# lsqnonlin

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

## Syntax

## Description

Nonlinear least-squares solver

Solves nonlinear least-squares curve fitting problems of the form

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

subject to the constraints

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

*x*, `lb`

, and `ub`

can be vectors or
matrices; see Matrix Arguments.

Do not specify the objective function as the scalar value $${\Vert f(x)\Vert}_{2}^{2}$$ (the sum of squares). `lsqnonlin`

requires the objective
function to be the *vector*-valued function

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

starts at the point `x`

= lsqnonlin(`fun`

,`x0`

)`x0`

and finds a minimum of the sum of squares of
the functions described in `fun`

. The function `fun`

should return a vector (or array) 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 `fun(x)`

, if necessary.

defines a set of lower and upper bounds on the design variables in `x`

= lsqnonlin(`fun`

,`x0`

,`lb`

,`ub`

)`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.

## Examples

## Input Arguments

## Output Arguments

## 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,`lsqnonlin`

uses the Levenberg-Marquardt algorithm.`lsqnonlin`

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*(where^{T}J*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*, can lead to a costly solution process for large problems.^{T}JIf 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.

You can use the trust-region reflective algorithm in `lsqnonlin`

, `lsqcurvefit`

,
and `fsolve`

with small- to medium-scale
problems without computing the Jacobian in `fun`

or
providing the Jacobian sparsity pattern. (This also applies to using `fmincon`

or `fminunc`

without
computing the Hessian or supplying the Hessian sparsity pattern.)
How small is small- to medium-scale? No absolute answer is available,
as it depends on the amount of virtual memory in your computer system
configuration.

Suppose your problem has `m`

equations and `n`

unknowns.
If the command `J = sparse(ones(m,n))`

causes
an `Out of memory`

error on your machine,
then this is certainly too large a problem. If it does not result
in an error, the problem might still be too large. You can find out
only by running it and seeing if MATLAB runs within the amount
of virtual memory available on your system.

## Algorithms

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

.

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

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

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 `lsqnonlin`

.

## 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

## Version History

**Introduced before R2006a**