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
subject to the constraints
x, lb
, and ub
can be vectors or
matrices; see Matrix Arguments.
Do not specify the objective function as the scalar value (the sum of squares). lsqnonlin
requires the objective
function to be the vector-valued function
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 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 ofub
(orlb
) be set toinf
(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.
More About
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.