# lsqnonlin

Solve nonlinear least-squares (nonlinear data-fitting) problems

## Equation

Solves nonlinear least-squares curve fitting problems of the form

$\underset{x}{\mathrm{min}}{‖f\left(x\right)‖}_{2}^{2}=\underset{x}{\mathrm{min}}\left({f}_{1}{\left(x\right)}^{2}+{f}_{2}{\left(x\right)}^{2}+...+{f}_{n}{\left(x\right)}^{2}\right)$

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.

## Syntax

```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(...) ```

## Description

`lsqnonlin` solves nonlinear least-squares problems, including nonlinear data-fitting problems.

Rather than compute the value ${‖f\left(x\right)‖}_{2}^{2}$ (the sum of squares), `lsqnonlin` requires the user-defined function to compute the vector-valued function

$f\left(x\right)=\left[\begin{array}{c}{f}_{1}\left(x\right)\\ {f}_{2}\left(x\right)\\ ⋮\\ {f}_{n}\left(x\right)\end{array}\right]$

Then, in vector terms, you can restate this optimization problem as

$\underset{x}{\mathrm{min}}{‖f\left(x\right)‖}_{2}^{2}=\underset{x}{\mathrm{min}}\left({f}_{1}{\left(x\right)}^{2}+{f}_{2}{\left(x\right)}^{2}+...+{f}_{n}{\left(x\right)}^{2}\right)$

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`. You can fix the solution component `x(i)` by specifying `lb(i) = ub(i)`.

`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 2-norm 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`.

 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.

## Input Arguments

Function Arguments contains general descriptions of arguments passed into `lsqnonlin`. This section provides function-specific details for `fun`, `options`, and `problem`:

`fun`

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 user-defined values for `x` and `F` are matrices, they are converted to a vector using linear indexing.

 Note   The sum of squares should not be formed explicitly. Instead, your function should return a vector of function values. See Examples.

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 `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`.)

`options`

Options provides the function-specific 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`

## Output Arguments

Function Arguments contains general descriptions of arguments returned by `lsqnonlin`. This section provides function-specific 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 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` (Levenberg-Marquardt algorithm) `algorithm` Optimization algorithm used `message` Exit message

## Options

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 trust-region-reflective algorithm, and others are only relevant when you are using the Levenberg-Marquardt algorithm. See Optimization Options Reference for detailed information.

### Algorithm Options

Both algorithms use the following options:

 `Algorithm` Choose between `'trust-region-reflective'` (default) and `'levenberg-marquardt'`. Set the initial Levenberg-Marquardt parameter λ by setting `Algorithm` to a cell array such as `{'levenberg-marquardt',.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 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`. The Levenberg-Marquardt algorithm does not handle bound constraints. For more information on choosing the algorithm, see Choosing the Algorithm. `DerivativeCheck` Compare user-supplied derivatives (gradients of objective or constraints) to finite-differencing 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 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:`'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. `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 user-defined 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 user-defined 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 (`[]`):`@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.For information on writing a custom plot function, see Plot Functions. `TolFun` Termination tolerance on the function value, a positive scalar. The default is `1e-6`. `TolX` Termination tolerance on `x`, a positive scalar. The default is `1e-6`. `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.

### Trust-Region-Reflective Algorithm Only

The trust-region-reflective algorithm uses the following options:

`JacobMult`

Function handle for Jacobian multiply function. 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. `lsqnonlin` uses `Jinfo` to compute the preconditioner. See Passing Extra Parameters for information on how to supply values for any additional parameters `jmfun` needs.

 Note   `'Jacobian'` must be set to `'on'` for `lsqnonlin` to pass `Jinfo` from `fun` to `jmfun`.

`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 finite-difference 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`.

### Levenberg-Marquardt Algorithm Only

The Levenberg-Marquardt algorithm uses the following options:

 `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'`.

## Examples

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 user-defined function, the function passed to `lsqnonlin` should instead compute the vector-valued function

${F}_{k}\left(x\right)=2+2k-{e}^{k{x}_{1}}-{e}^{k{x}_{2}},$

for k = 1 to 10 (that is, `F` should have `10` components).

First, write a file to compute the `10`-component vector `F`.

```function F = myfun(x) k = 1:10; F = 2 + 2*k-exp(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```

## Diagnostics

### Memory and Jacobians

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 only find out by running it and seeing if MATLAB runs within the amount of virtual memory available on your system.

### Trust-Region-Reflective Optimization

The trust-region-reflective 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.)

## Limitations

The function to be minimized must be continuous. `lsqnonlin` might only give local solutions.

`lsqnonlin` can solve complex-valued problems directly with the `levenberg-marquardt` 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 `trust-region-reflective` algorithm. See Fit a Model to Complex-Valued Data.

### Trust-Region-Reflective Optimization

The trust-region-reflective 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 Levenberg-Marquardt algorithm is used instead.

The preconditioner computation used in the preconditioned conjugate gradient part of the trust-region-reflective method forms JTJ (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 JTJ, 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.

Trust-Region-Reflective Problem Coverage and Requirements

For Large Problems
• Provide sparsity structure of the Jacobian or compute the Jacobian in `fun`.

• The Jacobian should be sparse.

### Levenberg-Marquardt Optimization

The Levenberg-Marquardt algorithm does not handle bound constraints.

Since the trust-region-reflective algorithm does not handle underdetermined systems and the Levenberg-Marquardt does not handle bound constraints, problems with both these characteristics cannot be solved by `lsqnonlin`.

collapse all

### Trust-Region-Reflective Optimization

By default, `lsqnonlin` chooses the trust-region-reflective algorithm. This 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, and in particular Large Scale Nonlinear Least Squares.

### Levenberg-Marquardt Optimization

If you set the `Algorithm` option to `'levenberg-marquardt'` using `optimoptions`, `lsqnonlin` uses the Levenberg-Marquardt method [4], [5], and [6]. See Levenberg-Marquardt Method.

## 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, pp. 418–445, 1996.

[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, pp. 189-224, 1994.

[3] Dennis, J.E., Jr., "Nonlinear Least-Squares," 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 Least-Squares," Quarterly Applied Math. 2, pp. 164–168, 1944.

[5] Marquardt, D., "An Algorithm for Least-Squares Estimation of Nonlinear Parameters," SIAM Journal Applied Math., Vol. 11, pp. 431–441, 1963.

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