# fsolve

Solve system of nonlinear equations

## Syntax

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

## 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.

example

x = fsolve(fun,x0) starts at x0 and tries to solve the equations fun(x) = 0, an array of zeros.

example

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

example

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.

example

[x,fval] = fsolve(___), for any syntax, returns the value of the objective function fun at the solution x.

example

[x,fval,exitflag,output] = fsolve(___) additionally returns a value exitflag that describes the exit condition of fsolve, and a structure output with information about the optimization process.
[x,fval,exitflag,output,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

Convert the equations to the form .

Write a function that 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; 

Save this code as a file named root2d.m on your MATLAB® path.

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. x = 0.3532 0.6061 

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

Convert the equations to the form .

Write a function that 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; 

Save this code as a file named root2d.m on your MATLAB® path.

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) 
x = 0.3532 0.6061 

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

Convert the equations to the form .

Write a function that 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; 

Save this code as a file named root2d.m on your MATLAB® path.

Create the remaining fields in the problem structure.

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

Solve the problem.

x = fsolve(problem) 
x = 0.3532 0.6061 

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

$\begin{array}{c}2{x}_{1}-{x}_{2}={e}^{-{x}_{1}}\\ -{x}_{1}+2{x}_{2}={e}^{-{x}_{2}}.\end{array}$

Rewrite the equations in the form :

$\begin{array}{c}2{x}_{1}-{x}_{2}-{e}^{-{x}_{1}}=0\\ -{x}_{1}+2{x}_{2}-{e}^{-{x}_{2}}=0.\end{array}$

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) [2*x(1) - x(2) - exp(-x(1)); -x(1) + 2*x(2) - exp(-x(2))];

Create the initial point x0.

x0 = [-5;-5];

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) 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. 
x = 2×1 0.5671 0.5671 
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

$X*X*X=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$,

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)x*x*x - [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 × -0.1618 0.0778 0.1160 -0.0474 
exitflag = 1 
output = struct with fields: iterations: 6 funcCount: 35 algorithm: 'trust-region-dogleg' firstorderopt: 2.4095e-10 message: '...' 

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.

sum(sum(fval.*fval))
ans = 4.7957e-20 

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

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 arrays, they are converted to vectors using linear indexing (see Array Indexing (MATLAB)).

If the Jacobian can also be computed and the 'SpecifyObjectiveGradient' option is true, 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 the optimoptions display. These options appear in italics in the following table. For details, see View Options.

All Algorithms
Algorithm

Choose between 'trust-region-dogleg' (default), 'trust-region', and 'levenberg-marquardt'.

The Algorithm option specifies a preference for which algorithm to use. It is only a preference because for the trust-region 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.

To set some algorithm options using optimset instead of optimoptions:

• Algorithm — Set the algorithm to 'trust-region-reflective' instead of 'trust-region'.

• InitDamping — Set the initial Levenberg-Marquardt parameter λ by setting Algorithm to a cell array such as {'levenberg-marquardt',.005}.

CheckGradients

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

For optimset, the name is DerivativeCheck and the values are 'on' or 'off'. See Current and Legacy Option Name Tables.

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 (see Iterative 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.

FiniteDifferenceStepSize

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

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

where sign′(x) = sign(x) except sign′(0) = 1. Central finite differences are

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

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

For optimset, the name is FinDiffRelStep. See Current and Legacy Option Name Tables.

FiniteDifferenceType

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.

For optimset, the name is FinDiffType. See Current and Legacy Option Name Tables.

FunctionTolerance

Termination tolerance on the function value, a positive scalar. The default is 1e-6. See Tolerances and Stopping Criteria.

For optimset, the name is TolFun. See Current and Legacy Option Name Tables.

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.

MaxFunctionEvaluations

Maximum number of function evaluations allowed, a positive integer. The default is 100*numberOfVariables. See Tolerances and Stopping Criteria and Iterations and Function Counts.

For optimset, the name is MaxFunEvals. See Current and Legacy Option Name Tables.

MaxIterations

Maximum number of iterations allowed, a positive integer. The default is 400. See Tolerances and Stopping Criteria and Iterations and Function Counts.

For optimset, the name is MaxIter. See Current and Legacy Option Name Tables.

OptimalityTolerance

Termination tolerance on the first-order optimality (a positive scalar). The default is 1e-6. See First-Order Optimality Measure.

Internally, the 'levenberg-marquardt' algorithm uses an optimality tolerance (stopping criterion) of 1e-4 times FunctionTolerance and does not use OptimalityTolerance.

OutputFcn

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

PlotFcn

Plots various measures of progress while the algorithm executes; select from predefined plots or write your own. Pass a built-in plot function name, a function handle, or a cell array of built-in plot function names or function handles. For custom plot functions, pass function handles. The default is none ([]):

• 'optimplotx' plots the current point.

• 'optimplotfunccount' plots the function count.

• 'optimplotfval' plots the function value.

• 'optimplotstepsize' plots the step size.

• 'optimplotfirstorderopt' plots the first-order optimality measure.

Custom plot functions use the same syntax as output functions. See Output Functions and Output Function Syntax.

For optimset, the name is PlotFcns. See Current and Legacy Option Name Tables.

SpecifyObjectiveGradient

If true, fsolve uses a user-defined Jacobian (defined in fun), or Jacobian information (when using JacobianMultiplyFcn), for the objective function. If false (default), fsolve approximates the Jacobian using finite differences.

For optimset, the name is Jacobian and the values are 'on' or 'off'. See Current and Legacy Option Name Tables.

StepTolerance

Termination tolerance on x, a positive scalar. The default is 1e-6. See Tolerances and Stopping Criteria.

For optimset, the name is TolX. See Current and Legacy Option Name Tables.

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.

UseParallel

When true, fsolve estimates gradients in parallel. Disable by setting to the default, false. See Parallel Computing.

trust-region Algorithm
JacobianMultiplyFcn

Jacobian multiply function, specified as a function handle. 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

'SpecifyObjectiveGradient' must be set to true for fsolve to pass Jinfo from fun to jmfun.

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

For optimset, the name is JacobMult. See Current and Legacy Option Name Tables.

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 Equation Solving 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.

SubproblemAlgorithm

Determines how the iteration step is calculated. The default, 'factorization', takes a slower but more accurate step than 'cg'. See Trust-Region Algorithm.

TolPCG

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

Levenberg-Marquardt Algorithm
InitDamping

Initial value of the Levenberg-Marquardt parameter, a positive scalar. Default is 1e-2. For details, see Levenberg-Marquardt Method.

ScaleProblem

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

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

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

Field NameEntry

objective

Objective function

x0

Initial point for x

solver

'fsolve'

options

Options created with optimoptions

The simplest way of obtaining a problem structure is to export the problem from the Optimization app.

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.

 1 Equation solved. First-order optimality is small. 2 Equation solved. Change in x smaller than the specified tolerance. 3 Equation solved. Change in residual smaller than the specified tolerance. 4 Equation solved. Magnitude of search direction smaller than specified tolerance. 0 Number of iterations exceeded options.MaxIterations or number of function evaluations exceeded options.MaxFunctionEvaluations. -1 Output function or plot function stopped the algorithm. -2 Equation not solved. The exit message can have more information. -3 Equation not solved. Trust region radius became too small (trust-region-dogleg algorithm).

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

 iterations Number of iterations taken funcCount Number of function evaluations algorithm Optimization algorithm used cgiterations Total number of PCG iterations ('trust-region' algorithm only) stepsize Final displacement in x (not in 'trust-region-dogleg') firstorderopt Measure of first-order optimality message Exit message

Jacobian at the solution, returned as a real matrix. jacobian(i,j) is the partial derivative of fun(i) with respect to x(j) at the solution x.

## 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.

## 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 the SubproblemAlgorithm 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.

## 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.