Read the last part of the test before working any of the problems.

1. The root mean square (abbreviated RMS) of a list of
numbers is defined to be the square
root of the average of the squares of the numbers.  (The
RMS concept has applications
in electrical engineering, such as in the phrase, "RMS voltage.")
For example, the root mean square of (2,4) is sqrt(10), approximately
3.16, since the average of 2^2 and 4^2 is 10.

Write a Fortran 90 subroutine that has three arguments.  When the
subroutine is called, the third argument is set equal to the
root mean square of the first and second arguments.  Thus, for example,
execution of the statement
CALL RMS(2,4,X) would cause the value of the variable X
to become sqrt(10).  Your subroutine code should make proper use
of IN, INOUT, or OUT, as needed.

If written in the way I think best, your code should be six lines long.


2. The following Fortran 90 function has a few errors in it, but you should
be able to tell what its purpose is.  Rewrite the function, correcting errors,
and using fewer lines by thinking of LOGICAL as a data type.

 function is_perfect_square(n)
  integer::n
  if n < 0 then
   .false.
  else if nint(sqrt(n))**2 .eqv. n then
   .true.
  else
   .false.
  end if
 end function is_perfect_square

3. If n is a non-negative integer, "n factorial" (usually written as n!)
is defined to be the product of all the positive integers up
through n.  Thus, 1! = 1, 2! = 1*2 = 2, 3! = 1*2*3 = 6, 
and 7! = 1*2*3*4*5*6*7 = 5040.  We define 0! = 1.
A programmer was attempting to write a recursive Fortran 90 function
to compute factorial.  Here is that programmer's code.
The code contains several errors.  Fix them.  (Either mark corrections
or rewrite the code completely.)

 function factorial(n)::integer
  intention in::n
  if n < 0 then
   print 10 "Error in function factorial: negative argument not allowed"
10 format(a10)
  else if n = 0 then
   factorial = 1
  else
   factorial = n * factorial(n-1)
  end if
 end function factorial

 ----------------- end of practice test ----------------------------------
 
Now, let us incorporate your answers into a program.

Suppose that all three of the above subprograms are inserted into the
program below in the correct place:

 program t03
  implicit none
  real::x
  integer::i
  call rms(2.0,4.0,x)
  print*,"The rms is",x
  do i = -20,20
   if (is_perfect_square(i)) print*,i,"is a perfect square"
  end do
  print*,"factoria(7) = ",factorial(7)
 contains
! Insert subprograms here
 end program t03

The output of the program, if the subprograms are written correctly, is:

 The rms is 3.1622777
 0 is a perfect square
 1 is a perfect square
 4 is a perfect square
 9 is a perfect square
 16 is a perfect square
 factoria(7) =  5040