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fzero

Root of nonlinear function

Syntax

  • [x,fval,exitflag,output] = fzero(___) example

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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Root Starting From One Point

Calculate π by finding the zero of the sine function near 3.

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

Root Starting From an Interval

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 cos(1) and cos(2) differ in sign.

Root of a Function Defined by a File

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

Root of Function with Extra Parameter

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

Nondefault Options

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 the options.

x = fzero(fun,x0,options)
x =

    1.8115

Solve Problem Structure

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

More Information from Solution

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 = 
    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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fun — Function to solvefunction handle

Function to solve, specified as a handle to a scalar-valued 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 Parameterizing Functions.

Example: @sin

Example: @myFunction

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

Data Types: function_handle

x0 — Initial valuescalar | 2-element vector

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 — Options for solution processstructure, typically created using optimset

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:

  • '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 Functions.

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.

For information on writing a custom plot function, see Plot Functions.

TolX

Termination tolerance on x, a positive scalar. The default is eps, 2.2204e–16.

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

Data Types: struct

problem — Root-finding problemstructure

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

objective

Objective function

x0

Initial point for x, real scalar or 2-element vector

solver

'fzero'

options

Options structure, typically created using optimset

For an example, see Solve Problem Structure.

Data Types: struct

Output Arguments

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x — Location of root or sign changereal scalar

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

fval — Function value at xreal scalar

Function value at x, returned as a scalar.

exitflag — Integer encoding the exit conditioninteger

Integer encoding the exit condition, meaning the reason fsolve 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.

output — Information about root-finding processstructure

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

More About

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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].

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.

See Also

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