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,A,b,Aeq,beq)

x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,A,b,Aeq,beq,nonlcon)

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

x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,A,b,Aeq,beq,nonlcon,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 are subject to the constraints

$$\begin{array}{c}\text{lb}\le x\\ x\le \text{ub}\\ Ax\le b\\ \text{Aeq}x=\text{beq}\\ c(x)\le 0\\ \text{ceq}(x)=0.\end{array}$$

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

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

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,A,b,Aeq,beq) constrains the solution to satisfy the linear constraints

A x ≤ b

Aeq x = beq.

x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,A,b,Aeq,beq,nonlcon) constrain the solution to satisfy the nonlinear constraints in the nonlcon(x) function. nonlcon returns two outputs, c and ceq. The solver attempts to satisfy the constraints

c ≤ 0

ceq = 0.

x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,options) or x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,A,b,Aeq,beq,nonlcon,options) minimizes with the optimization options specified in options. Use optimoptions to set these options. Pass empty matrices for lb and ub and for other input arguments if the arguments do not 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')

Nonlinear Data Fitting with Linear Constraints

Create artificial data for a nonlinear model $$y=a+b\arctan (t-t_0)+ct$$ with parameters $$a$$, $$b$$, $$t_0$$, and $$c$$, for time $$t$$ from 2 to 7. Add noise to the data using randn.

a = 2; % x(1)
b = 4; % x(2)
t0 = 5; % x(3)
c = 1/2; % x(4)
xdata = linspace(2,7);
rng default
ydata = a + b*atan(xdata - t0) + c*xdata + 1/10*randn(size(xdata));

Plot the data.

plot(xdata,ydata,'ro')

Fit a nonlinear model to the data with the following constraints:

• All coefficients are between 0 and 7.

• $$x_1+x_2\ge x_3+x_4$$. You can write this constraint in the form A*x <= b using A = [-1 -1 1 1] and b = 0.

lb = zeros(4,1);
ub = 7*ones(4,1);
A = [-1 -1 1 1];
b = 0;

The myfun function at the end of this example creates the objective function for this model.

Solve the fitting problem starting from the point [1 2 3 1].

startpt = [1 2 3 1];
Aeq = [];
beq = [];
[x,res] = lsqcurvefit(@myfun,startpt,xdata,ydata,lb,ub,A,b,Aeq,beq)

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

x = 1×4

    2.3447    4.0972    4.9979    0.4303

res = 
1.2682

The returned solution is not far from the original point [2 4 5 1/2]. Plot the data against the curve from the solution point.

plot(xdata,ydata,'ro',xdata,myfun(x,xdata),'b-')

The returned solution matches the data pretty well. Is the constraint active?

A*x(:)

ans = 
-1.0137

The constraint is not active, because A*x < 0.

function F = myfun(x,xdata)
a = x(1);
b = x(2);
t0 = x(3);
c = x(4);
F = a + b*atan(xdata - t0) + c*xdata;
end

Nonlinear Data-Fitting with Nonlinear Constraints

Create artificial data for a nonlinear model $$y=a+b\arctan (t-t_0)+ct$$ with parameters $$a$$, $$b$$, $$t_0$$, and $$c$$, for time $$t$$ from 2 to 7. Add noise to the data using randn.

a = 2; % x(1)
b = 4; % x(2)
t0 = 5; % x(3)
c = 1/2; % x(4)
xdata = linspace(2,7);
rng default
ydata = a + b*atan(xdata - t0) + c*xdata + 1/10*randn(size(xdata));

Plot the data.

plot(xdata,ydata,'ro')

Fit a nonlinear model to the data with the following constraints:

• All coefficients are between 0 and 7.

• $$x_1^2+x_2^2\le 4^2$$

lb = zeros(4,1);
ub = 7*ones(4,1);

The problem has no linear constraints.

A = [];
b = [];
Aeq = [];
beq = [];

The myfun function at the end of this example creates the objective function for this model. The nlcon function at the end of this example creates the nonlinear constraint function.

Solve the fitting problem starting from the point [1 2 3 1].

startpt = [1 2 3 1];
[x,res] = lsqcurvefit(@myfun,startpt,xdata,ydata,lb,ub,A,b,Aeq,beq,@nlcon)

Feasible point with lower objective function value found, but optimality criteria not satisfied. See output.bestfeasible..


Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

x = 1×4

    1.3806    3.7542    5.0169    0.6337

res = 
1.6018

The returned solution x is not at the original point [2 4 5 1/2] because the nonlinear constraint is violated at that point. Plot the data against the curve from the solution point and compute the constraint function.

plot(xdata,ydata,'ro',xdata,myfun(x,xdata),'b-')

[c,ceq] = nlcon(x)

c = 
-3.1307e-06

ceq =

     []

The nonlinear inequality constraint is active at the solution because c = 0 at the solution.

Even though the solution point is not at the original point, the solution curve matches the data pretty well.

function F = myfun(x,xdata)
a = x(1);
b = x(2);
t0 = x(3);
c = x(4);
F = a + b*atan(xdata - t0) + c*xdata;
end

function [c,ceq] = nlcon(x)
ceq = [];
c = x(1)^2 + x(2)^2 - 4^2;
end

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 = 
2.0642e-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. For the 'interior-point' algorithm, fun must be a function handle. 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.

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.

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.

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: single | 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: single | double

A — Linear inequality constraints

real matrix

Linear inequality constraints, specified as a real matrix. A is an M-by-N matrix, where M is the number of inequalities, and N is the number of variables (number of elements in x0). For large problems, pass A as a sparse matrix.

A encodes the M linear inequalities

A*x <= b,

where x is the column vector of N variables x(:), and b is a column vector with M elements.

For example, consider these inequalities:

x₁ + 2x₂ ≤ 10
3x₁ + 4x₂ ≤ 20
5x₁ + 6x₂ ≤ 30,

Specify the inequalities by entering the following constraints.

A = [1,2;3,4;5,6];
b = [10;20;30];

Example: To specify that the x components sum to 1 or less, use A = ones(1,N) and b = 1.

Data Types: single | double

b — Linear inequality constraints

real vector

Linear inequality constraints, specified as a real vector. b is an M-element vector related to the A matrix. If you pass b as a row vector, solvers internally convert b to the column vector b(:). For large problems, pass b as a sparse vector.

b encodes the M linear inequalities

A*x <= b,

where x is the column vector of N variables x(:), and A is a matrix of size M-by-N.

For example, consider these inequalities:

x₁ + 2x₂ ≤ 10
3x₁ + 4x₂ ≤ 20
5x₁ + 6x₂ ≤ 30.

Specify the inequalities by entering the following constraints.

A = [1,2;3,4;5,6];
b = [10;20;30];

Example: To specify that the x components sum to 1 or less, use A = ones(1,N) and b = 1.

Data Types: single | double

Aeq — Linear equality constraints

real matrix

Linear equality constraints, specified as a real matrix. Aeq is an Me-by-N matrix, where Me is the number of equalities, and N is the number of variables (number of elements in x0). For large problems, pass Aeq as a sparse matrix.

Aeq encodes the Me linear equalities

Aeq*x = beq,

where x is the column vector of N variables x(:), and beq is a column vector with Me elements.

For example, consider these inequalities:

x₁ + 2x₂ + 3x₃ = 10
2x₁ + 4x₂ + x₃ = 20,

Specify the inequalities by entering the following constraints.

Aeq = [1,2,3;2,4,1];
beq = [10;20];

Example: To specify that the x components sum to 1, use Aeq = ones(1,N) and beq = 1.

Data Types: single | double

beq — Linear equality constraints

real vector

Linear equality constraints, specified as a real vector. beq is an Me-element vector related to the Aeq matrix. If you pass beq as a row vector, solvers internally convert beq to the column vector beq(:). For large problems, pass beq as a sparse vector.

beq encodes the Me linear equalities

Aeq*x = beq,

where x is the column vector of N variables x(:), and Aeq is a matrix of size Me-by-N.

For example, consider these equalities:

x₁ + 2x₂ + 3x₃ = 10
2x₁ + 4x₂ + x₃ = 20.

Specify the equalities by entering the following constraints.

Aeq = [1,2,3;2,4,1];
beq = [10;20];

Example: To specify that the x components sum to 1, use Aeq = ones(1,N) and beq = 1.

Data Types: single | double

nonlcon — Nonlinear constraints

function handle

Nonlinear constraints, specified as a function handle. nonlcon is a function that accepts a vector or array x and returns two arrays, c(x) and ceq(x).

• c(x) is the array of nonlinear inequality constraints at x. lsqcurvefit attempts to satisfy

  c(x) <= 0 for all entries of c.

  (1)

• ceq(x) is the array of nonlinear equality constraints at x. lsqcurvefit attempts to satisfy

  ceq(x) = 0 for all entries of ceq.

  (2)

For example,

x = lsqcurvefit(@myfun,x0,xdata,ydata,lb,ub,A,b,Aeq,beq,@mycon,options)

where mycon is a MATLAB function such as

function [c,ceq] = mycon(x)
c = ...     % Compute nonlinear inequalities at x.
ceq = ...   % Compute nonlinear equalities at x.

If the gradients of the constraints can also be computed and the SpecifyConstraintGradient option is true, as set by

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

then nonlcon must also return, in the third and fourth output arguments, GC, the gradient of c(x), and GCeq, the gradient of ceq(x). GC and GCeq can be sparse or dense. If GC or GCeq is large, with relatively few nonzero entries, save running time and memory in the 'interior-point' algorithm by representing them as sparse matrices. For more information, see Nonlinear Constraints.

Data Types: function_handle

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 Optimization Options.

All Algorithms
Algorithm	Choose between 'trust-region-reflective' (default), 'levenberg-marquardt', and 'interior-point'. 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 number of elements of F returned by fun must be at least as many as the length of x. The 'interior-point' algorithm is the only algorithm that can solve problems with linear or nonlinear constraints. If you include these constraints in your problem and do not specify an algorithm, the solver automatically switches to the 'interior-point' algorithm. The 'interior-point' algorithm calls a modified version of the fmincon 'interior-point' algorithm. 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. The CheckGradients option will be removed in a future release. To check derivatives, use the checkGradients function.
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); A 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 nonnegative 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 nonnegative integer. The default is 100*numberOfVariables for the 'trust-region-reflective' algorithm, 200*numberOfVariables for the 'levenberg-marquardt' algorithm, and 3000 for the 'interior-point' 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 nonnegative integer. The default is 400 for the 'trust-region-reflective' and 'levenberg-marquardt' algorithms, and 1000 for the 'interior-point' algorithm. 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 nonnegative 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 nonnegative scalar. The default is 1e-6 for the 'trust-region-reflective' and 'levenberg-marquardt' algorithms, and 1e-10 for the 'interior-point' algorithm. 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. For lsqnonlin the function is of the form W = jmfun(Jinfo,Y,flag) where Jinfo contains the data that helps to compute J*Y (or J'*Y, or J'*(J*Y)). For lsqcurvefit the function is of the form W = jmfun(Jinfo,Y,flag,xdata) where xdata is the data passed in the xdata argument. The data Jinfo is the second argument returned by the objective function fun: [F,Jinfo] = fun(x) % or [F,Jinfo] = fun(x,xdata) lsqcurvefit passes the data Jinfo, Y, flag, and, for lsqcurvefit, xdata, and your function jmfun computes a result as specified next. Y is a matrix whose size depends on the value of flag. Let m specify the number of components of the objective function fun, and let n specify the number of problem variables in x. The Jacobian is of size m-by-n as described in fun. The jmfun function returns one of these results: If flag == 0 then W = J'*(J*Y) and Y has size n-by-2. If flag > 0 then W = J*Y and Y has size n-by-1. If flag < 0 then W = J'*Y and Y has size m-by-1. In each case, J is not formed explicitly. The solver uses Jinfo to compute the multiplications. 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'.
Interior-Point Algorithm
BarrierParamUpdate	Specifies how fmincon updates the barrier parameter (see fmincon Interior Point Algorithm). The options are: 'monotone' (default) 'predictor-corrector' This option can affect the speed and convergence of the solver, but the effect is not easy to predict.
ConstraintTolerance	Tolerance on the constraint violation, a nonnegative scalar. The default is 1e-6. See Tolerances and Stopping Criteria. For optimset, the name is TolCon. See Current and Legacy Option Names.
InitBarrierParam	Initial barrier value, a positive scalar. Sometimes it might help to try a value above the default 0.1, especially if the objective or constraint functions are large.
SpecifyConstraintGradient	Gradient for nonlinear constraint functions defined by the user. When set to the default, false, lsqcurvefit estimates gradients of the nonlinear constraints by finite differences. When set to true, lsqcurvefit expects the constraint function to have four outputs, as described in nonlcon. For optimset, the name is GradConstr and the values are 'on' or 'off'. See Current and Legacy Option Names.
SubproblemAlgorithm	Determines how the iteration step is calculated. The default, 'factorization', is usually faster than 'cg' (conjugate gradient), though 'cg' might be faster for large problems with dense Hessians. See fmincon Interior Point Algorithm. For optimset, the values are 'cg' and 'ldl-factorization'. See Current and Legacy Option Names.

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
x0	Initial point for x
xdata	Input data for objective function
ydata	Output data to be matched by objective function
Aineq	Matrix for linear inequality constraints
bineq	Vector for linear inequality constraints
Aeq	Matrix for linear equality constraints
beq	Vector for linear equality constraints
lb	Vector of lower bounds
ub	Vector of upper bounds
nonlcon	Nonlinear constraint function
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	No feasible point found. The bounds lb and ub are inconsistent, or the solver stopped at an infeasible point.

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' and 'interior-point' algorithms)
stepsize	Final displacement in x
constrviolation	Maximum of constraint functions ('interior-point' algorithm)
bestfeasible	Best (lowest objective function) feasible point encountered ('interior-point' algorithm). A structure with these fields: x fval firstorderopt constrviolation If no feasible point is found, the bestfeasible field is empty. For this purpose, a point is feasible when the maximum of the constraint functions does not exceed options.ConstraintTolerance. The bestfeasible point can differ from the returned solution point x for a variety of reasons. For an example, see Obtain Best Feasible Point.
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 corresponding to lb
upper	Upper bounds corresponding to ub
ineqlin	Linear inequalities corresponding to A and b
eqlin	Linear equalities corresponding to Aeq and beq
ineqnonlin	Nonlinear inequalities corresponding to the c in nonlcon
eqnonlin	Nonlinear equalities corresponding to the ceq in nonlcon

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.

For problems with active constraints at the solution, jacobian is not useful for estimating confidence intervals.

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 constraints do not make sense for complex values, because complex numbers are not well-ordered; asking whether one complex value is greater or less than another complex value is nonsensical. For a complex problem with bound constraints, split the variables into real and imaginary parts. Do not use the 'interior-point' algorithm with complex data. 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.

More About

Enhanced Exit Messages

The next few items list the possible enhanced exit messages from lsqcurvefit. Enhanced exit messages give a link for more information as the first sentence of the message.

Local Minimum Found

The solver located a point that seems to be a local minimum of the sum of squares, since the first-order optimality measure is close to 0.

For suggestions on how to proceed, see When the Solver Succeeds.

Local Minimum Possible

The solver may have reached a local minimum but cannot be certain because the first-order optimality measure is not less than the OptimalityTolerance tolerance (1e-4*OptimalityTolerance for the Levenberg-Marquardt algorithm).

For suggestions on how to proceed, see Local Minimum Possible.

Initial Point is a Local Minimum

The initial point seems to be a local minimum of the sum of squares, because the first-order optimality measure is close to 0.

For suggestions on how to proceed, see Final Point Equals Initial Point.

Definitions for Exit Messages

The next few items contain definitions for terms in the lsqcurvefit exit messages.

local minimum

A local minimum of a function is a point where the function value is smaller than at nearby points, but possibly greater than at a distant point.

A global minimum is a point where the function value is smaller than at all other feasible points.

Solvers try to find a local minimum. The result can be a global minimum. For more information, see Local vs. Global Optima.

first-order optimality measure

For unconstrained problems, the first-order optimality measure is the maximum of the absolute value of the components of the gradient vector (also known as the infinity norm of the gradient). This should be zero at a minimizing point.

For problems with bounds, the first-order optimality measure is the maximum over i of |v_i*g_i|. Here g_i is the ith component of the gradient, x is the current point, and

$$v_i=\{\begin{array}{ll}\left|x_i-b_i\right|\hfill & \text{if the negative gradient points toward bound }{b}_i\hfill \\ 1\hfill & \text{otherwise}\text{.}\hfill \end{array}$$

If x_i is at a bound, v_i is zero. If x_i is not at a bound, then at a minimizing point the gradient g_i should be zero. Therefore the first-order optimality measure should be zero at a minimizing point.

For more information, see First-Order Optimality Measure.

tolerance

Generally, a tolerance is a threshold which, if crossed, stops the iterations of a solver. For more information on tolerances, see Tolerances and Stopping Criteria.

OptimalityTolerance

The tolerance called OptimalityTolerance relates to the first-order optimality measure. Iterations end when the first-order optimality measure is less than OptimalityTolerance.

FunctionTolerance

The function tolerance called FunctionTolerance relates to the size of the latest change in objective function value.

Gradient Size

The gradient vector is the gradient of the sum of squares. For unconstrained problems, the gradient size is the maximum of the absolute value of the components of the gradient vector (also known as the infinity norm of the gradient). This should be zero at a minimizing point.

For problems with bounds, the gradient size is the maximum over i of |v_i*g_i|. Here g_i is the ith component of the gradient, x is the current point, and

$$v_i=\{\begin{array}{ll}\left|x_i-b_i\right|\hfill & \text{if the negative gradient points toward bound }{b}_i\hfill \\ 1\hfill & \text{otherwise}\text{.}\hfill \end{array}$$

If x_i is at a bound, v_i is zero. If x_i is not at a bound, then at a minimizing point the gradient g_i should be zero. Therefore the gradient size should be zero at a minimizing point.

For more information, see First-Order Optimality Measure.

Size of the Current Step

The size of the current step is the magnitude of the change in location of the current point in the final iteration.

For more information, see Tolerances and Stopping Criteria.

Locally Singular

The Levenberg-Marquardt regularization parameter is related to the inverse of a trust-region radius. It becomes large when the sum of squares of function values is not close to a quadratic model. For more information, see Levenberg-Marquardt Method.

Jacobian Calculation is Undefined

Solvers estimate the Jacobian of your objective vector function by taking finite differences. A finite difference calculation stepped outside the region where the objective function is well-defined, returning Inf, NaN, or a complex result.

For more information about how solvers use the Jacobian J, see Levenberg-Marquardt Method. For suggestions on how to proceed, see 6. Provide Gradient or Jacobian.

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.

The 'interior-point' algorithm uses the fmincon 'interior-point' algorithm with some modifications. For details, see Modified fmincon Algorithm for Constrained Least Squares.

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.

• Code generation for lsqcurvefit and lsqnonlin currently does not support linear or nonlinear constraints.

• 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 [].

• If your target hardware does not support infinite bounds, use optim.coder.infbound.

• 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

R2023b: JacobianMultiplyFcn option accepts any data type

The syntax for the JacobianMultiplyFcn option is

W = jmfun(Jinfo,Y,flag,xdata)

The Jinfo data, which MATLAB passes to your function jmfun, can now be of any data type. For example, you can now have Jinfo be a structure. In previous releases, Jinfo had to be a standard double array.

The Jinfo data is the second output of your objective function:

[F,Jinfo] = myfun(x,xdata)

R2023b: CheckGradients option will be removed

The CheckGradients option will be removed in a future release. To check the first derivatives of objective functions or nonlinear constraint functions, use the checkGradients function.

R2023a: Linear and Nonlinear Constraint Support

lsqcurvefit gains support for both linear and nonlinear constraints. To enable constraint satisfaction, the solver uses the "interior-point" algorithm from fmincon.

• If you specify constraints but do not specify an algorithm, the solver automatically switches to the "interior-point" algorithm.

• If you specify constraints and an algorithm, you must specify the "interior-point" algorithm.

For algorithm details, see Modified fmincon Algorithm for Constrained Least Squares. For an example, see Compare lsqnonlin and fmincon for Constrained Nonlinear Least Squares.