# fzero

Root of nonlinear function

## Description

example

x = fzero(fun,x0) tries to find a point x where fun(x) = 0. This solution is where fun(x) changes sign—fzero cannot find a root of a function such as x^2.

example

x = fzero(fun,x0,options) uses options to modify the solution process.

example

x = fzero(problem) solves a root-finding problem specified by problem.

example

[x,fval,exitflag,output] = fzero(___) returns fun(x) in the fval output, exitflag encoding the reason fzero stopped, and an output structure containing information on the solution process.

## Examples

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Calculate $\pi$ by finding the zero of the sine function near 3.

fun = @sin; % function
x0 = 3; % initial point
x = fzero(fun,x0)
x = 3.1416

Find the zero of cosine between 1 and 2.

fun = @cos; % function
x0 = [1 2]; % initial interval
x = fzero(fun,x0)
x = 1.5708

Note that $\mathrm{cos}\left(1\right)$ and $\mathrm{cos}\left(2\right)$ differ in sign.

Find a zero of the function f(x) = x3 – 2x – 5.

First, write a file called f.m.

function y = f(x)
y = x.^3-2*x-5;

Save f.m on your MATLAB® path.

Find the zero of f(x) near 2.

fun = @f; % function
x0 = 2; % initial point
z = fzero(fun,x0)
z =
2.0946

Since f(x) is a polynomial, you can find the same real zero, and a complex conjugate pair of zeros, using the roots command.

roots([1 0 -2 -5])
ans =
2.0946
-1.0473 + 1.1359i
-1.0473 - 1.1359i

Find the root of a function that has an extra parameter.

myfun = @(x,c) cos(c*x);  % parameterized function
c = 2;                    % parameter
fun = @(x) myfun(x,c);    % function of x alone
x = fzero(fun,0.1)
x = 0.7854

Plot the solution process by setting some plot functions.

Define the function and initial point.

fun = @(x)sin(cosh(x));
x0 = 1;

Examine the solution process by setting options that include plot functions.

options = optimset('PlotFcns',{@optimplotx,@optimplotfval});

Run fzero including options.

x = fzero(fun,x0,options)

x = 1.8115

Solve a problem that is defined by a problem structure.

Define a structure that encodes a root-finding problem.

problem.objective = @(x)sin(cosh(x));
problem.x0 = 1;
problem.solver = 'fzero'; % a required part of the structure
problem.options = optimset(@fzero); % default options

Solve the problem.

x = fzero(problem)
x = 1.8115

Find the point where exp(-exp(-x)) = x, and display information about the solution process.

fun = @(x) exp(-exp(-x)) - x; % function
x0 = [0 1]; % initial interval
options = optimset('Display','iter'); % show iterations
[x fval exitflag output] = fzero(fun,x0,options)

Func-count    x          f(x)             Procedure
2               1     -0.307799        initial
3        0.544459     0.0153522        interpolation
4        0.566101    0.00070708        interpolation
5        0.567143  -1.40255e-08        interpolation
6        0.567143   1.50013e-12        interpolation
7        0.567143             0        interpolation

Zero found in the interval [0, 1]
x = 0.5671
fval = 0
exitflag = 1
output = struct with fields:
intervaliterations: 0
iterations: 5
funcCount: 7
algorithm: 'bisection, interpolation'
message: 'Zero found in the interval [0, 1]'

fval = 0 means fun(x) = 0, as desired.

## Input Arguments

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Function to solve, specified as a handle to a scalar-valued function or the name of such a function. fun accepts a scalar x and returns a scalar fun(x).

fzero solves fun(x) = 0. To solve an equation fun(x) = c(x), instead solve fun2(x) = fun(x) - c(x) = 0.

To include extra parameters in your function, see the example Root of Function with Extra Parameter and the section Passing Extra Parameters.

Example: 'sin'

Example: @myFunction

Example: @(x)(x-a)^5 - 3*x + a - 1

Data Types: char | function_handle | string

Initial value, specified as a real scalar or a 2-element real vector.

• Scalar — fzero begins at x0 and tries to locate a point x1 where fun(x1) has the opposite sign of fun(x0). Then fzero iteratively shrinks the interval where fun changes sign to reach a solution.

• 2-element vector — fzero checks that fun(x0(1)) and fun(x0(2)) have opposite signs, and errors if they do not. It then iteratively shrinks the interval where fun changes sign to reach a solution. An interval x0 must be finite; it cannot contain ±Inf.

Tip

Calling fzero with an interval (x0 with two elements) is often faster than calling it with a scalar x0.

Example: 3

Example: [2,17]

Data Types: double

Options for solution process, specified as a structure. Create or modify the options structure using optimset. fzero uses these options structure fields.

 Display Level of display (see Iterative Display):'off' displays no output.'iter' displays output at each iteration.'final' displays just the final output.'notify' (default) displays output only if the function does not converge. 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. 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 and Plot Function Syntax. PlotFcns Plot 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.@optimplotfval plots the function value. Custom plot functions use the same syntax as output functions. See Output Functions for Optimization Toolbox and Output Function and Plot Function Syntax. TolX Termination tolerance on x, a positive scalar. The default is eps, 2.2204e–16. See Tolerances and Stopping Criteria.

Example: options = optimset('FunValCheck','on')

Data Types: struct

Root-finding problem, specified as a structure with all of the following fields.

 objective Objective function x0 Initial point for x, scalar or 2-D vector solver 'fzero' options Options structure, typically created using optimset

Data Types: struct

## Output Arguments

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Location of root or sign change, returned as a scalar.

Function value at x, returned as a scalar.

Integer encoding the exit condition, meaning the reason fzero stopped its iterations.

 1 Function converged to a solution x. -1 Algorithm was terminated by the output function or plot function. -3 NaN or Inf function value was encountered while searching for an interval containing a sign change. -4 Complex function value was encountered while searching for an interval containing a sign change. -5 Algorithm might have converged to a singular point. -6 fzero did not detect a sign change.

Information about root-finding process, returned as a structure. The fields of the structure are:

 intervaliterations Number of iterations taken to find an interval containing a root iterations Number of zero-finding iterations funcCount Number of function evaluations algorithm 'bisection, interpolation' message Exit message

## Algorithms

The fzero command is a function file. The algorithm, created by T. Dekker, uses a combination of bisection, secant, and inverse quadratic interpolation methods. An Algol 60 version, with some improvements, is given in [1]. A Fortran version, upon which fzero is based, is in [2].

## Alternative Functionality

### App

The Optimize Live Editor task provides a visual interface for fzero.

## References

[1] Brent, R., Algorithms for Minimization Without Derivatives, Prentice-Hall, 1973.

[2] Forsythe, G. E., M. A. Malcolm, and C. B. Moler, Computer Methods for Mathematical Computations, Prentice-Hall, 1976.

## Version History

Introduced before R2006a