Numerical Methods with Programming

Section 5.2 Named functions

Named functions can do the same thing as anonymous functions. Here is \(\exp(-x^2)\) again, this time as a named function.

function y = f(x) 
    y = exp(-x.^2);
end

But there may be multiple lines within the body of a named function, including loops and conditional statements. In general, it looks like this.

function y = myFunction(x)
    (do something);
    (do more things);
    (y = result of computations);  
end

A named functions can have multiple inputs: function w = myFunction(x, y, z). It can also have multiple outputs: function [u, v] = myFunction(x, y, z) in which case both u and v must be assigned some values within the function.

function z = agm(x, y, tol) 
   while abs(x - y) >= tol
       a = (x + y)/2;
       g = sqrt(x*y);
       x = a;
       y = g;
   end
   z = x;
end

def agm(x, y, tol): 
    # some Python code computing z
return z

Example 5.2.1 illustrates an important point: a function has limited communication with the code outside of it. It receives input and produces certain output; nothing else that it does has any influence on the outside world. Suppose that the following sequence of commands was executed.

x = 0.3;
y = 12;
disp(agm(x, y, 1e-6))
disp(x)
disp(a)

The third line will display the AGM which happens to be 3.7136. The fourth line will display the original value of x, which is 0.3. Why is it still 0.3, even though the function agm changed the value of x in the loop? This is because a function operates like a separate program with its own variables. The variable x inside of the function is its own local variable x, and changing it does not change x in other places. For the same reason, the last command disp(a) produces an error: the variable a is undefined. A variable with that name was introduces inside of function agm but it does not exist outside of it.

One can have a function that does not return any values. Here is a function whose job is to plot the sine function on the interval \([a, b]\) using a given number of points.

function plotsin(a, b, num) 
    x = linspace(a, b, num);
    plot(x, sin(x))
end

One can also have a function with no arguments (no input values). For example, this function computes an approximate value of the Golden Ratio by iterating \(\sqrt{x+1}\text{.}\)

function z = goldenratio() 
   z = 1;
   for k = 1:100
       z = sqrt(z+1);
   end
end

One can use this function same as others: x = goldenratio(); or disp(goldenratio()).

Back in the days when Matlab was designed, the convention was to put every named function in a separate “M-file” named as that function: for example, goldenratio.m. Modern versions allow multiple named functions appear in the same file, along with code using those functions. However, one arbitrary restriction remains: “Function definitions in a script must appear at the end of the file.” For example, the following is allowed:

x = 0.3;
y = 12;
disp(agm(x, y, 1e-6))

function z = agm(x, y, tol) 
   while abs(x - y) >= tol
       a = (x + y)/2;
       g = sqrt(x*y);
       x = a;
       y = g;
   end
   z = x;
end

Anonymous functions, described in Section 5.1, have no such restrictions.

function k = smallestprime(n)
    k = n + 1;
    while ~isprime(k)
        k = k + 1;
    end
end