fsolve - Solve system of nonlinear equations

Equation

Solves a problem specified by

F(x) = 0

for x, where x is a vector and F(x) is a function that returns a vector value.

Syntax

x = fsolve(fun,x0)
x = fsolve(fun,x0,options)
x = fsolve(problem)
[x,fval] = fsolve(fun,x0)
[x,fval,exitflag] = fsolve(...)
[x,fval,exitflag,output] = fsolve(...)
[x,fval,exitflag,output,jacobian] = fsolve(...)

Description

fsolve finds a root (zero) of a system of nonlinear equations.

Note   Passing Extra Parameters explains how to pass extra parameters to the system of equations, if necessary.

x = fsolve(fun,x0) starts at x0 and tries to solve the equations described in fun.

x = fsolve(fun,x0,options) solves the equations with the optimization options specified in the structure options. Use optimset to set these options.

x = fsolve(problem) solves problem, where problem is a structure described in Input Arguments.

Create the structure problem by exporting a problem from Optimization Tool, as described in Exporting to the MATLAB Workspace.

[x,fval] = fsolve(fun,x0) returns the value of the objective function fun at the solution x.

[x,fval,exitflag] = fsolve(...) returns a value exitflag that describes the exit condition.

[x,fval,exitflag,output] = fsolve(...) returns a structure output that contains information about the optimization.

[x,fval,exitflag,output,jacobian] = fsolve(...) returns the Jacobian of fun at the solution x.

Input Arguments

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

x = fsolve(@myfun,x0)

function F = myfun(x)
F = ...            % Compute function values at x

x = fsolve(@(x)sin(x.*x),x0);

options = optimset('Jacobian','on')

Output Arguments

Function Arguments contains general descriptions of arguments returned by fsolve. For more information on the output headings for fsolve, see Function-Specific Output Headings.

This section provides function-specific details for exitflag and output:

Options

Optimization options used by fsolve. Some options apply to all algorithms, some are only relevant when using the trust-region-reflective algorithm, and others are only relevant when using the other algorithms. You can use optimset to set or change the values of these fields in the options structure, options. See Optimization Options for detailed information.

All Algorithms

All algorithms use the following options:

Trust-Region-Reflective Algorithm Only

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

W = jmfun(Jinfo,Y,flag)

[F,Jinfo] = fun(x)

Levenberg-Marquardt Algorithm Only

The Levenberg-Marquardt algorithm uses the following option:

Gauss-Newton Algorithm Only

The Gauss-Newton algorithm uses the following option:

Examples

Example 1

This example finds a zero of the system of two equations and two unknowns:

You want to solve the following system for x

starting at x0 = [-5 -5].

First, write an M-file that computes F, the values of the equations at x.

function F = myfun(x)
F = [2*x(1) - x(2) - exp(-x(1));
      -x(1) + 2*x(2) - exp(-x(2))];

Next, call an optimization routine.

x0 = [-5; -5];           % Make a starting guess at the solution
options=optimset('Display','iter');   % Option to display output
[x,fval] = fsolve(@myfun,x0,options)  % Call optimizer

After 33 function evaluations, a zero is found.

                                  Norm of  First-order Trust-region
Iteration Func-count    f(x)        step   optimality       radius
    0        3       23535.6                2.29e+004        1
    1        6       6001.72           1    5.75e+003        1
    2        9       1573.51           1    1.47e+003        1
    3       12       427.226           1          388        1
    4       15       119.763           1          107        1
    5       18       33.5206           1         30.8        1
    6       21       8.35208           1         9.05        1
    7       24       1.21394           1         2.26        1
    8       27      0.016329    0.759511        0.206      2.5
    9       30  3.51575e-006    0.111927      0.00294      2.5
   10       33  1.64763e-013  0.00169132    6.36e-007      2.5

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the default value of the function tolerance, and
the problem appears regular as measured by the gradient.

x =
    0.5671
    0.5671

fval =
  1.0e-006 *
      -0.4059
      -0.4059

Example 2

Find a matrix x that satisfies the equation

starting at the point x= [1,1; 1,1].

First, write an M-file that computes the equations to be solved.

function F = myfun(x)
F = x*x*x-[1,2;3,4];

Next, invoke an optimization routine.

x0 = ones(2,2);  % Make a starting guess at the solution
options = optimset('Display','off');  % Turn off Display
[x,Fval,exitflag] = fsolve(@myfun,x0,options)

The solution is

x =
    -0.1291    0.8602
     1.2903    1.1612 

Fval =
  1.0e-009 *
   -0.1619    0.0777
    0.1162   -0.0466

exitflag =
     1

and the residual is close to zero.

sum(sum(Fval.*Fval))
ans = 
     4.7935e-020

Notes

If the system of equations is linear, use\ (matrix left division) for better speed and accuracy. For example, to find the solution to the following linear system of equations:

3x₁ + 11x₂ – 2x₃ = 7
x₁ + x₂ – 2x₃ = 4
x₁ – x₂ + x₃ = 19.

Formulate and solve the problem as

A = [ 3 11 -2; 1 1 -2; 1 -1 1];
b = [ 7; 4; 19];
x = A\b
x =
   13.2188
   -2.3438
    3.4375

Algorithm

The Gauss-Newton, Levenberg-Marquardt, and trust-region-reflective methods are based on the nonlinear least-squares algorithms also used in lsqnonlin. Use one of these methods if the system may not have a zero. The algorithm still returns a point where the residual is small. However, if the Jacobian of the system is singular, the algorithm might converge to a point that is not a solution of the system of equations (see Limitations and Diagnostics following).

• By default fsolve chooses the trust-region dogleg algorithm. The algorithm is a variant of the Powell dogleg method described in [8]. It is similar in nature to the algorithm implemented in [7]. It is described in the User's Guide in Trust-Region Dogleg Method.

• The 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 fsolve Algorithm.

• The Levenberg-Marquardt method is described in [4], [5], and [6]. It is described in the User's Guide in Levenberg-Marquardt Method.

• The medium-scale Gauss-Newton method [3] with line search is described in the User's Guide in Gauss-Newton Method. The default line search algorithm for the Gauss-Newton algorithm , i.e., the LineSearchType option, is 'quadcubic'. This is a safeguarded mixed quadratic and cubic polynomial interpolation and extrapolation method. A safeguarded cubic polynomial method can be selected by setting LineSearchType to 'cubicpoly'. This method generally requires fewer function evaluations but more gradient evaluations. Thus, if gradients are being supplied and can be calculated inexpensively, the cubic polynomial line search method is preferable.

Diagnostics

All Algorithms

fsolve may converge to a nonzero point and give this message:

Optimizer is stuck at a minimum that is not a root
Try again with a new starting guess

In this case, run fsolve again with other starting values.

Trust-Region-Dogleg Algorithm

For the trust-region dogleg method, fsolve stops if the step size becomes too small and it can make no more progress. fsolve gives this message:

The optimization algorithm can make no further progress:
 Trust region radius less than 10*eps

In this case, run fsolve again with other starting values.

Limitations

The function to be solved must be continuous. When successful, fsolve only gives one root. fsolve may converge to a nonzero point, in which case, try other starting values.

fsolve only handles real variables. When x has complex variables, the variables must be split into real and imaginary parts.

Trust-Region-Reflective Algorithm

The preconditioner computation used in the preconditioned conjugate gradient part of the trust-region-reflective algorithm forms J^TJ (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^TJ, might lead to a costly solution process for large problems.

Large-Scale Problem Coverage and Requirements

Number of Equations

The default trust-region dogleg method can only be used when the system of equations is square, i.e., the number of equations equals the number of unknowns. For the Levenberg-Marquardt and Gauss-Newton methods, the system of equations need not be square.

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.

[4] Levenberg, K., "A Method for the Solution of Certain Problems in Least-Squares," Quarterly Applied Mathematics 2, pp. 164-168, 1944.

[5] Marquardt, D., "An Algorithm for Least-squares Estimation of Nonlinear Parameters," SIAM Journal Applied Mathematics, 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.

[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.

See Also

@ (function_handle), \ (matrix left division), lsqcurvefit, lsqnonlin, optimset, optimtool, Anonymous Functions

For more details about the fsolve algorithms, see Equation Solving. For more examples of equation solving, see Equation Solving Examples.

fseminf

fzero

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