What is the equation of a nonlinear system?

A system of nonlinear equations is a system of two or more equations in two or more variables containing at least one equation that is not linear . Recall that a linear equation can take the form Ax+By+C=0 A x + B y + C = 0 . Any equation that cannot be written in this form in nonlinear.

What is an example of a nonlinear system?

Examples of nonlinear differential equations are the Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology . One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions.

Can you solve a non linear equation?

We can also use elimination to solve systems of nonlinear equations . It works well when the equations have both variables squared. When using elimination, we try to make the coefficients of one variable to be opposites, so when we add the equations together, that variable is eliminated.

Solve system of nonlinear equations (2024)

Solve system of nonlinear equations

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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) startsat 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.

example

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

example

x = fsolve(problem) solves problem, a structure described in problem.

example

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

example

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

[x,fval,exitflag,output,jacobian]= fsolve(___) returns the Jacobian of fun atthe solution x.

Examples

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Solution of 2-D Nonlinear System

Open Live Script

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

$\begin{array}{c} e^{- e^{- (x_{1} + x_{2})}} = x_{2} (1 + x_{1}^{2}) \\ x_{1} \cos (x_{2}) + x_{2} \sin (x_{1}) = \frac{1}{2} . \end{array}$

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

$\begin{array}{c} e^{- e^{- (x_{1} + x_{2})}} - x_{2} (1 + x_{1}^{2}) = 0 \\ x_{1} \cos (x_{2}) + x_{2} \sin (x_{1}) - \frac{1}{2} = 0 . \end{array}$

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 zeroas measured by the value of the function tolerance, andthe problem appears regular as measured by the gradient.

x = 1×2 0.3532 0.6061

Solution with Nondefault Options

Open Live Script

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

$\begin{array}{c} e^{- e^{- (x_{1} + x_{2})}} = x_{2} (1 + x_{1}^{2}) \\ x_{1} \cos (x_{2}) + x_{2} \sin (x_{1}) = \frac{1}{2} . \end{array}$

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

$\begin{array}{c} e^{- e^{- (x_{1} + x_{2})}} - x_{2} (1 + x_{1}^{2}) = 0 \\ x_{1} \cos (x_{2}) + x_{2} \sin (x_{1}) - \frac{1}{2} = 0 . \end{array}$

The root2d function computes the left-hand 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

Open Live Script

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 $c$ :

$\begin{array}{c} 2 x_{1} + x_{2} = \exp (c x_{1}) \\ - x_{1} + 2 x_{2} = \exp (c x_{2}) . \end{array}$

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 zeroas measured by the value of the function tolerance, andthe 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 valuex = fsolve(fun,x0)

Equation solved.fsolve completed because the vector of function values is near zeroas measured by the value of the function tolerance, andthe 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

Open Live Script

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

$\begin{array}{c} e^{- e^{- (x_{1} + x_{2})}} = x_{2} (1 + x_{1}^{2}) \\ x_{1} \cos (x_{2}) + x_{2} \sin (x_{1}) = \frac{1}{2} . \end{array}$

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

$\begin{array}{c} e^{- e^{- (x_{1} + x_{2})}} - x_{2} (1 + x_{1}^{2}) = 0 \\ x_{1} \cos (x_{2}) + x_{2} \sin (x_{1}) - \frac{1}{2} = 0 . \end{array}$

The root2d function computes the left-hand 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

Open Live Script

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 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 2.5Equation solved.fsolve completed because the vector of function values is near zeroas measured by the value of the function tolerance, andthe problem appears regular as measured by the gradient.

x = 2×1 0.5671 0.5671

fval = 2×110^-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).

Examine Matrix Equation Solution

Open Live Script

Find a matrix $X$ that satisfies

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

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 = 2×210^-9 × -0.2742 0.1258 0.1876 -0.0864

exitflag = 1

output = struct with fields: iterations: 11 funcCount: 52 algorithm: 'trust-region-dogleg' firstorderopt: 4.0197e-10 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 = 1.3367e-19

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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`fun` — Nonlinear equations to solve
function handle | function name

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

x = fsolve(@myfun,x0)

where myfun is a MATLAB^® function suchas

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

fun can also be a function handle for ananonymous 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 is true, set by

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

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

If fun returns a vector (matrix) of m componentsand x has length n, where n isthe length of x0, the Jacobian J isan m-by-n matrix where J(i,j) isthe partial derivative of F(i) with respect to x(j).(The Jacobian J is the transpose of the gradientof 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 usesthe number of elements in and size of x0 to determinethe 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 ora structure such as optimset returns.

Some options apply to all algorithms, and others are relevantfor 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 Optimization Options.

All Algorithms
`Algorithm`	Choose between `'trust-region-dogleg'` (default), `'trust-region'`,and `'levenberg-marquardt'`. The `Algorithm` optionspecifies a preference for which algorithm to use. It is only a preferencebecause for the trust-region algorithm, the nonlinear system of equationscannot be underdetermined; that is, the number of equations (the numberof 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 mustbe the same as the length of `x`. `fsolve` usesthe Levenberg-Marquardt algorithm when the selected algorithm is unavailable.For more information on choosing the algorithm, see Choosing the Algorithm. Toset some algorithm options using `optimset` insteadof `optimoptions`: `Algorithm` — Set the algorithmto `'trust-region-reflective'` instead of `'trust-region'`. InitDamping — Set theinitial Levenberg-Marquardt parameter λ bysetting `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 Names. The `CheckGradients` option will be removed in a future release. To check derivatives, use the checkGradients function.
Diagnostics	Display diagnostic informationabout the function to be minimized or solved. The choices are `'on'` orthe default `'off'`.
DiffMaxChange	Maximum change in variables forfinite-difference gradients (a positive scalar). The default is `Inf`.
DiffMinChange	Minimum change in variables forfinite-difference gradients (a positive scalar). The default is `0`.
`Display`	Level of display (see Iterative Display): `'off'` or `'none'` displaysno output. `'iter'` displays output at eachiteration, and gives the default exit message. `'iter-detailed'` displays outputat each iteration, and gives the technical exit message. `'final'` (default) displays justthe final output, and gives the default exit message. `'final-detailed'` displays justthe 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);` A 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 Names.
`FiniteDifferenceType`	Finite differences, used to estimate gradients,are either `'forward'` (default), or `'central'` (centered). `'central'` takestwice as many function evaluations, but should be more accurate. Thealgorithm is careful to obey bounds when estimating both types offinite differences. So, for example, it could take a backward, ratherthan a forward, difference to avoid evaluating at a point outsidebounds. For `optimset`, the name is `FinDiffType`. See Current and Legacy Option Names.
`FunctionTolerance`	Termination tolerance on the function value, a nonnegative scalar. The default is `1e-6`. See Tolerances and Stopping Criteria. For `optimset`, the name is `TolFun`. See Current and Legacy Option Names.
FunValCheck	Check whether objective functionvalues are valid. `'on'` displays an error when theobjective function returns a value that is `complex`, `Inf`,or `NaN`. The default, `'off'`,displays no error.
`MaxFunctionEvaluations`	Maximum number of function evaluations allowed, a nonnegative integer. The default is `100numberOfVariables` for the `'trust-region-dogleg'` and `'trust-region'` algorithms, and `200numberOfVariables` for the `'levenberg-marquardt'` algorithm. See Tolerances and Stopping Criteria and Iterations and Function Counts. For `optimset`, the name is `MaxFunEvals`. See Current and Legacy Option Names.
`MaxIterations`	Maximum number of iterations allowed, a nonnegative 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 Names.
`OptimalityTolerance`	Termination tolerance on the first-order optimality (a nonnegative scalar). The default is `1e-6`. See First-Order Optimality Measure. Internally,the `'levenberg-marquardt'` algorithm uses an optimalitytolerance (stopping criterion) of `1e-4` times `FunctionTolerance` anddoes 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 and Plot 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 for Optimization Toolbox and Output Function and Plot Function Syntax. For `optimset`, the name is `PlotFcns`. See Current and Legacy Option Names.
`SpecifyObjectiveGradient`	If `true`, `fsolve` usesa user-defined Jacobian (defined in fun), or Jacobian information (when using `JacobianMultiplyFcn`),for the objective function. If `false` (default), `fsolve` approximatesthe Jacobian using finite differences. For `optimset`, the name is `Jacobian` and the values are `'on'` or `'off'`. See Current and Legacy Option Names.
`StepTolerance`	Termination tolerance on `x`, a nonnegative scalar. The default is `1e-6`. See Tolerances and Stopping Criteria. For `optimset`, the name is `TolX`. See Current and Legacy Option Names.
`TypicalX`	Typical `x` values.The number of elements in `TypicalX` is equal tothe number of elements in `x0`, the starting point.The default value is `ones(numberofvariables,1)`. `fsolve` uses `TypicalX` forscaling finite differences for gradient estimation. The `trust-region-dogleg` algorithmuses `TypicalX` as the diagonal terms of a scalingmatrix.
`UseParallel`	When `true`, `fsolve` estimatesgradients 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 `JY`, `J'Y`, or `J'(JY)` without actually forming `J`. The function is of the form W = jmfun(Jinfo,Y,flag) where `Jinfo` contains data used to compute `JY` (or `J'Y`, or `J'(JY)`). The first argument `Jinfo` is 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'(JY)`. If `flag > 0`, `W = JY`. 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 Names.
JacobPattern	Sparsity pattern of the Jacobianfor finite differencing. Set `JacobPattern(i,j) = 1` when `fun(i)` dependson `x(j)`. Otherwise, set `JacobPattern(i,j)= 0`. In other words, `JacobPattern(i,j) = 1` whenyou can have ∂`fun(i)`/∂`x(j)`≠0. Use `JacobPattern` whenit is inconvenient to compute the Jacobian matrix `J` in `fun`,though you can determine (say, by inspection) when `fun(i)` dependson `x(j)`. `fsolve` can approximate `J` viasparse finite differences when you give `JacobPattern`. Inthe worst case, if the structure is unknown, do not set `JacobPattern`.The default behavior is as if `JacobPattern` is adense matrix of ones. Then `fsolve` computes afull finite-difference approximation in each iteration. This can bevery expensive for large problems, so it is usually better to determinethe sparsity structure.
MaxPCGIter	Maximum number of PCG (preconditionedconjugate gradient) iterations, a positive scalar. The default is `max(1,floor(numberOfVariables/2))`.For more information, see Equation Solving Algorithms.
PrecondBandWidth	Upper bandwidth of preconditionerfor PCG, a nonnegative integer. The default `PrecondBandWidth` is `Inf`,which means a direct factorization (Cholesky) is used rather thanthe conjugate gradients (CG). The direct factorization is computationallymore expensive than CG, but produces a better quality step towardsthe solution. Set `PrecondBandWidth` to `0` fordiagonal preconditioning (upper bandwidth of 0). For some problems,an intermediate bandwidth reduces the number of PCG iterations.
`SubproblemAlgorithm`	Determines how the iteration stepis calculated. The default, `'factorization'`, takesa slower but more accurate step than `'cg'`. See Trust-Region Algorithm.
TolPCG	Termination tolerance on the PCGiteration, 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 theconvergence of a poorly scaled problem. The default is `'none'`.

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

`problem` — Problem structure
structure

Problem structure, specified as a structure with the followingfields:

Field Name	Entry
`objective`	Objective function
`x0`	Initial point for `x`
`solver`	`'fsolve'`
`options`	Options created with optimoptions

Data Types: struct

Output Arguments

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`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.

`1`	Equation solved. First-order optimality is small.
`2`	Equation solved. Change in `x` smaller than the specified tolerance, or Jacobian at `x` is undefined.
`3`	Equation solved. Change in residual smaller than thespecified tolerance.
`4`	Equation solved. Magnitude of search direction smallerthan specified tolerance.
	Number of iterations exceeded `options.MaxIterations` ornumber 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).

`output` — Information about the optimization process
structure

Information about the optimization process, returned as a structurewith fields:

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

Limitations

The function to be solved must be continuous.
When successful, fsolve onlygives one root.
The default trust-region dogleg method can only beused when the system of equations is square, i.e., the number of equationsequals 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 onthe nonlinear least-squares algorithms also used in lsqnonlin. Use one of these methods ifthe system may not have a zero. The algorithm still returns a pointwhere the residual is small. However, if the Jacobian of the systemis singular, the algorithm might converge to a point that is not asolution of the system of equations (see Limitations).

By default fsolve chooses thetrust-region dogleg algorithm. The algorithm is a variant of the Powelldogleg 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-regionmethod and is based on the interior-reflective Newton method describedin [1] and [2]. Each iteration involvesthe approximate solution of a large linear system using the methodof 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,” SIAMJournal on Optimization, Vol. 6, pp. 418-445, 1996.

[2] Coleman, T.F. and Y. Li, “On theConvergence of Reflective Newton Methods for Large-Scale NonlinearMinimization Subject to Bounds,” Mathematical Programming,Vol. 67, Number 2, pp. 189-224, 1994.

[3] Dennis, J. E. Jr., “Nonlinear Least-Squares,” Stateof the Art in Numerical Analysis, ed. D. Jacobs, AcademicPress, pp. 269-312.

[4] Levenberg, K., “A Method for theSolution of Certain Problems in Least-Squares,” QuarterlyApplied Mathematics 2, pp. 164-168, 1944.

[5] Marquardt, D., “An Algorithm forLeast-squares Estimation of Nonlinear Parameters,” SIAMJournal Applied Mathematics, Vol. 11, pp. 431-441, 1963.

[6] Moré, J. J., “The Levenberg-MarquardtAlgorithm: Implementation and Theory,” NumericalAnalysis, 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, ArgonneNational Laboratory, Rept. ANL-80-74, 1980.

[8] Powell, M. J. D., “A Fortran Subroutinefor Solving Systems of Nonlinear Algebraic Equations,” NumericalMethods 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 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); % Recommendedopts.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

expand all

R2023b: `JacobianMultiplyFcn` accepts any data type

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:

[F,Jinfo] = myfun(x)

R2023b: `CheckGradients` option will be removed

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.

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Solve system of nonlinear equations (2024)

Syntax

Description

Examples

Solution of 2-D Nonlinear System

Solution with Nondefault Options

Solve Parameterized Equation

Solve a Problem Structure

Solution Process of Nonlinear System

Examine Matrix Equation Solution

Input Arguments

`fun` — Nonlinear equations to solve
function handle | function name

`x0` — Initial point
real vector | real array

`options` — Optimization options
output of `optimoptions` | structure as `optimset` returns

`problem` — Problem structure
structure

Output Arguments

`fval` — Objective function value at the solution
real vector

`exitflag` — Reason `fsolve` stopped
integer

`output` — Information about the optimization process
structure

Limitations

Tips

Algorithms

Alternative Functionality

App

References

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Automatic Parallel Support
Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

Version History

R2023b: `JacobianMultiplyFcn` accepts any data type

R2023b: `CheckGradients` option will be removed

See Also

Topics

MATLAB Command

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Europe

Asia Pacific

FAQs