fsolve

Solve system of nonlinear equations

Equation

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.

Syntax

`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(...)`

Description

`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`.

Input Arguments

Function Arguments contains general descriptions of arguments passed into `fsolve`. This section provides function-specific details for `fun` and `problem`:

 `fun` 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 user-defined 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 `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`.) `problem` `objective` Objective function `x0` Initial point for `x` `solver` `'fsolve'` `options` Options created with `optimoptions`

Output Arguments

Function Arguments contains general descriptions of arguments returned by `fsolve`. For more information on the output headings for `fsolve`, see Function-Specific Headings.

This section provides function-specific 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 (`trust-region-dogleg` 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 (trust-region-reflective algorithm only) `stepsize` Final displacement in `x` (Levenberg-Marquardt algorithm) `firstorderopt` Measure of first-order optimality `message` Exit message

Options

Optimization options used by `fsolve`. Some options apply to all algorithms, some are only relevant when using the trust-region-reflective 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

All algorithms use the following options:

 `Algorithm` Choose between `'trust-region-dogleg'` (default), `'trust-region-reflective'`, and `'levenberg-marquardt'`. Set the initial Levenberg-Marquardt parameter λ by setting `Algorithm` to a cell array such as `{'levenberg-marquardt',.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 trust-region-reflective 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. `DerivativeCheck` Compare user-supplied derivatives (gradients of objective or constraints) to finite-differencing 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 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:`'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. `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 user-defined 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 user-defined 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 (`[]`):`@optimplotx` plots the current point.`@optimplotfunccount` plots the function count.`@optimplotfval` plots the function value.`@optimplotresnorm` plots the norm of the residuals.`@optimplotstepsize` plots the step size.`@optimplotfirstorderopt` plots the first-order optimality measure.For information on writing a custom plot function, see Plot Functions. `TolFun` Termination tolerance on the function value, a positive scalar. The default is `1e-6`. `TolX` Termination tolerance on `x`, a positive scalar. The default is `1e-6`. `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.

Trust-Region-Reflective Algorithm Only

The trust-region-reflective algorithm uses the following options:

`JacobMult`

Function handle for Jacobian multiply function. 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   `'Jacobian'` must be set to `'on'` for `fsolve` to pass `Jinfo` from `fun` to `jmfun`.

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 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 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`.

Levenberg-Marquardt Algorithm Only

The Levenberg-Marquardt algorithm uses the following option:

 `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'`.

Examples

Example 1

This example solves 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 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 First-order Trust-region Iteration Func-count 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 3.51575e-006 0.111927 0.00294 2.5 10 33 1.64763e-013 0.00169132 6.36e-007 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 = 1.0e-006 * -0.4059 -0.4059```

Example 2

Find a matrix `x` that satisfies the equation

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

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 = 1.0e-009 * -0.1621 0.0780 0.1167 -0.0465 exitflag = 1```

and the residual is close to zero.

```sum(sum(Fval.*Fval)) ans = 4.8133e-20```

Notes

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:

3x1 + 11x2 – 2x3 = 7
x1 + x2 – 2x3 = 4
x1x2 + x3 = 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```

Diagnostics

Memory and Jacobians

You can use the trust-region reflective algorithm in `lsqnonlin`, `lsqcurvefit`, and `fsolve` with small- to medium-scale problems without computing the Jacobian in `fun` or providing the Jacobian sparsity pattern. (This also applies to using `fmincon` or `fminunc` without computing the Hessian or supplying the Hessian sparsity pattern.) How small is small- to medium-scale? No absolute answer is available, as it depends on the amount of virtual memory in your computer system configuration.

Suppose your problem has `m` equations and `n` unknowns. If the command `J = sparse(ones(m,n))` causes an `Out of memory` error on your machine, then this is certainly too large a problem. If it does not result in an error, the problem might still be too large. You can only find out by running it and seeing if MATLAB runs within the amount of virtual memory available on your system.

All Algorithms

`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.

Trust-Region-Dogleg Algorithm

For the trust-region 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.

Limitations

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.

Trust-Region-Reflective Algorithm

The preconditioner computation used in the preconditioned conjugate gradient part of the trust-region-reflective algorithm forms JTJ (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 JTJ, might lead to a costly solution process for large problems.

Trust-Region-Reflective Problem Coverage and Requirements

For Large Problems
• Provide sparsity structure of the Jacobian or compute the Jacobian in `fun`.

• The Jacobian should be sparse.

Number of Equations

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.

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Algorithms

The Levenberg-Marquardt and trust-region-reflective 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 and Diagnostics following).

• 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 Method.

• The trust-region-reflective 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 Reflective fsolve 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.