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fsolve

Solve system of nonlinear equations

Equation

Solves a problem specified by

F(x) = 0

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

x is a vector or a matrix; see Matrix Arguments.

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.

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 options. Use optimoptions to set these options.

x = fsolve(problem) solves 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.

[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:

fun

The nonlinear system of equations to solve. fun is a function that accepts a vector x and returns a vector F, the nonlinear equations evaluated at x. The function fun can be specified as a function handle for a file

x = fsolve(@myfun,x0)

where myfun is a MATLAB® function such as

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

fun can also be a function handle for an anonymous function.

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

If the user-defined values for x and F are matrices, they are converted to a vector using linear indexing.

If the Jacobian can also be computed and the Jacobian option is 'on', set by

options = optimoptions('fsolve','Jacobian','on')

the function fun must return, in a second output argument, the Jacobian value J, a matrix, at x.

If fun returns a vector (matrix) of m components and x has length n, where n is the length of x0, the Jacobian J is an m-by-n matrix where J(i,j) is the partial derivative of F(i) with respect to x(j). (The Jacobian J is the transpose of the gradient of F.)

problem

objective

Objective function

x0

Initial point for x

solver

'fsolve'

options

Options created with optimoptions

Output Arguments

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

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

exitflag

Integer identifying the reason the algorithm terminated. The following lists the values of exitflag and the corresponding reasons the algorithm terminated.

1

Function converged to a solution x.

2

Change in x was smaller than the specified tolerance.

3

Change in the residual was smaller than the specified tolerance.

4

Magnitude of search direction was smaller than the specified tolerance.

0

Number of iterations exceeded options.MaxIter or number of function evaluations exceeded options.MaxFunEvals.

-1

Output function terminated the algorithm.

-2

Algorithm appears to be converging to a point that is not a root.

-3

Trust region radius became too small (trust-region-dogleg algorithm).

-4

Line search cannot sufficiently decrease the residual along the current search direction.

output

Structure containing information about the optimization. The fields of the structure are

iterations

Number of iterations taken

funcCount

Number of function evaluations

algorithm

Optimization algorithm used

cgiterations

Total number of PCG iterations (trust-region-reflective algorithm only)

stepsize

Final displacement in x (Levenberg-Marquardt algorithm)

firstorderopt

Measure of first-order optimality (dogleg or trust-region-reflective algorithm, [ ] for others)

message

Exit message

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. Use optimoptions to set or change options. See Optimization Options Reference for detailed information.

All Algorithms

All algorithms use the following options:

Algorithm

Choose between 'trust-region-dogleg' (default), 'trust-region-reflective', and 'levenberg-marquardt'. Set the initial Levenberg-Marquardt parameter λ by setting Algorithm to a cell array such as {'levenberg-marquardt',.005}. The default λ = 0.01.

The Algorithm option specifies a preference for which algorithm to use. It is only a preference because for the trust-region-reflective algorithm, the nonlinear system of equations cannot be underdetermined; that is, the number of equations (the number of elements of F returned by fun) must be at least as many as the length of x. Similarly, for the trust-region-dogleg algorithm, the number of equations must be the same as the length of x. fsolve uses the Levenberg-Marquardt algorithm when the selected algorithm is unavailable. For more information on choosing the algorithm, see Choosing the Algorithm.

DerivativeCheck

Compare user-supplied derivatives (gradients of objective or constraints) to finite-differencing derivatives. The choices are 'on' or the default 'off'.

Diagnostics

Display diagnostic information about the function to be minimized or solved. The choices are 'on' or the default 'off'.

DiffMaxChange

Maximum change in variables for finite-difference gradients (a positive scalar). The default is Inf.

DiffMinChange

Minimum change in variables for finite-difference gradients (a positive scalar). The default is 0.

Display

Level of display:

  • 'off' or 'none' displays no output.

  • 'iter' displays output at each iteration, and gives the default exit message.

  • 'iter-detailed' displays output at each iteration, and gives the technical exit message.

  • 'final' (default) displays just the final output, and gives the default exit message.

  • 'final-detailed' displays just the final output, and gives the technical exit message.

FinDiffRelStep

Scalar or vector step size factor. When you set FinDiffRelStep to a vector v, forward finite differences delta are

delta = v.*sign(x).*max(abs(x),TypicalX);

and central finite differences are

delta = v.*max(abs(x),TypicalX);

Scalar FinDiffRelStep expands to a vector. The default is sqrt(eps) for forward finite differences, and eps^(1/3) for central finite differences.

FinDiffType

Finite differences, used to estimate gradients, are either 'forward' (default), or 'central' (centered). 'central' takes twice as many function evaluations, but should be more accurate.

The algorithm is careful to obey bounds when estimating both types of finite differences. So, for example, it could take a backward, rather than a forward, difference to avoid evaluating at a point outside bounds.

FunValCheck

Check whether objective function values are valid. 'on' displays an error when the objective function returns a value that is complex, Inf, or NaN. The default, 'off', displays no error.

Jacobian

If 'on', fsolve uses a user-defined Jacobian (defined in fun), or Jacobian information (when using JacobMult), for the objective function. If 'off' (default), fsolve approximates the Jacobian using finite differences.

MaxFunEvals

Maximum number of function evaluations allowed, a positive integer. The default is 100*numberOfVariables.

MaxIter

Maximum number of iterations allowed, a positive integer. The default is 400.

OutputFcn

Specify one or more user-defined functions that an optimization function calls at each iteration, either as a function handle or as a cell array of function handles. The default is none ([]). See Output Function.

PlotFcns

Plots various measures of progress while the algorithm executes. Select from predefined plots or write your own. Pass a function handle or a cell array of function handles. The default is none ([]):

  • @optimplotx plots the current point.

  • @optimplotfunccount plots the function count.

  • @optimplotfval plots the function value.

  • @optimplotresnorm plots the norm of the residuals.

  • @optimplotstepsize plots the step size.

  • @optimplotfirstorderopt plots the first-order optimality measure.

For information on writing a custom plot function, see Plot Functions.

TolFun

Termination tolerance on the function value, a positive scalar. The default is 1e-6.

TolX

Termination tolerance on x, a positive scalar. The default is 1e-6.

TypicalX

Typical x values. The number of elements in TypicalX is equal to the number of elements in x0, the starting point. The default value is ones(numberofvariables,1). fsolve uses TypicalX for scaling finite differences for gradient estimation.

The trust-region-dogleg algorithm uses TypicalX as the diagonal terms of a scaling matrix.

Trust-Region-Reflective Algorithm Only

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

JacobMult

Function handle for Jacobian multiply function. For large-scale structured problems, this function computes the Jacobian matrix product J*Y, J'*Y, or J'*(J*Y) without actually forming J. The function is of the form

W = jmfun(Jinfo,Y,flag)

where Jinfo contains a matrix used to compute J*Y (or J'*Y, or J'*(J*Y)). The first argument Jinfo must be the same as the second argument returned by the objective function fun, for example, in

[F,Jinfo] = fun(x)

Y is a matrix that has the same number of rows as there are dimensions in the problem. flag determines which product to compute:

  • If flag == 0, W = J'*(J*Y).

  • If flag > 0, W = J*Y.

  • If flag < 0, W = J'*Y.

In each case, J is not formed explicitly. fsolve uses Jinfo to compute the preconditioner. See Passing Extra Parameters for information on how to supply values for any additional parameters jmfun needs.

    Note   'Jacobian' must be set to 'on' for fsolve to pass Jinfo from fun to jmfun.

See Minimization with Dense Structured Hessian, Linear Equalities for a similar example.

 
 

JacobPattern

Sparsity pattern of the Jacobian for finite differencing. Set JacobPattern(i,j) = 1 when fun(i) depends on x(j). Otherwise, set JacobPattern(i,j) = 0. In other words, JacobPattern(i,j) = 1 when you can have ∂fun(i)/∂x(j) ≠ 0.

Use JacobPattern when it is inconvenient to compute the Jacobian matrix J in fun, though you can determine (say, by inspection) when fun(i) depends on x(j). fsolve can approximate J via sparse finite differences when you give JacobPattern.

In the worst case, if the structure is unknown, do not set JacobPattern. The default behavior is as if JacobPattern is a dense matrix of ones. Then fsolve computes a full finite-difference approximation in each iteration. This can be very expensive for large problems, so it is usually better to determine the sparsity structure.

 

MaxPCGIter

Maximum number of PCG (preconditioned conjugate gradient) iterations, a positive scalar. The default is max(1,floor(numberOfVariables/2)). For more information, see Algorithms.

 

PrecondBandWidth

Upper bandwidth of preconditioner for PCG, a nonnegative integer. The default PrecondBandWidth is Inf, which means a direct factorization (Cholesky) is used rather than the conjugate gradients (CG). The direct factorization is computationally more expensive than CG, but produces a better quality step towards the solution. Set PrecondBandWidth to 0 for diagonal preconditioning (upper bandwidth of 0). For some problems, an intermediate bandwidth reduces the number of PCG iterations.

 

TolPCG

Termination tolerance on the PCG iteration, a positive scalar. The default is 0.1.

 

Levenberg-Marquardt Algorithm Only

The Levenberg-Marquardt algorithm uses the following option:

ScaleProblem

'Jacobian' can sometimes improve the convergence of a poorly scaled problem. The default is 'none'.

Examples

Example 1

This example solves the system of two equations and two unknowns:

Rewrite the equations in the form F(x) = 0:

Start your search for a solution at x0 = [-5 -5].

First, write a 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))];

Save this function file as myfun.m somewhere on your MATLAB path. Next, set up the initial point and options and call fsolve:

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

After several iterations, fsolve finds an answer:

                                  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 a file that computes the equations to be solved.

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

Save this function file as myfun.m somewhere on your MATLAB path. Next, set up an initial point and options and call fsolve:

x0 = ones(2,2);  % Make a starting guess at the solution
options = optimoptions('fsolve','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 *
   -0.1621    0.0780
    0.1167   -0.0465

exitflag =
     1

and the residual is close to zero.

sum(sum(Fval.*Fval))
ans = 
   4.8133e-20

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
x1x2 + 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

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 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, might lead to a costly solution process for large problems.

Trust-Region-Reflective Problem Coverage and Requirements

For Large Problems
  • Provide sparsity structure of the Jacobian or compute the Jacobian in fun.

  • The Jacobian should be sparse.

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 method, the system of equations need not be square.

More About

expand all

Algorithms

The 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]. See 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 references [4], [5], and [6]. See Levenberg-Marquardt Method.

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

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