Fzreo matlab

fzero

Root of nonlinear function

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Syntax

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

• x = fzero(problem) example

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

Description

example

x = fzero(fun,x0) triesto find a point x where fun(x) = 0. This solution is where fun(x) changessign—fzero cannot find a root of a functionsuch as x^2.

example

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

example

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

example

[x,fval,exitflag,output]= fzero(___) returns fun(x) inthe fval output, exitflag encodingthe reason fzero stopped, and an output structurecontaining information on the solution process.

Examples

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Root Starting From One Point

Calculate 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 and 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 canfind the same real zero, and a complex conjugate pair of zeros, usingthe 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 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.

Related Examples

• Roots of Scalar Functions
• Parameterizing Functions

Input Arguments

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fun — Function to solvefunction handle

Function to solve, specified as a handle to a scalar-valuedfunction. fun accepts a scalar x andreturns 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 andthe 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 realvector.

• Scalar — fzero begins at x0 andtries to locate a point x1 where fun(x1) hasthe opposite sign of fun(x0). Then fzero iterativelyshrinks the interval where fun changes sign toreach a solution.

• 2-element vector — fzero checksthat fun(x0(1)) and fun(x0(2)) haveopposite signs, and errors if they do not. It then iteratively shrinksthe interval where fun changes sign to reach asolution. An interval x0 must be finite; it cannotcontain ±Inf.

Tip  Calling fzero with an interval (x0 withtwo 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. Createor modify the options structure using optimset. fzero usesthese options structure fields.

Display	Level of display: 'off' displays no output. 'iter' displays output at eachiteration. 'final' displays just the finaloutput. 'notify' (default) displays outputonly if the function does not converge.
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 noerror.
OutputFcn	Specify one or more user-definedfunctions that an optimization function calls at each iteration, eitheras a function handle or as a cell array of function handles. The defaultis none ([]). See Output Functions.
PlotFcns	Plot various measures of progresswhile the algorithm executes. Select from predefined plots or writeyour own. Pass a function handle or a cell array of function handles.The default is none ([]). @optimplotx plots the current point. @optimplotfval plots the functionvalue. 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 thefollowing fields.

objective	Objective function
x0	Initial point for x,real scalar or 2-element vector
solver	'fzero'
options	Options structure, typically createdusing 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 stoppedits iterations.

1	Function converged to a solution x.
-1	Algorithm was terminated by the output function or plotfunction.
-3	NaN or Inf functionvalue was encountered while searching for an interval containing asign change.
-4	Complex function value was encountered while searchingfor 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 containinga 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 commandis a function file. The algorithm, created by T. Dekker,uses a combination of bisection, secant, and inverse quadratic interpolationmethods. An Algol 60 version, with some improvements, is given in [1]. A Fortran version, upon which fzero isbased, is in [2].

References

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

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