fsolve - Solve system of nonlinear equations - MATLAB (original) (raw)

Solve system of nonlinear equations

Syntax

Description

Nonlinear system solver

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.

[x](#buta%5F%5Fs%5Fsep%5Fshared-x) = fsolve([fun](#buta%5F%5Fs-fun),[x0](#buta%5F%5Fs-x0)) starts at x0 and tries to solve the equations fun(x) = 0, an array of zeros.

example

[x](#buta%5F%5Fs%5Fsep%5Fshared-x) = fsolve([fun](#buta%5F%5Fs-fun),[x0](#buta%5F%5Fs-x0),[options](#buta%5F%5Fs-options)) solves the equations with the optimization options specified in options. Use optimoptions to set these options.

example

[[x](#buta%5F%5Fs%5Fsep%5Fshared-x),[fval](#buta%5F%5Fs-fval)] = fsolve(___), for any syntax, returns the value of the objective function fun at the solution x.

example

[[x](#buta%5F%5Fs%5Fsep%5Fshared-x),[fval](#buta%5F%5Fs-fval),[exitflag](#buta%5F%5Fs-exitflag),[output](#buta%5F%5Fs-output)] = fsolve(___) additionally returns a value exitflag that describes the exit condition of fsolve, and a structure output with information about the optimization process.

example

[[x](#buta%5F%5Fs%5Fsep%5Fshared-x),[fval](#buta%5F%5Fs-fval),[exitflag](#buta%5F%5Fs-exitflag),[output](#buta%5F%5Fs-output),[jacobian](#buta%5F%5Fs%5Fsep%5Fshared-jacobian)] = fsolve(___) returns the Jacobian of fun at the solution x.

Examples

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This example shows how to solve two nonlinear equations in two variables. The equations are

e-e-(x1+x2)=x2(1+x12)x1cos(x2)+x2sin(x1)=12.

Convert the equations to the form F(x)=0.

e-e-(x1+x2)-x2(1+x12)=0x1cos(x2)+x2sin(x1)-12=0.

The root2d.m function, which is available when you run this example, computes the values.

function F = root2d(x)

F(1) = exp(-exp(-(x(1)+x(2)))) - x(2)*(1+x(1)^2); F(2) = x(1)*cos(x(2)) + x(2)*sin(x(1)) - 0.5;

Solve the system of equations starting at the point [0,0].

fun = @root2d; x0 = [0,0]; x = fsolve(fun,x0)

Equation solved.

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

Examine the solution process for a nonlinear system.

Set options to have no display and a plot function that displays the first-order optimality, which should converge to 0 as the algorithm iterates.

options = optimoptions('fsolve','Display','none','PlotFcn',@optimplotfirstorderopt);

The equations in the nonlinear system are

e-e-(x1+x2)=x2(1+x12)x1cos(x2)+x2sin(x1)=12.

Convert the equations to the form F(x)=0.

e-e-(x1+x2)-x2(1+x12)=0x1cos(x2)+x2sin(x1)-12=0.

The root2d function computes the left-hand side of these two equations.

function F = root2d(x)

F(1) = exp(-exp(-(x(1)+x(2)))) - x(2)*(1+x(1)^2); F(2) = x(1)*cos(x(2)) + x(2)*sin(x(1)) - 0.5;

Solve the nonlinear system starting from the point [0,0] and observe the solution process.

fun = @root2d; x0 = [0,0]; x = fsolve(fun,x0,options)

You can parameterize equations as described in the topic Passing Extra Parameters. For example, the paramfun helper function at the end of this example creates the following equation system parameterized by c:

2x1+x2=exp(cx1)-x1+2x2=exp(cx2).

To solve the system for a particular value, in this case c=-1, set c in the workspace and create an anonymous function in x from paramfun.

c = -1; fun = @(x)paramfun(x,c);

Solve the system starting from the point x0 = [0 1].

x0 = [0 1]; x = fsolve(fun,x0)

Equation solved.

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

To solve for a different value of c, enter c in the workspace and create the fun function again, so it has the new c value.

c = -2; fun = @(x)paramfun(x,c); % fun now has the new c value x = fsolve(fun,x0)

Equation solved.

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

Helper Function

This code creates the paramfun helper function.

function F = paramfun(x,c) F = [ 2x(1) + x(2) - exp(cx(1)) -x(1) + 2x(2) - exp(cx(2))]; end

Create a problem structure for fsolve and solve the problem.

Solve the same problem as in Solution with Nondefault Options, but formulate the problem using a problem structure.

Set options for the problem to have no display and a plot function that displays the first-order optimality, which should converge to 0 as the algorithm iterates.

problem.options = optimoptions('fsolve','Display','none','PlotFcn',@optimplotfirstorderopt);

The equations in the nonlinear system are

e-e-(x1+x2)=x2(1+x12)x1cos(x2)+x2sin(x1)=12.

Convert the equations to the form F(x)=0.

e-e-(x1+x2)-x2(1+x12)=0x1cos(x2)+x2sin(x1)-12=0.

The root2d function computes the left-hand side of these two equations.

function F = root2d(x)

F(1) = exp(-exp(-(x(1)+x(2)))) - x(2)*(1+x(1)^2); F(2) = x(1)*cos(x(2)) + x(2)*sin(x(1)) - 0.5;

Create the remaining fields in the problem structure.

problem.objective = @root2d; problem.x0 = [0,0]; problem.solver = 'fsolve';

Solve the problem.

This example returns the iterative display showing the solution process for the system of two equations and two unknowns

2x1-x2=e-x1-x1+2x2=e-x2.

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

2x1-x2-e-x1=0-x1+2x2-e-x2=0.

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

First, write a function that computes F, the values of the equations at x.

F = @(x) [2x(1) - x(2) - exp(-x(1)); -x(1) + 2x(2) - exp(-x(2))];

Create the initial point x0.

Set options to return iterative display.

options = optimoptions('fsolve','Display','iter');

Solve the equations.

[x,fval] = fsolve(F,x0,options)

                                         Norm of      First-order   Trust-region

Iteration Func-count ||f(x)||^2 step optimality radius 0 3 47071.2 2.29e+04 1 1 6 12003.4 1 5.75e+03 1 2 9 3147.02 1 1.47e+03 1 3 12 854.452 1 388 1 4 15 239.527 1 107 1 5 18 67.0412 1 30.8 1 6 21 16.7042 1 9.05 1 7 24 2.42788 1 2.26 1 8 27 0.032658 0.759511 0.206 2.5 9 30 7.03149e-06 0.111927 0.00294 2.5 10 33 3.29525e-13 0.00169132 6.36e-07 2.5

Equation solved.

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

fval = 2×1 10-6 ×

-0.4059 -0.4059

The iterative display shows f(x), which is the square of the norm of the function F(x). This value decreases to near zero as the iterations proceed. The first-order optimality measure likewise decreases to near zero as the iterations proceed. These entries show the convergence of the iterations to a solution. For the meanings of the other entries, see Iterative Display.

The fval output gives the function value F(x), which should be zero at a solution (to within the FunctionTolerance tolerance).

Find a matrix X that satisfies

X*X*X=[1234],

starting at the point x0 = [1,1;1,1]. Create an anonymous function that calculates the matrix equation and create the point x0.

fun = @(x)xxx - [1,2;3,4]; x0 = ones(2);

Set options to have no display.

options = optimoptions('fsolve','Display','off');

Examine the fsolve outputs to see the solution quality and process.

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

x = 2×2

-0.1291 0.8602 1.2903 1.1612

fval = 2×2 10-9 ×

-0.2742 0.1258 0.1876 -0.0864

output = struct with fields: iterations: 11 funcCount: 52 algorithm: 'trust-region-dogleg' firstorderopt: 4.0197e-10 message: 'Equation solved.↵↵fsolve completed because the vector of function values is near zero↵as measured by the value of the function tolerance, and↵the problem appears regular as measured by the gradient.↵↵↵↵Equation solved. The sum of squared function values, r = 1.336702e-19, is less than↵sqrt(options.FunctionTolerance) = 1.000000e-03. The relative norm of the gradient of r,↵4.019681e-10, is less than options.OptimalityTolerance = 1.000000e-06.'

The exit flag value 1 indicates that the solution is reliable. To verify this manually, calculate the residual (sum of squares of fval) to see how close it is to zero.

This small residual confirms that x is a solution.

You can see in the output structure how many iterations and function evaluations fsolve performed to find the solution.

Input Arguments

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Nonlinear equations to solve, specified as a function handle or function name. fun is a function that accepts a vector x and returns a vector F, the nonlinear equations evaluated at x. The equations to solve are F = 0 for all components of F. The function fun can be specified as a function handle for a file

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);

fsolve passes x to your objective function in the shape of the x0 argument. For example, if x0 is a 5-by-3 array, then fsolve passes x to fun as a 5-by-3 array.

If the Jacobian can also be computed and the'SpecifyObjectiveGradient' option istrue, set by

options = optimoptions('fsolve','SpecifyObjectiveGradient',true)

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.)

Example: fun = @(x)x*x*x-[1,2;3,4]

Data Types: char | function_handle | string

Initial point, specified as a real vector or real array. fsolve uses the number of elements in and size of x0 to determine the number and size of variables that fun accepts.

Example: x0 = [1,2,3,4]

Data Types: double

Optimization options, specified as the output of optimoptions or a structure such as optimset returns.

Some options apply to all algorithms, and others are relevant for particular algorithms. See Optimization Options Reference for detailed information.

Some options are absent from theoptimoptions display. These options appear in italics in the following table. For details, see View Optimization Options.

Example: options = optimoptions('fsolve','FiniteDifferenceType','central')

Problem structure, specified as a structure with the following fields:

Data Types: struct

Output Arguments

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Solution, returned as a real vector or real array. The size of x is the same as the size of x0. Typically, x is a local solution to the problem when exitflag is positive. For information on the quality of the solution, see When the Solver Succeeds.

Objective function value at the solution, returned as a real vector. Generally,fval = fun(x).

Reason fsolve stopped, returned as an integer.

Information about the optimization process, returned as a structure with fields:

Limitations

• The function to be solved must be continuous.
• When successful, fsolve only gives one root.
• 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

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The next few items list the possible enhanced exit messages fromfsolve. Enhanced exit messages give a link for more information as the first sentence of the message.

The initial point seems to be a solution of the equation, because the sum of squares of function values is less than the square root of the FunctionTolerance tolerance. The size of the gradient of the sum of squares is also less than OptimalityTolerance (1e-4*OptimalityTolerance for the Levenberg-Marquardt algorithm).

For suggestions on how to proceed, see Final Point Equals Initial Point.

The solver found a point where the sum of squares of function values is less than the square root of the FunctionTolerance tolerance. However, the last step was less than the StepTolerance tolerance, indicating the function may be changing rapidly, or that the function is not smooth near the final point. This is the meaning of stalled.

For suggestions on how to proceed, see Local Minimum Possible.

The solver found a point where the sum of squares of function values is less than the square root of the FunctionTolerance tolerance. However, the sum of squares changed very little in the last step, even though the gradient of the sum was larger than OptimalityTolerance (1e-4*OptimalityTolerance for the Levenberg-Marquardt algorithm). This can indicate that the reported point is not near a solution.

For suggestions on how to proceed, see Local Minimum Possible.

The next few items contain definitions for terms in the fsolve exit messages.

To solve a system of equations F(x) = 0, the solver generally attempts to minimize the sum of squared function values r = Σ(Fi(x))2. Both r and ∇r should be zero at a solution.

Generally, a tolerance is a threshold which, if crossed, stops the iterations of a solver. For more information on tolerances, see Tolerances and Stopping Criteria.

The tolerance called OptimalityTolerance relates to the first-order optimality measure. Iterations end when the first-order optimality measure is less than OptimalityTolerance.

The first-order optimality measure is the size of the gradient of the sum of squares of function values. This should be zero at the root of a smooth function.

The function tolerance calledFunctionTolerance relates to the size of the latest change in sum of squares of function values.

StepTolerance is a tolerance for the size of the last step, meaning the size of the change in location wherefsolve was evaluated.

The problem appears to be regular means the size of the gradient of the sum of squares of function values is less than the OptimalityTolerance tolerance (1e-4*OptimalityTolerance for the Levenberg-Marquardt algorithm).

The solver was unable to reduce the sum of squares of function values to below the square root of the FunctionTolerance tolerance. Its last iteration did not reduce the sum of squares enough to warrant further attempts.

For suggestions on how to proceed, see fsolve Could Not Solve Equation.

The trust region is too small to continue. This could be because the sum of squares of function values is not close to a quadratic model. For more information, see Trust-Region-Dogleg Algorithm.

The trust region is too small to continue. This could be because the sum of squares of function values is not close to a quadratic model. For more information, see Trust-Region-Dogleg Algorithm.

The Levenberg-Marquardt regularization parameter is related to the inverse of a trust-region radius. It becomes large when the sum of squares of function values is not close to a quadratic model. For more information, see Levenberg-Marquardt Method.

Tips

• For large problems, meaning those with thousands of variables or more, save memory (and possibly save time) by setting the Algorithm option to 'trust-region' and theSubproblemAlgorithm option to'cg'.

Algorithms

The Levenberg-Marquardt and trust-region 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).

• 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 Algorithm.
• The trust-region 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 Algorithm.
• The Levenberg-Marquardt method is described in references [4], [5], and [6]. See Levenberg-Marquardt Method.

Alternative Functionality

App

The Optimize Live Editor task provides a visual interface for fsolve.

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.

Extended Capabilities

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• fsolve supports code generation using either the codegen (MATLAB Coder) function or the MATLAB Coder™ app. You must have a MATLAB Coder license to generate code.
• The target hardware must support standard double-precision floating-point computations. You cannot generate code for single-precision or fixed-point computations.
• Code generation targets do not use the same math kernel libraries as MATLAB solvers. Therefore, code generation solutions can vary from solver solutions, especially for poorly conditioned problems.
• All code for generation must be MATLAB code. In particular, you cannot use a custom black-box function as an objective function for fsolve. You can use coder.ceval to evaluate a custom function coded in C or C++. However, the custom function must be called in a MATLAB function.
• fsolve does not support the problem argument for code generation.
  [x,fval] = fsolve(problem) % Not supported
• You must specify the objective function by using function handles, not strings or character names.
  x = fsolve(@fun,x0,options) % Supported
  % Not supported: fsolve('fun',...) or fsolve("fun",...)
• For advanced code optimization involving embedded processors, you also need an Embedded Coder® license.
• You must include options for fsolve and specify them using optimoptions. The options must include the Algorithm option, set to 'levenberg-marquardt'.
  options = optimoptions('fsolve','Algorithm','levenberg-marquardt');
  [x,fval,exitflag] = fsolve(fun,x0,options);
• Code generation supports these options:
  • Algorithm — Must be'levenberg-marquardt'
  • FiniteDifferenceStepSize
  • FiniteDifferenceType
  • FunctionTolerance
  • MaxFunctionEvaluations
  • MaxIterations
  • SpecifyObjectiveGradient
  • StepTolerance
  • TypicalX
• Generated code has limited error checking for options. The recommended way to update an option is to use optimoptions, not dot notation.
  opts = optimoptions('fsolve','Algorithm','levenberg-marquardt');
  opts = optimoptions(opts,'MaxIterations',1e4); % Recommended
  opts.MaxIterations = 1e4; % Not recommended
• Do not load options from a file. Doing so can cause code generation to fail. Instead, create options in your code.
• Usually, if you specify an option that is not supported, the option is silently ignored during code generation. However, if you specify a plot function or output function by using dot notation, code generation can issue an error. For reliability, specify only supported options.
• Because output functions and plot functions are not supported, solvers do not return the exit flag –1.

For an example, see Generate Code for fsolve.

Version History

Introduced before R2006a

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The syntax for the JacobianMultiplyFcn option is

W = jmfun(Jinfo, Y, flag)

The Jinfo data, which MATLAB passes to your function jmfun, can now be of any data type. For example, you can now have Jinfo be a structure. In previous releases, Jinfo had to be a standard double array.

The Jinfo data is the second output of your objective function:

The CheckGradients option will be removed in a future release. To check the first derivatives of objective functions or nonlinear constraint functions, use the checkGradients function.