returned as a nonnegative real. resnorm is the squared 2-norm of the residual at x: sum((fun(x,xdata)-ydata).^2).

residual — Value of objective function at solution

array

Value of objective function at solution, returned as an array. In general, residual = fun(x,xdata)-ydata.

exitflag — Reason the solver stopped

integer

Reason the solver stopped, returned as an integer.

1	Function converged to a solution x.
2	Change in x is less than the specified tolerance, or Jacobian at x is undefined.
3	Change in the residual is less than the specified tolerance.
4	Relative magnitude of search direction is smaller than the step tolerance.
0	Number of iterations exceeds options.MaxIterations or number of function evaluations exceeded options.MaxFunctionEvaluations.
-1	A plot function or output function stopped the solver.
-2	Problem is infeasible: the bounds lb and ub are inconsistent.

output — Information about the optimization process

structure

Information about the optimization process, returned as a structure with fields:

firstorderopt	Measure of first-order optimality
iterations	Number of iterations taken
funcCount	The number of function evaluations
cgiterations	Total number of PCG iterations (trust-region-reflective algorithm only)
stepsize	Final displacement in x
algorithm	Optimization algorithm used
message	Exit message

lambda — Lagrange multipliers at the solution

structure

Lagrange multipliers at the solution, returned as a structure with fields:

lower	Lower bounds lb
upper	Upper bounds ub

jacobian — Jacobian at the solution

real matrix

Jacobian at the solution, returned as a real matrix. jacobian(i,j) is the partial derivative of fun(i) with respect to x(j) at the solution x.

Limitations

• The trust-region-reflective algorithm does not solve underdetermined systems; it requires that the number of equations, i.e., the row dimension of F, be at least as great as the number of variables. In the underdetermined case, lsqcurvefit uses the Levenberg-Marquardt algorithm.

• lsqcurvefit can solve complex-valued problems directly. Note that bound constraints do not make sense for complex values. For a complex problem with bound constraints, split the variables into real and imaginary parts. See Fit a Model to Complex-Valued Data.

• The preconditioner computation used in the preconditioned conjugate gradient part of the trust-region-reflective method forms J^TJ (where J is the Jacobian matrix) before computing the preconditioner. Therefore, a row of J with many nonzeros, which results in a nearly dense product J^TJ, can lead to a costly solution process for large problems.

• If components of x have no upper (or lower) bounds, lsqcurvefit prefers that the corresponding components of ub (or lb) be set to inf (or -inf for lower bounds) as opposed to an arbitrary but very large positive (or negative for lower bounds) number.

You can use the trust-region reflective algorithm in lsqnonlin, lsqcurvefit, and fsolve with small- to medium-scale problems without computing the Jacobian in fun or providing the Jacobian sparsity pattern. (This also applies to using fmincon or fminunc without computing the Hessian or supplying the Hessian sparsity pattern.) How small is small- to medium-scale? No absolute answer is available, as it depends on the amount of virtual memory in your computer system configuration.

Suppose your problem has m equations and n unknowns. If the command J = sparse(ones(m,n)) causes an Out of memory error on your machine, then this is certainly too large a problem. If it does not result in an error, the problem might still be too large. You can find out only by running it and seeing if MATLAB runs within the amount of virtual memory available on your system.

Algorithms

The Levenberg-Marquardt and trust-region-reflective methods are based on the nonlinear least-squares algorithms also used in fsolve.

• The default trust-region-reflective algorithm is a subspace trust-region method and is based on the interior-reflective Newton method described in [1] and [2]. Each iteration involves the approximate solution of a large linear system using the method of preconditioned conjugate gradients (PCG). See Trust-Region-Reflective Least Squares.

• The Levenberg-Marquardt method is described in references [4], [5], and [6]. See Levenberg-Marquardt Method.

Alternative Functionality

App

The Optimize Live Editor task provides a visual interface for lsqcurvefit.

References

[1] Coleman, T.F. and Y. Li. “An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds.” SIAM Journal on Optimization, Vol. 6, 1996, pp. 418–445.

[2] Coleman, T.F. and Y. Li. “On the Convergence of Reflective Newton Methods for Large-Scale Nonlinear Minimization Subject to Bounds.” Mathematical Programming, Vol. 67, Number 2, 1994, pp. 189–224.

[3] Dennis, J. E. Jr. “Nonlinear Least-Squares.” State of the Art in Numerical Analysis, ed. D. Jacobs, Academic Press, pp. 269–312.

[4] Levenberg, K. “A Method for the Solution of Certain Problems in Least-Squares.” Quarterly Applied Mathematics 2, 1944, pp. 164–168.

[5] Marquardt, D. “An Algorithm for Least-squares Estimation of Nonlinear Parameters.” SIAM Journal Applied Mathematics, Vol. 11, 1963, pp. 431–441.

[6] Moré, J. J. “The Levenberg-Marquardt Algorithm: Implementation and Theory.” Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics 630, Springer Verlag, 1977, pp. 105–116.

[7] Moré, J. J., B. S. Garbow, and K. E. Hillstrom. User Guide for MINPACK 1. Argonne National Laboratory, Rept. ANL–80–74, 1980.

[8] Powell, M. J. D. “A Fortran Subroutine for Solving Systems of Nonlinear Algebraic Equations.” Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed., Ch.7, 1970.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

• lsqcurvefit and lsqnonlin support code generation using either the codegen (MATLAB Coder) function or the MATLAB Coder™ app. You must have a MATLAB Coder license to generate code.

• The target hardware must support standard double-precision floating-point computations. You cannot generate code for single-precision or fixed-point computations.

• Code generation targets do not use the same math kernel libraries as MATLAB solvers. Therefore, code generation solutions can vary from solver solutions, especially for poorly conditioned problems.

• All code for generation must be MATLAB code. In particular, you cannot use a custom black-box function as an objective function for lsqcurvefit or lsqnonlin. You can use coder.ceval to evaluate a custom function coded in C or C++. However, the custom function must be called in a MATLAB function.

• lsqcurvefit and lsqnonlin do not support the problem argument for code generation.

  [x,fval] = lsqnonlin(problem) % Not supported

• You must specify the objective function by using function handles, not strings or character names.

  x = lsqnonlin(@fun,x0,lb,ub,options) % Supported
  % Not supported: lsqnonlin('fun',...) or lsqnonlin("fun",...)

• All input matrices lb and ub must be full, not sparse. You can convert sparse matrices to full by using the full function.

• The lb and ub arguments must have the same number of entries as the x0 argument or must be empty [].

• For advanced code optimization involving embedded processors, you also need an Embedded Coder^® license.

• You must include options for lsqcurvefit or lsqnonlin and specify them using optimoptions. The options must include the Algorithm option, set to 'levenberg-marquardt'.

  options = optimoptions('lsqnonlin','Algorithm','levenberg-marquardt');
  [x,fval,exitflag] = lsqnonlin(fun,x0,lb,ub,options);

• Code generation supports these options:

  • Algorithm — Must be 'levenberg-marquardt'

  • FiniteDifferenceStepSize

  • FiniteDifferenceType

  • FunctionTolerance

  • MaxFunctionEvaluations

  • MaxIterations

  • SpecifyObjectiveGradient

  • StepTolerance

  • TypicalX

• Generated code has limited error checking for options. The recommended way to update an option is to use optimoptions, not dot notation.

  opts = optimoptions('lsqnonlin','Algorithm','levenberg-marquardt');
  opts = optimoptions(opts,'MaxIterations',1e4); % Recommended
  opts.MaxIterations = 1e4; % Not recommended

• Do not load options from a file. Doing so can cause code generation to fail. Instead, create options in your code.

• Usually, if you specify an option that is not supported, the option is silently ignored during code generation. However, if you specify a plot function or output function by using dot notation, code generation can issue an error. For reliability, specify only supported options.

• Because output functions and plot functions are not supported, solvers do not return the exit flag –1.

For an example, see Generate Code for lsqcurvefit or lsqnonlin.

Automatic Parallel Support

Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

To run in parallel, set the 'UseParallel' option to true.

options = optimoptions('solvername','UseParallel',true)

For more information, see Using Parallel Computing in Optimization Toolbox.

Version History

Introduced before R2006a