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

Nonlinear least-squares solver

Solves nonlinear least-squares curve fitting problems of the form

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

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.

Rather than compute the value $${\Vert f(x)\Vert}_{2}^{2}$$ (the
sum of squares), `lsqnonlin`

requires the user-defined
function to compute 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)`

.)

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)`

.

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.

finds
the minimum for `x`

= lsqnonlin(`problem`

)`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.

The Levenberg-Marquardt algorithm does not handle bound constraints.

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

`lsqnonlin`

.`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.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.

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.

[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.