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);

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

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

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)

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)

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

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