# fzero

Root of nonlinear function

## Syntax

``x = fzero(fun,x0)``
``x = fzero(fun,x0,options)``
``x = fzero(problem)``
``````[x,fval,exitflag,output] = fzero(___)``````

## 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.