Using functions as arguments

Your assignment this week is to write a routine (actually, several functions) which finds the root of a function. For example, suppose the function is

       y = sqrt(x) - 3.8

You will need to write a Scilab routine which evaluates this function repeatedly, and takes some action based on the results. But how do to that?

One way would be to hardwire the function into your routine, like this:


      // set the bounds of the interval
      starting_x = 0;
      ending_x   = 10;

      // now evaluate the function in the middle of the interval
      mid_x = (starting_x + ending_x) / 2;
      y = sqrt(mid_x) - 3.8;

      // take the appropriate action ...

But such a routine will work ONLY for the function sqrt(x) - 3.8. If you need to find the roots of a different function, like cos(x) - sqrt(x), then you need to edit your routine and change the line at which y is calculated.

Editing source code every time you need to run a routine in a slightly different way is poor programming style. It's much better to find a way to leave the source code as-is, and act upon some input supplied by the user.

So, it would be nice if we could just call an arbitrary function in the root-finding routine, like this:


      // set the bounds of the interval
      starting_x = 0;
      ending_x   = 10;

      // now evaluate the function in the middle of the interval
      mid_x = (starting_x + ending_x) / 2;
      y = call_the_function_I_want(mid_x);

      // take the appropriate action ...

So, in order to find the root of sqrt(x) - 3.8, one would somehow write

       call_the_function_I_want(x)    means      sqrt(x) - 3.8

and to find the root of cos(x) - sqrt(x), somehow tell Scilab that now

       call_the_function_I_want(x)    means      cos(x) - sqrt(x)

There is a way to do this! The builtin Scilab routine feval will call a function for you, and it will even supply the appropriate arguments to the function. All you have to do is to provide it with

the first argument*: the values to be passed to that function
the second argument: the name of a function to evaluate

* Actually, the feval function may take either one or two initial numerical arguments, before the name of the function to evaluate. See the on-line documentation.

It will do all the work, and then return to you the results.

Example of using feval

Suppose we know we'll have to find the roots of three different functions:

quadratic_function involves a quadratic equation

  function y = quadratic_function(x)

    y = x*x - 4*x + 3;

cubic_function involves a cubic equation

  function y = cubic_function(x)

    y = 12*x*x*x + x*x - 4*x + 3;

trig_function involves a trigonometric equation

  function y = trig_function(x)

    y = sin(x)*cos(x) - sin(x)*sin(x);

We create a general root-finding program -- perhaps it uses the bisection method. We design this program so that it take 3 arguments: the starting and ending points of the interval on which to seek a root, and the name of the function whose root we're seeking.

      function y = root_finder(starting_x, ending_x, routine_to_root)

Inside this program is a loop, which iterates to get closer and closer to the true root. It contains a section which looks something like this:

      // find the middle of the interval
      mid_x = (starting_x + ending_x) / 2;

      // and evaluate the function at that point
      y = feval(mid_x, routine_to_root);

      // take the appropriate action ...

Now, suppose we are interested in the interval from 0 to 25. To look for a root of the quadratic function, we would type into the MATLAB command window

  >>  root_finder(0, 25, quadratic_function)

To find the root of the cubic function,

  >>  root_finder(0, 25, cubic_function)

To find the root of the trigonometric function,

  >>  root_finder(0, 25, trig_function)

Unfortunately, in Scilab, we can't use this facility to call built-in functions such as sqrt or sin. Rats. In MATLAB, on the other hand, the feval function will do this job as well.

If you want some practice ...

write a function called square which takes one argument, calculates the square of the argument, and returns it.
write another function called cube which takes one argument, calculates the cube of the argument, and returns it.
write a function called plusone which takes one argument, calculates the sum of the argument plus one, and returns it.
now, write a function called calculate, which takes two arguments:
- the first is the name of a function
- the second is a value at which to evaluate the function
You should be able to call calculate like this:
```
        calculate(cube, 5.5)
        calculate(square, 10)
        calculate(plusone, -33.3)
```
What is the result of each?

Here are my versions of these programs:

Last modified 3/27/2007 by MWR.