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Solve system of nonlinear equations
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 = 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(...)
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 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.
Function Arguments contains general descriptions of arguments passed into fsolve. This section provides functionspecific details for fun and problem:
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 userdefined 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 mbyn 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 
Function Arguments contains general descriptions of arguments returned by fsolve. For more information on the output headings for fsolve, see FunctionSpecific Headings.
This section provides functionspecific 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 (trustregiondogleg 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 (trustregionreflective algorithm only)  
stepsize  Final displacement in x (LevenbergMarquardt algorithm)  
firstorderopt  Measure of firstorder optimality (dogleg or trustregionreflective algorithm, [ ] for others)  
message  Exit message 
Optimization options used by fsolve. Some options apply to all algorithms, some are only relevant when using the trustregionreflective 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 use the following options:
Algorithm  Choose between 'trustregiondogleg' (default), 'trustregionreflective', and 'levenbergmarquardt'. Set the initial LevenbergMarquardt parameter λ by setting Algorithm to a cell array such as {'levenbergmarquardt',.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 trustregionreflective 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 trustregiondogleg algorithm, the number of equations must be the same as the length of x. fsolve uses the LevenbergMarquardt algorithm when the selected algorithm is unavailable. For more information on choosing the algorithm, see Choosing the Algorithm. 
DerivativeCheck  Compare usersupplied derivatives (gradients of objective or constraints) to finitedifferencing 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 finitedifference gradients (a positive scalar). The default is Inf. 
DiffMinChange  Minimum change in variables for finitedifference gradients (a positive scalar). The default is 0. 
Display  Level of display:

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 userdefined 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 userdefined 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 ([]):
For information on writing a custom plot function, see Plot Functions. 
TolFun  Termination tolerance on the function value, a positive scalar. The default is 1e6. 
TolX  Termination tolerance on x, a positive scalar. The default is 1e6. 
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 trustregiondogleg algorithm uses TypicalX as the diagonal terms of a scaling matrix. 
The trustregionreflective algorithm uses the following options:
JacobMult  Function handle for Jacobian multiply function. For largescale 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:
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. 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 finitedifference 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. 
The LevenbergMarquardt algorithm uses the following option:
ScaleProblem  'Jacobian' can sometimes improve the convergence of a poorly scaled problem. The default is 'none'. 
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 Firstorder Trustregion Iteration Funccount 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 3.51575e006 0.111927 0.00294 2.5 10 33 1.64763e013 0.00169132 6.36e007 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 = 1.0e006 * 0.4059 0.4059
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 = 1.0e009 * 0.1621 0.0780 0.1167 0.0465 exitflag = 1
and the residual is close to zero.
sum(sum(Fval.*Fval)) ans = 4.8133e20
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:
3x_{1} + 11x_{2} –
2x_{3} = 7
x_{1} + x_{2} –
2x_{3} = 4
x_{1} – x_{2} + x_{3} =
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
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.
For the trustregion 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.
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.
The preconditioner computation used in the preconditioned conjugate gradient part of the trustregionreflective algorithm forms J^{T}J (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^{T}J, might lead to a costly solution process for large problems.
TrustRegionReflective Problem Coverage and Requirements
For Large Problems 


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.
[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.
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