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
$$f(x)=\left[\begin{array}{c}{f}_{1}(x)\\ {f}_{2}(x)\\ \vdots \\ {f}_{n}(x)\end{array}\right].$$
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,exitflag,output]
= lsqnonlin(___)
[x,resnorm,residual,exitflag,output,lambda,jacobian]
= lsqnonlin(___)
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 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 |
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 Components of |
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^{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.
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.