fsolve
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.
starts
at x
= fsolve(fun
,x0
)x0
and tries to solve the equations fun(x) =
0,
an array of zeros.
Note
Passing Extra Parameters explains how
to pass extra parameters to the vector function fun(x)
,
if necessary. See Solve Parameterized Equation.
solves
the equations with the optimization options specified in x
= fsolve(fun
,x0
,options
)options
.
Use optimoptions
to set these
options.
Examples
Solution of 2D Nonlinear System
This example shows how to solve two nonlinear equations in two variables. The equations are
$$\begin{array}{c}{e}^{{e}^{({x}_{1}+{x}_{2})}}={x}_{2}\left(1+{x}_{1}^{2}\right)\\ {x}_{1}\mathrm{cos}\left({x}_{2}\right)+{x}_{2}\mathrm{sin}\left({x}_{1}\right)=\frac{1}{2}.\end{array}$$
Convert the equations to the form $$F(x)=0$$.
$$\begin{array}{c}{e}^{{e}^{({x}_{1}+{x}_{2})}}{x}_{2}\left(1+{x}_{1}^{2}\right)=0\\ {x}_{1}\mathrm{cos}\left({x}_{2}\right)+{x}_{2}\mathrm{sin}\left({x}_{1}\right)\frac{1}{2}=0.\end{array}$$
The root2d.m
function, which is available when you run this example, computes the values.
type root2d.m
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.
x = 1×2
0.3532 0.6061
Solution with Nondefault Options
Examine the solution process for a nonlinear system.
Set options to have no display and a plot function that displays the firstorder optimality, which should converge to 0 as the algorithm iterates.
options = optimoptions('fsolve','Display','none','PlotFcn',@optimplotfirstorderopt);
The equations in the nonlinear system are
$$\begin{array}{c}{e}^{{e}^{({x}_{1}+{x}_{2})}}={x}_{2}\left(1+{x}_{1}^{2}\right)\\ {x}_{1}\mathrm{cos}\left({x}_{2}\right)+{x}_{2}\mathrm{sin}\left({x}_{1}\right)=\frac{1}{2}.\end{array}$$
Convert the equations to the form $$F(x)=0$$.
$$\begin{array}{c}{e}^{{e}^{({x}_{1}+{x}_{2})}}{x}_{2}\left(1+{x}_{1}^{2}\right)=0\\ {x}_{1}\mathrm{cos}\left({x}_{2}\right)+{x}_{2}\mathrm{sin}\left({x}_{1}\right)\frac{1}{2}=0.\end{array}$$
The root2d
function computes the lefthand side of these two equations.
type root2d.m
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)
x = 1×2
0.3532 0.6061
Solve Parameterized Equation
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 $$c$$:
$$\begin{array}{c}2{x}_{1}+{x}_{2}=\mathrm{exp}(c{x}_{1})\\ {x}_{1}+2{x}_{2}=\mathrm{exp}(c{x}_{2}).\end{array}$$
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.
x = 1×2
0.1976 0.4255
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.
x = 1×2
0.1788 0.3418
Helper Function
This code creates the paramfun
helper function.
function F = paramfun(x,c) F = [ 2*x(1) + x(2)  exp(c*x(1)) x(1) + 2*x(2)  exp(c*x(2))]; end
Solve a Problem Structure
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 firstorder 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
$$\begin{array}{c}{e}^{{e}^{({x}_{1}+{x}_{2})}}={x}_{2}\left(1+{x}_{1}^{2}\right)\\ {x}_{1}\mathrm{cos}\left({x}_{2}\right)+{x}_{2}\mathrm{sin}\left({x}_{1}\right)=\frac{1}{2}.\end{array}$$
Convert the equations to the form $$F(x)=0$$.
$$\begin{array}{c}{e}^{{e}^{({x}_{1}+{x}_{2})}}{x}_{2}\left(1+{x}_{1}^{2}\right)=0\\ {x}_{1}\mathrm{cos}\left({x}_{2}\right)+{x}_{2}\mathrm{sin}\left({x}_{1}\right)\frac{1}{2}=0.\end{array}$$
The root2d
function computes the lefthand side of these two equations.
type root2d
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.
x = fsolve(problem)
x = 1×2
0.3532 0.6061
Solution Process of Nonlinear System
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 $$F(x)=0$$:
$$\begin{array}{c}2{x}_{1}{x}_{2}{e}^{{x}_{1}}=0\\ {x}_{1}+2{x}_{2}{e}^{{x}_{2}}=0.\end{array}$$
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 Firstorder Trustregion Iteration Funccount 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 7.03149e06 0.111927 0.00294 2.5 10 33 3.29525e13 0.00169132 6.36e07 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 firstorder 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).
Examine Matrix Equation Solution
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} ×
0.2742 0.1258
0.1876 0.0864
exitflag = 1
output = struct with fields:
iterations: 11
funcCount: 52
algorithm: 'trustregiondogleg'
firstorderopt: 4.0197e10
message: 'Equation solved....'
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 = 1.3367e19
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
fun
— Nonlinear equations to solve
function handle  function name
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);
fsolve
passes x
to your objective function in the shape of the x0
argument. For example, if x0
is a 5by3 array, then fsolve
passes x
to fun
as a 5by3 array.
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
byn
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
x0
— Initial point
real vector  real array
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
options
— Optimization options
output of optimoptions
 structure as optimset
returns
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 The To
set some algorithm options using

CheckGradients  Compare usersupplied derivatives
(gradients of objective or constraints) to finitedifferencing derivatives.
The choices are For 
Diagnostics  Display diagnostic information
about the function to be minimized or solved. The choices are 
DiffMaxChange  Maximum change in variables for
finitedifference gradients (a positive scalar). The default is 
DiffMinChange  Minimum change in variables for
finitedifference gradients (a positive scalar). The default is 
Display  Level of display (see Iterative Display):

FiniteDifferenceStepSize  Scalar or vector step size factor for finite differences. When
you set
sign′(x) = sign(x) except sign′(0) = 1 .
Central finite differences are
FiniteDifferenceStepSize expands to a vector. The default
is sqrt(eps) for forward finite differences, and eps^(1/3)
for central finite differences.
For 
FiniteDifferenceType  Finite differences, used to estimate gradients,
are either 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 
FunctionTolerance  Termination tolerance on the function
value, a positive scalar. The default is For 
FunValCheck  Check whether objective function
values are valid. 
MaxFunctionEvaluations  Maximum number of function evaluations allowed, a positive
integer. The default is
For 
MaxIterations  Maximum number of iterations allowed,
a positive integer. The default is For 
OptimalityTolerance  Termination tolerance on the firstorder optimality (a positive
scalar). The default is Internally,
the 
OutputFcn  Specify one or more userdefined functions that an optimization
function calls at each iteration. Pass a function handle
or a cell array of function handles. The default is none
( 
PlotFcn  Plots various measures of progress while the algorithm executes;
select from predefined plots or write your own. Pass a
builtin plot function name, a function handle, or a
cell array of builtin plot function names or function
handles. For custom plot functions, pass function
handles. The default is none
(
Custom plot functions use the same syntax as output functions. See Output Functions for Optimization Toolbox and Output Function and Plot Function Syntax. For

SpecifyObjectiveGradient  If For 
StepTolerance  Termination tolerance on For 
TypicalX  Typical The 
UseParallel  When 
trustregion Algorithm  
JacobianMultiplyFcn  Jacobian multiply function, specified as a function handle. For
largescale structured problems, this function computes
the Jacobian matrix product W = jmfun(Jinfo,Y,flag) where
[F,Jinfo] = fun(x)
In each case, Note
See Minimization with Dense Structured Hessian, Linear Equalities for a similar example. For

JacobPattern  Sparsity pattern of the Jacobian
for finite differencing. Set Use In
the worst case, if the structure is unknown, do not set 
MaxPCGIter  Maximum number of PCG (preconditioned
conjugate gradient) iterations, a positive scalar. The default is 
PrecondBandWidth  Upper bandwidth of preconditioner
for PCG, a nonnegative integer. The default 
SubproblemAlgorithm  Determines how the iteration step
is calculated. The default, 
TolPCG  Termination tolerance on the PCG
iteration, a positive scalar. The default is 
LevenbergMarquardt Algorithm  
InitDamping  Initial value of the LevenbergMarquardt parameter,
a positive scalar. Default is 
ScaleProblem 

Example: options = optimoptions('fsolve','FiniteDifferenceType','central')
problem
— Problem structure
structure
Problem structure, specified as a structure with the following fields:
Field Name  Entry 

 Objective function 
 Initial point for x 
 'fsolve' 
 Options created with optimoptions 
Data Types: struct
Output Arguments
x
— Solution
real vector  real array
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.
fval
— Objective function value at the solution
real vector
Objective function value at the solution, returned as a real vector. Generally,
fval
= fun(x)
.
exitflag
— Reason fsolve
stopped
integer
Reason fsolve
stopped, returned as an integer.
 Equation solved. Firstorder optimality is small. 
 Equation solved. Change in 
 Equation solved. Change in residual smaller than the specified tolerance. 
 Equation solved. Magnitude of search direction smaller than specified tolerance. 
 Number of iterations exceeded 
 Output function or plot function stopped the algorithm. 
 Equation not solved. The exit message can have more information. 
 Equation not solved. Trust region radius became too small
( 
output
— Information about the optimization process
structure
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 ( 
stepsize  Final displacement in 
firstorderopt  Measure of firstorder optimality 
message  Exit message 
jacobian
— Jacobian at the solution
real matrix
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 trustregion 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 LevenbergMarquardt 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'trustregion'
and theSubproblemAlgorithm
option to'cg'
.
Algorithms
The LevenbergMarquardt and trustregion methods are based on
the nonlinear leastsquares 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 trustregion 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 TrustRegionDogleg Algorithm.The trustregion algorithm is a subspace trustregion method and is based on the interiorreflective 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 TrustRegion Algorithm.
The LevenbergMarquardt method is described in references [4], [5], and [6]. See LevenbergMarquardt 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. 418445, 1996.
[2] Coleman, T.F. and Y. Li, “On the Convergence of Reflective Newton Methods for LargeScale Nonlinear Minimization Subject to Bounds,” Mathematical Programming, Vol. 67, Number 2, pp. 189224, 1994.
[3] Dennis, J. E. Jr., “Nonlinear LeastSquares,” State of the Art in Numerical Analysis, ed. D. Jacobs, Academic Press, pp. 269312.
[4] Levenberg, K., “A Method for the Solution of Certain Problems in LeastSquares,” Quarterly Applied Mathematics 2, pp. 164168, 1944.
[5] Marquardt, D., “An Algorithm for Leastsquares Estimation of Nonlinear Parameters,” SIAM Journal Applied Mathematics, Vol. 11, pp. 431441, 1963.
[6] Moré, J. J., “The LevenbergMarquardt Algorithm: Implementation and Theory,” Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics 630, Springer Verlag, pp. 105116, 1977.
[7] Moré, J. J., B. S. Garbow, and K. E. Hillstrom, User Guide for MINPACK 1, Argonne National Laboratory, Rept. ANL8074, 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
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
fsolve
supports code generation using either thecodegen
(MATLAB Coder) function or the MATLAB Coder™ app. You must have a MATLAB Coder license to generate code.The target hardware must support standard doubleprecision floatingpoint computations. You cannot generate code for singleprecision or fixedpoint 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 blackbox function as an objective function for
fsolve
. You can usecoder.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 theproblem
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 usingoptimoptions
. The options must include theAlgorithm
option, set to'levenbergmarquardt'
.options = optimoptions('fsolve','Algorithm','levenbergmarquardt'); [x,fval,exitflag] = fsolve(fun,x0,options);
Code generation supports these options:
Algorithm
— Must be'levenbergmarquardt'
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','levenbergmarquardt'); 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.
Automatic Parallel Support
Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.
To run in parallel, set the 'UseParallel'
option to true
.
options = optimoptions('
solvername
','UseParallel',true)
For more information, see Using Parallel Computing in Optimization Toolbox.
Version History
Introduced before R2006a
See Also
fzero
 lsqcurvefit
 lsqnonlin
 optimoptions
 Optimize
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