OPTSTRUCT

function y = computeSquare(x)

y = x.^2;

end

Create a handle and call the function to compute the square of four.

f = @computeSquare;

a = 4;

b = f(a)

b =

    16

If the function does not require any inputs, then you can call the function with empty parentheses, such as

h = @ones;

a = h()

a =

    1

Without the parentheses, the assignment creates another function handle.

a = h

a = 

    @ones

Function handles are variables that you can pass to other functions. For example, calculate the integral of x2 on the range [0,1].

q = integral(f,0,1);

Function handles store their absolute path, so when you have a valid handle, you can invoke the function from any location. You do not have to specify the path to the function when creating the handle, only the function name.

Keep the following in mind when creating handles to functions:

Name length — Each part of the function name (including package and class names) must be less than the number specified by namelengthmax. Otherwise, MATLAB truncates the latter part of the name.

Scope — The function must be in scope at the time you create the handle. Therefore, the function must be on the MATLAB path or in the current folder. Or, for handles to local or nested functions, the function must be in the current file.

Precedence — When there are multiple functions with the same name, MATLAB uses the same precedence rules to define function handles as it does to call functions. For more information, see Function Precedence Order.

Overloading — If the function you specify overloads a function in a class that is not a fundamental MATLAB class, the function is not associated with the function handle at the time it is constructed. Instead, MATLAB considers the input arguments and determines which implementation to call at the time of evaluation.

Anonymous Functions

You can create handles to anonymous functions. An anonymous function is a one-line expression-based MATLAB function that does not require a program file. Construct a handle to an anonymous function by defining the body of the function, anonymous_function, and a comma-separated list of input arguments to the anonymous function, arglist. The syntax is:

h = @(arglist)anonymous_function

For example, create a handle, sqr, to an anonymous function that computes the square of a number, and call the anonymous function using its handle.

sqr = @(n) n.^2;

x = sqr(3)

x =

     9

For more information, see Anonymous Functions.

Arrays of Function Handles

You can create an array of function handles by collecting them into a cell or structure array. For example, use a cell array:

C = {@sin, @cos, @tan};

C{2}(pi)

ans =

    -1

Or use a structure array:

S.a = @sin;  S.b = @cos;  S.c = @tan;

S.a(pi/2)

ans =

     1

Saving and Loading Function Handles

You can save and load function handles in MATLAB, as you would any other variable. In other words, use the save and load functions. If you save a function handle, MATLAB does not save the path information. If you load a function handle, and the function file no longer exists on the path, the handle is invalid. An invalid handle occurs if the file location or file name has changed since you created the handle. If a handle is invalid, MATLAB still performs the load successfully and without displaying a warning. However, when you invoke the handle, MATLAB issues an error.

Nested Functions

What Are Nested Functions?

A nested function is a function that is completely contained within a parent function. Any function in a program file can include a nested function.

For example, this function named parent contains a nested function named nestedfx:

function parent

disp('This is the parent function')

nestedfx

   function nestedfx

      disp('This is the nested function')

   end

end

The primary difference between nested functions and other types of functions is that they can access and modify variables that are defined in their parent functions. As a result:

Nested functions can use variables that are not explicitly passed as input arguments.

In a parent function, you can create a handle to a nested function that contains the data necessary to run the nested function.

Requirements for Nested Functions

Typically, functions do not require an end statement. However, to nest any function in a program file, all functions in that file must use an end statement.

You cannot define a nested function inside any of the MATLAB® program control statements, such as if/elseif/else, switch/case, for, while, or try/catch.

You must call a nested function either directly by name (without using feval), or using a function handle that you created using the @ operator (and not str2func).

All of the variables in nested functions or the functions that contain them must be explicitly defined. That is, you cannot call a function or script that assigns values to variables unless those variables already exist in the function workspace. (For more information, see Variables in Nested and Anonymous Functions.)

Sharing Variables Between Parent and Nested Functions

In general, variables in one function workspace are not available to other functions. However, nested functions can access and modify variables in the workspaces of the functions that contain them.

This means that both a nested function and a function that contains it can modify the same variable without passing that variable as an argument. For example, in each of these functions, main1 and main2, both the main function and the nested function can access variable x:

function main1

x = 5;

nestfun1

   function nestfun1 

     x = x + 1;

   end 

end

function main2

nestfun2

   function nestfun2

      x = 5;

   end 

x = x + 1;

end

When parent functions do not use a given variable, the variable remains local to the nested function. For example, in this function named main, the two nested functions have their own versions of x that cannot interact with each other:

function main

   nestedfun1

   nestedfun2

   function nestedfun1

      x = 1;

   end

   function nestedfun2

      x = 2;

   end

end

Functions that return output arguments have variables for the outputs in their workspace. However, parent functions only have variables for the output of nested functions if they explicitly request them. For example, this function parentfun does not have variable y in its workspace:

function parentfun

x = 5;

nestfun;

   function y = nestfun

     y = x + 1;

   end 

end

If you modify the code as follows, variable z is in the workspace of parentfun:

function parentfun

x = 5;

z = nestfun;

   function y = nestfun

     y = x + 1;

   end 

end

Using Handles to Store Function Parameters

Nested functions can use variables from three sources:

Input arguments

Variables defined within the nested function

Variables defined in a parent function, also called externally scoped variables

When you create a function handle for a nested function, that handle stores not only the name of the function, but also the values of externally scoped variables.

For example, create a function in a file named makeParabola.m. This function accepts several polynomial coefficients, and returns a handle to a nested function that calculates the value of that polynomial.

function p = makeParabola(a,b,c)

p = @parabola;

   function y = parabola(x)

   y = a*x.^2 + b*x + c;

   end

end

The makeParabola function returns a handle to the parabola function that includes values for coefficients a, b, and c.

At the command line, call the makeParabola function with coefficient values of 1.3, .2, and 30. Use the returned function handle p to evaluate the polynomial at a particular point:

p = makeParabola(1.3,.2,30);

X = 25;

Y = p(X)

Y =

  847.5000

Many MATLAB functions accept function handle inputs to evaluate functions over a range of values. For example, plot the parabolic equation from -25 to +25:

fplot(p,[-25,25])

You can create multiple handles to the parabola function that each use different polynomial coefficients:

firstp = makeParabola(0.8,1.6,32);

secondp = makeParabola(3,4,50);

range = [-25,25];

figure

hold on

fplot(firstp,range)

fplot(secondp,range,'r:')

hold off

Visibility of Nested Functions

Every function has a certain scope, that is, a set of other functions to which it is visible. A nested function is available:

From the level immediately above it. (In the following code, function A can call B or D, but not C or E.)

From a function nested at the same level within the same parent function. (Function B can call D, and D can call B.)

From a function at any lower level. (Function C can call B or D, but not E.)

function A(x, y)              % Main function

B(x,y)

D(y)

   function B(x,y)            % Nested in A

   C(x)

   D(y)

      function C(x)           % Nested in B

      D(x)

      end

   end

   function D(x)              % Nested in A

   E(x)

      function E(x)           % Nested in D

      disp(x)

      end

   end

end

The easiest way to extend the scope of a nested function is to create a function handle and return it as an output argument, as shown in Using Handles to Store Function Parameters. Only functions that can call a nested function can create a handle to it.

Anonymous Functions

What Are Anonymous Functions?

An anonymous function is a function that is not stored in a program file, but is associated with a variable whose data type is function_handle. Anonymous functions can accept inputs and return outputs, just as standard functions do. However, they can contain only a single executable statement.

For example, create a handle to an anonymous function that finds the square of a number:

sqr = @(x) x.^2;

Variable sqr is a function handle. The @ operator creates the handle, and the parentheses () immediately after the @ operator include the function input arguments. This anonymous function accepts a single input x, and implicitly returns a single output, an array the same size as x that contains the squared values.

Find the square of a particular value (5) by passing the value to the function handle, just as you would pass an input argument to a standard function.

a = sqr(5)

a =

   25

Many MATLAB® functions accept function handles as inputs so that you can evaluate functions over a range of values. You can create handles either for anonymous functions or for functions in program files. The benefit of using anonymous functions is that you do not have to edit and maintain a file for a function that requires only a brief definition.

For example, find the integral of the sqr function from 0 to 1 by passing the function handle to the integral function:

q = integral(sqr,0,1);

You do not need to create a variable in the workspace to store an anonymous function. Instead, you can create a temporary function handle within an expression, such as this call to the integral function:

q = integral(@(x) x.^2,0,1);

Variables in the Expression

Function handles can store not only an expression, but also variables that the expression requires for evaluation.

For example, create a function handle to an anonymous function that requires coefficients a, b, and c.

a = 1.3;

b = .2;

c = 30;

parabola = @(x) a*x.^2 + b*x + c;

Because a, b, and c are available at the time you create parabola, the function handle includes those values. The values persist within the function handle even if you clear the variables:

clear a b c

x = 1;

y = parabola(x)

y =

   31.5000

To supply different values for the coefficients, you must create a new function handle:

a = -3.9;

b = 52;

c = 0;

parabola = @(x) a*x.^2 + b*x + c;

x = 1;

y = parabola(1)

y =

   48.1000

You can save function handles and their associated values in a MAT-file and load them in a subsequent MATLAB session using the save and load functions, such as

save myfile.mat parabola

Use only explicit variables when constructing anonymous functions. If an anonymous function accesses any variable or nested function that is not explicitly referenced in the argument list or body, MATLAB throws an error when you invoke the function. Implicit variables and function calls are often encountered in the functions such as eval, evalin, assignin, and load.  Avoid using these functions in the body of anonymous functions.

Multiple Anonymous Functions

The expression in an anonymous function can include another anonymous function. This is useful for passing different parameters to a function that you are evaluating over a range of values. For example, you can solve the equation

for varying values of c by combining two anonymous functions:

g = @(c) (integral(@(x) (x.^2 + c*x + 1),0,1));

Here is how to derive this statement:

Write the integrand as an anonymous function,

@(x) (x.^2 + c*x + 1)

Evaluate the function from zero to one by passing the function handle to integral,

integral(@(x) (x.^2 + c*x + 1),0,1)

Supply the value for c by constructing an anonymous function for the entire equation,

g = @(c) (integral(@(x) (x.^2 + c*x + 1),0,1));

The final function allows you to solve the equation for any value of c. For example:

g(2)

ans =

   2.3333

Functions with No Inputs

If your function does not require any inputs, use empty parentheses when you define and call the anonymous function. For example:

t = @() datestr(now);

d = t()

d =

26-Jan-2012 15:11:47

Omitting the parentheses in the assignment statement creates another function handle, and does not execute the function:

d = t

d = 

    @() datestr(now)

Functions with Multiple Inputs or Outputs

Open This Example

Anonymous functions require that you explicitly specify the input arguments as you would for a standard function, separating multiple inputs with commas. For example, this function accepts two inputs, x and y:

myfunction = @(x,y) (x^2 + y^2 + x*y);

x = 1;

y = 10;

z = myfunction(x,y)

z =

   111

However, you do not explicitly define output arguments when you create an anonymous function. If the expression in the function returns multiple outputs, then you can request them when you call the function. Enclose multiple output variables in square brackets.

For example, the ndgrid function can return as many outputs as the number of input vectors. This anonymous function that calls ndgrid can also return multiple outputs:

c = 10;

mygrid = @(x,y) ndgrid((-x:x/c:x),(-y:y/c:y));

[x,y] = mygrid(pi,2*pi);

You can use the output from mygrid to create a mesh or surface plot:

z = sin(x) + cos(y);

mesh(x,y,z)

Arrays of Anonymous Functions

Although most MATLAB fundamental data types support multidimensional arrays, function handles must be scalars (single elements). However, you can store multiple function handles using a cell array or structure array. The most common approach is to use a cell array, such as

f = {@(x)x.^2;

     @(y)y+10;

     @(x,y)x.^2+y+10};

When you create the cell array, keep in mind that MATLAB interprets spaces as column separators. Either omit spaces from expressions, as shown in the previous code, or enclose expressions in parentheses, such as

f = {@(x) (x.^2);

     @(y) (y + 10);

     @(x,y) (x.^2 + y + 10)};

Access the contents of a cell using curly braces. For example, f{1} returns the first function handle. To execute the function, pass input values in parentheses after the curly braces:

x = 1;

y = 10;

f{1}(x)

f{2}(y)

f{3}(x,y)

ans =

     1

ans =

    20

ans =

    21

Root of nonlinear functioncollapse all in page

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) 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

collapse all

Root Starting From One Point

Open This Example

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

Root Starting From an Interval

Open This Example

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

Open This Example

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

Open This Example

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

Open This Example

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

Open This Example

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

collapse all

fun — Function to solve

function 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 value

scalar | 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 process

structure, 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 problem

structure

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

collapse all

x — Location of root or sign change

real scalar

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

fval — Function value at x

real scalar

Function value at x, returned as a scalar.

exitflag — Integer encoding the exit condition

integer

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.

output — Information about root-finding process

structure

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

Solving a Nonlinear Equation in One Variable

The fzero function attempts to find a root of one equation with one variable. You can call this function with either a one-element starting point or a two-element vector that designates a starting interval. If you give fzero a starting point x0, fzero first searches for an interval around this point where the function changes sign. If the interval is found, fzero returns a value near where the function changes sign. If no such interval is found, fzero returns NaN. Alternatively, if you know two points where the function value differs in sign, you can specify this starting interval using a two-element vector; fzero is guaranteed to narrow down the interval and return a value near a sign change.

The following sections contain two examples that illustrate how to find a zero of a function using a starting interval and a starting point. The examples use the function humps.m, which is provided with MATLAB®. The following figure shows the graph of humps.

x = -1:.01:2;

y = humps(x);

plot(x,y)

xlabel('x');

ylabel('humps(x)')

grid on

Setting Options For fzero

You can control several aspects of the fzero function by setting options. You set options using optimset. Options include:

Choosing the amount of display fzero generates — see Set Options, Using a Starting Interval, and Using a Starting Point.

Choosing various tolerances that control how fzero determines it is at a root — see Set Options.

Choosing a plot function for observing the progress of fzero towards a root — see Plot Functions.

Using a custom-programmed output function for observing the progress of fzero towards a root — see Output Functions.

Using a Starting Interval

The graph of humps indicates that the function is negative at x = -1 and positive at x = 1. You can confirm this by calculating humps at these two points.

humps(1)

ans =

    16

humps(-1)

ans =

   -5.1378

Consequently, you can use [-1 1] as a starting interval for fzero.

The iterative algorithm for fzero finds smaller and smaller subintervals of [-1 1]. For each subinterval, the sign of humps differs at the two endpoints. As the endpoints of the subintervals get closer and closer, they converge to zero for humps.

To show the progress of fzero at each iteration, set the Display option to iter using the optimset function.

options = optimset('Display','iter');

Then call fzero as follows:

a = fzero(@humps,[-1 1],options)

 Func-count    x          f(x)             Procedure

    2              -1      -5.13779        initial

    3       -0.513876      -4.02235        interpolation

    4       -0.513876      -4.02235        bisection

    5       -0.473635      -3.83767        interpolation

    6       -0.115287      0.414441        bisection

    7       -0.115287      0.414441        interpolation

    8       -0.132562    -0.0226907        interpolation

    9       -0.131666    -0.0011492        interpolation

   10       -0.131618   1.88371e-07        interpolation

   11       -0.131618   -2.7935e-11        interpolation

   12       -0.131618   8.88178e-16        interpolation

   13       -0.131618   8.88178e-16        interpolation

Zero found in the interval [-1, 1]

a =

   -0.1316

Each value x represents the best endpoint so far. The Procedure column tells you whether each step of the algorithm uses bisection or interpolation.

You can verify that the function value at a is close to zero by entering

humps(a)

ans =

   8.8818e-16

Using a Starting Point

Suppose you do not know two points at which the function values of humps differ in sign. In that case, you can choose a scalar x0 as the starting point for fzero. fzero first searches for an interval around this point on which the function changes sign. If fzero finds such an interval, it proceeds with the algorithm described in the previous section. If no such interval is found, fzero returns NaN.

For example, set the starting point to -0.2, the Display option to Iter, and call fzero:

options = optimset('Display','iter');

a = fzero(@humps,-0.2,options)

Search for an interval around -0.2 containing a sign change:

 Func-count    a          f(a)             b          f(b)        Procedure

    1            -0.2      -1.35385          -0.2      -1.35385   initial interval

    3       -0.194343      -1.26077     -0.205657      -1.44411   search

    5          -0.192      -1.22137        -0.208       -1.4807   search

    7       -0.188686      -1.16477     -0.211314      -1.53167   search

    9          -0.184      -1.08293        -0.216      -1.60224   search

   11       -0.177373     -0.963455     -0.222627      -1.69911   search

   13          -0.168     -0.786636        -0.232      -1.83055   search

   15       -0.154745      -0.51962     -0.245255      -2.00602   search

   17          -0.136     -0.104165        -0.264      -2.23521   search

   18        -0.10949      0.572246        -0.264      -2.23521   search

Search for a zero in the interval [-0.10949, -0.264]:

 Func-count    x          f(x)             Procedure

   18        -0.10949      0.572246        initial

   19       -0.140984     -0.219277        interpolation

   20       -0.132259    -0.0154224        interpolation

   21       -0.131617   3.40729e-05        interpolation

   22       -0.131618  -6.79505e-08        interpolation

   23       -0.131618  -2.98428e-13        interpolation

   24       -0.131618   8.88178e-16        interpolation

   25       -0.131618   8.88178e-16        interpolation

Zero found in the interval [-0.10949, -0.264]

a =

   -0.1316

The endpoints of the current subinterval at each iteration are listed under the headings a and b, while the corresponding values of humps at the endpoints are listed under f(a) and f(b), respectively.

Note:   The endpoints a and b are not listed in any specific order: a can be greater than b or less than b.

For the first nine steps, the sign of humps is negative at both endpoints of the current subinterval, which is shown in the output. At the tenth step, the sign of humps is positive at the endpoint, -0.10949, but negative at the endpoint, -0.264. From this point on, the algorithm continues to narrow down the interval [-0.10949 -0.264], as described in the previous section, until it reaches the value -0.1316.

Parameterizing Functions

Overview

This topic explains how to store or access extra parameters for mathematical functions that you pass to MATLAB® function functions, such as fzero or integral.

MATLAB function functions evaluate mathematical expressions over a range of values. They are called function functions because they are functions that accept a function handle (a pointer to a function) as an input. Each of these functions expects that your objective function has a specific number of input variables. For example, fzero and integral accept handles to functions that have exactly one input variable.

Suppose you want to find the zero of the cubic polynomial x3 + bx + c for different values of the coefficients b and c. Although you could create a function that accepts three input variables (x, b, and c), you cannot pass a function handle that requires all three of those inputs to fzero. However, you can take advantage of properties of anonymous or nested functions to define values for additional inputs.

Parameterizing Using Nested Functions

One approach for defining parameters is to use a nested function—a function completely contained within another function in a program file. For this example, create a file named findzero.m that contains a parent function findzero and a nested function poly:

function y = findzero(b,c,x0)

y = fzero(@poly,x0);

   function y = poly(x)

   y = x^3 + b*x + c;

   end

end

The nested function defines the cubic polynomial with one input variable, x. The parent function accepts the parameters b and c as input values. The reason to nest poly within findzero is that nested functions share the workspace of their parent functions. Therefore, the poly function can access the values of b and c that you pass to findzero.

To find a zero of the polynomial with b = 2 and c = 3.5, using the starting point x0 = 0, you can call findzero from the command line:

x = findzero(2,3.5,0)

x =

   -1.0945

Parameterizing Using Anonymous Functions

Another approach for accessing extra parameters is to use an anonymous function. Anonymous functions are functions that you can define in a single command, without creating a separate program file. They can use any variables that are available in the current workspace.

For example, create a handle to an anonymous function that describes the cubic polynomial, and find the zero:

b = 2;

c = 3.5;

cubicpoly = @(x) x^3 + b*x + c;

x = fzero(cubicpoly,0)

x =

   -1.0945

Variable cubicpoly is a function handle for an anonymous function that has one input, x. Inputs for anonymous functions appear in parentheses immediately following the @ symbol that creates the function handle. Because b and c are in the workspace when you create cubicpoly, the anonymous function does not require inputs for those coefficients.

You do not need to create an intermediate variable, cubicpoly, for the anonymous function. Instead, you can include the entire definition of the function handle within the call to fzero:

b = 2;

c = 3.5;

x = fzero(@(x) x^3 + b*x + c,0)

x =

   -1.0945

You also can use anonymous functions to call more complicated objective functions that you define in a function file. For example, suppose you have a file named cubicpoly.m with this function definition:

function y = cubicpoly(x,b,c)

y = x^3 + b*x + c;

end

At the command line, define b and c, and then call fzero with an anonymous function that invokes cubicpoly:

b = 2;

c = 3.5;

x = fzero(@(x) cubicpoly(x,b,c),0)

x =

   -1.0945

Note:   To change the values of the parameters, you must create a new anonymous function. For example:

b = 10;

c = 25;

x = fzero(@(x) x^3 + b*x + c,0);