Documentation 
Solve nonlinear leastsquares (nonlinear datafitting) problems
Solves nonlinear leastsquares curve fitting problems of the form
$$\underset{x}{\mathrm{min}}{\Vert f(x)\Vert}_{2}^{2}=\underset{x}{\mathrm{min}}\left({f}_{1}{(x)}^{2}+{f}_{2}{(x)}^{2}+\mathrm{...}+{f}_{n}{(x)}^{2}\right)$$
with optional lower and upper bounds lb and ub on the components of x.
x, lb, and ub can be vectors or matrices; see Matrix Arguments.
x = lsqnonlin(fun,x0)
x = lsqnonlin(fun,x0,lb,ub)
x = lsqnonlin(fun,x0,lb,ub,options)
x = lsqnonlin(problem)
[x,resnorm] = lsqnonlin(...)
[x,resnorm,residual] = lsqnonlin(...)
[x,resnorm,residual,exitflag] = lsqnonlin(...)
[x,resnorm,residual,exitflag,output]
= lsqnonlin(...)
[x,resnorm,residual,exitflag,output,lambda]
= lsqnonlin(...)
[x,resnorm,residual,exitflag,output,lambda,jacobian]
= lsqnonlin(...)
lsqnonlin solves nonlinear leastsquares problems, including nonlinear datafitting problems.
Rather than compute the value $${\Vert f(x)\Vert}_{2}^{2}$$ (the sum of squares), lsqnonlin requires the userdefined function to compute the vectorvalued function
$$f(x)=\left[\begin{array}{c}{f}_{1}(x)\\ {f}_{2}(x)\\ \vdots \\ {f}_{n}(x)\end{array}\right]$$
Then, in vector terms, you can restate this optimization problem as
$$\underset{x}{\mathrm{min}}{\Vert f(x)\Vert}_{2}^{2}=\underset{x}{\mathrm{min}}\left({f}_{1}{(x)}^{2}+{f}_{2}{(x)}^{2}+\mathrm{...}+{f}_{n}{(x)}^{2}\right)$$
where x is a vector or matrix and f(x) is a function that returns a vector or matrix value. For details of matrix values, see Matrix Arguments.
x = lsqnonlin(fun,x0) starts at the point x0 and finds a minimum of the sum of squares of the functions described in fun. fun should return a vector of values and not the sum of squares of the values. (The algorithm implicitly computes the sum of squares of the components of fun(x).)
Note: Passing Extra Parameters explains how to pass extra parameters to the vector function f, if necessary. 
x = lsqnonlin(fun,x0,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.
x = lsqnonlin(fun,x0,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 = lsqnonlin(problem) finds the minimum for problem, where problem is a structure described in Input Arguments.
Create the problem structure by exporting a problem from Optimization app, as described in Exporting Your Work.
[x,resnorm] = lsqnonlin(...) returns the value of the squared 2norm of the residual at x: sum(fun(x).^2).
[x,resnorm,residual] = lsqnonlin(...) returns the value of the residual fun(x) at the solution x.
[x,resnorm,residual,exitflag] = lsqnonlin(...) returns a value exitflag that describes the exit condition.
[x,resnorm,residual,exitflag,output] = lsqnonlin(...) returns a structure output that contains information about the optimization.
[x,resnorm,residual,exitflag,output,lambda] = lsqnonlin(...) returns a structure lambda whose fields contain the Lagrange multipliers at the solution x.
[x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqnonlin(...) returns the Jacobian of fun at the solution x.
Function Arguments contains general descriptions of arguments passed into lsqnonlin. This section provides functionspecific details for fun, options, and problem:
The function whose sum of squares is minimized. fun is a function that accepts a vector x and returns a vector F, the objective functions evaluated at x. The function fun can be specified as a function handle to a file: x = lsqnonlin(@myfun,x0) where myfun is a MATLAB^{®} function such as function F = myfun(x) F = ... % Compute function values at x fun can also be a function handle for an anonymous function. x = lsqnonlin(@(x)sin(x.*x),x0); If the userdefined values for x and F are matrices, they are converted to a vector using linear indexing.
If the Jacobian can also be computed and the Jacobian option is 'on', set by options = optimoptions('lsqnonlin','Jacobian','on') the function fun must return, in a second output argument, 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) 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 length n, where n is the length of x0, the Jacobian J is an mbyn 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.)  
options  Options provides the functionspecific details for the options values.  
problem  objective  Objective function  
x0  Initial point for x  
lb  Vector of lower bounds  
ub  Vector of upper bounds  
solver  'lsqnonlin'  
options  Options created with optimoptions 
Function Arguments contains general descriptions of arguments returned by lsqnonlin. This section provides functionspecific details for exitflag, lambda, and output:
exitflag  Integer identifying the reason the algorithm terminated. The following lists the values of exitflag and the corresponding reasons the algorithm terminated:  
1  Function converged to a solution x.  
2  Change in x was less than the specified tolerance.  
3  Change in the residual was less than the specified tolerance.  
4  Magnitude of search direction was smaller than the specified tolerance.  
0  Number of iterations exceeded options.MaxIter or number of function evaluations exceeded options.MaxFunEvals.  
1  Output function terminated the algorithm.  
2  Problem is infeasible: the bounds lb and ub are inconsistent.  
4  Line search could not sufficiently decrease the residual along the current search direction.  
lambda  Structure containing the Lagrange multipliers at the solution x (separated by constraint type). The fields are  
lower  Lower bounds lb  
upper  Upper bounds ub  
output  Structure containing information about the optimization. The fields of the structure are  
firstorderopt  Measure of firstorder optimality  
iterations  Number of iterations taken  
funcCount  The number of function evaluations  
cgiterations  Total number of PCG iterations (trustregionreflective algorithm only)  
stepsize  Final displacement in x (LevenbergMarquardt algorithm)  
algorithm  Optimization algorithm used  
message  Exit message 
Optimization options. Set or change options using the optimoptions function. Some options apply to all algorithms, some are only relevant when you are using the trustregionreflective algorithm, and others are only relevant when you are using the LevenbergMarquardt algorithm. See Optimization Options Reference for detailed information.
Both algorithms use the following options:
Algorithm  Choose between 'trustregionreflective' (default) and 'levenbergmarquardt'. Set the initial LevenbergMarquardt parameter λ by setting Algorithm to a cell array such as {'levenbergmarquardt',.005}. The default λ = 0.01. 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 trustregionreflective 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. The LevenbergMarquardt algorithm does not handle bound constraints. For more information on choosing the algorithm, see Choosing the Algorithm. 
DerivativeCheck  Compare usersupplied derivatives (gradients of objective or constraints) to finitedifferencing derivatives. The choices are 'on' or the default 'off'. 
Diagnostics  Display diagnostic information about the function to be minimized or solved. The choices are 'on' or the default 'off'. 
DiffMaxChange  Maximum change in variables for finitedifference gradients (a positive scalar). The default is Inf. 
DiffMinChange  Minimum change in variables for finitedifference gradients (a positive scalar). The default is 0. 
Display  Level of display:

FinDiffRelStep  Scalar or vector step size factor. When you set FinDiffRelStep to a vector v, forward finite differences delta are delta = v.*sign(x).*max(abs(x),TypicalX); and central finite differences are delta = v.*max(abs(x),TypicalX); Scalar FinDiffRelStep expands to a vector. The default is sqrt(eps) for forward finite differences, and eps^(1/3) for central finite differences. 
FinDiffType  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. 
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. 
Jacobian  If 'on', lsqnonlin uses a userdefined Jacobian (defined in fun), or Jacobian information (when using JacobMult), for the objective function. If 'off' (default), lsqnonlin approximates the Jacobian using finite differences. 
MaxFunEvals  Maximum number of function evaluations allowed, a positive integer. The default is 100*numberOfVariables. 
MaxIter  Maximum number of iterations allowed, a positive integer. The default is 400. 
OutputFcn  Specify one or more userdefined functions that an optimization function calls at each iteration, either as a function handle or as a cell array of function handles. The default is none ([]). See Output Function. 
PlotFcns  Plots various measures of progress while the algorithm executes, select from predefined plots or write your own. Pass a function handle or a cell array of function handles. The default is none ([]):
For information on writing a custom plot function, see Plot Functions. 
TolFun  Termination tolerance on the function value, a positive scalar. The default is 1e6. 
TolX  Termination tolerance on x, a positive scalar. The default is 1e6. 
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). lsqnonlin uses TypicalX for scaling finite differences for gradient estimation. 
The trustregionreflective algorithm uses the following options:
JacobMult  Function handle for Jacobian multiply function. For largescale 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:
In each case, J is not formed explicitly. lsqnonlin uses Jinfo to compute the preconditioner. See Passing Extra Parameters for information on how to supply values for any additional parameters jmfun needs. See Minimization with Dense Structured Hessian, Linear Equalities and Jacobian Multiply Function with Linear Least Squares for similar examples.  
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). lsqnonlin can approximate J via sparse finite differences when you give JacobPattern. In the worst case, if the structure is unknown, do not set JacobPattern. The default behavior is as if JacobPattern is a dense matrix of ones. Then lsqnonlin computes a full finitedifference approximation in each iteration. This can be very 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 Algorithms.  
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.  
TolPCG  Termination tolerance on the PCG iteration, a positive scalar. The default is 0.1. 
The LevenbergMarquardt algorithm uses the following options:
InitDamping  Initial value of the LevenbergMarquardt parameter, a positive scalar. Default is 1e2. For details, see LevenbergMarquardt Method. 
ScaleProblem  'Jacobian' can sometimes improve the convergence of a poorly scaled problem; the default is 'none'. 
Find x that minimizes
$$\sum _{k=1}^{10}{\left(2+2k{e}^{k{x}_{1}}{e}^{k{x}_{2}}\right)}^{2}},$$
starting at the point x = [0.3, 0.4].
Because lsqnonlin assumes that the sum of squares is not explicitly formed in the userdefined function, the function passed to lsqnonlin should instead compute the vectorvalued function
$${F}_{k}(x)=2+2k{e}^{k{x}_{1}}{e}^{k{x}_{2}},$$
for k = 1 to 10 (that is, F should have 10 components).
First, write a file to compute the 10component vector F.
function F = myfun(x) k = 1:10; F = 2 + 2*kexp(k*x(1))exp(k*x(2));
Next, invoke an optimization routine.
x0 = [0.3 0.4] % Starting guess [x,resnorm] = lsqnonlin(@myfun,x0); % Invoke optimizer
After about 24 function evaluations, this example gives the solution
x,resnorm x = 0.2578 0.2578 resnorm = 124.3622
You can use the trustregion reflective algorithm in lsqnonlin, lsqcurvefit, and fsolve with small to mediumscale 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 mediumscale? 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 only find out by running it and seeing if MATLAB runs within the amount of virtual memory available on your system.
The trustregionreflective method does not allow equal upper and lower bounds. For example, if lb(2)==ub(2), lsqlin gives the error
Equal upper and lower bounds not permitted.
(lsqnonlin does not handle equality constraints, which is another way to formulate equal bounds. If equality constraints are present, use fmincon, fminimax, or fgoalattain for alternative formulations where equality constraints can be included.)
The function to be minimized must be continuous. lsqnonlin might only give local solutions.
lsqnonlin can solve complexvalued problems directly with the levenbergmarquardt algorithm. However, this algorithm does not accept bound constraints. For a complex problem with bound constraints, split the variables into real and imaginary parts, and use the trustregionreflective algorithm. See Fit a Model to ComplexValued Data.
The trustregionreflective algorithm for lsqnonlin 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, the LevenbergMarquardt algorithm is used instead.
The preconditioner computation used in the preconditioned conjugate gradient part of the trustregionreflective method forms J^{T}J (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^{T}J, can lead to a costly solution process for large problems.
If components of x have no upper (or lower) bounds, lsqnonlin 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.
TrustRegionReflective Problem Coverage and Requirements
For Large Problems 


The LevenbergMarquardt algorithm does not handle bound constraints.
Since the trustregionreflective algorithm does not handle underdetermined systems and the LevenbergMarquardt does not handle bound constraints, problems with both these characteristics cannot be solved by lsqnonlin.
[1] Coleman, T.F. and Y. Li, "An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds," SIAM Journal on Optimization, Vol. 6, pp. 418–445, 1996.
[2] Coleman, T.F. and Y. Li, "On the Convergence of Reflective Newton Methods for LargeScale Nonlinear Minimization Subject to Bounds," Mathematical Programming, Vol. 67, Number 2, pp. 189224, 1994.
[3] Dennis, J.E., Jr., "Nonlinear LeastSquares," State of the Art in Numerical Analysis, ed. D. Jacobs, Academic Press, pp. 269–312, 1977.
[4] Levenberg, K., "A Method for the Solution of Certain Problems in LeastSquares," Quarterly Applied Math. 2, pp. 164–168, 1944.
[5] Marquardt, D., "An Algorithm for LeastSquares Estimation of Nonlinear Parameters," SIAM Journal Applied Math., Vol. 11, pp. 431–441, 1963.
[6] Moré, J.J., "The LevenbergMarquardt Algorithm: Implementation and Theory," Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics 630, Springer Verlag, pp. 105–116, 1977.