# 9. Recursion

A completely new capability of Fortran 90 is recursion. Note that
it requires that you assign a new property `RESULT ` to the output variable
in the function declaration. This output variable is
required inside the function as the "old" function name in order to
store the value of the function. At the actual call of the function,
both externally and internally, you use the outer or "old" function
name. The user can therefore ignore the output variable.
Here follows two examples: first the recursive calculation of
factorials, then the recursive calculations of the Fibonacci-numbers. The
later is very inefficient.
Brainerd,
Goldberg and Adams (1990), page 226, propose
an efficient, but non-recursive
method.

The listings of the routines follow. The output variables are called
`FAC_RESULT ` and `FIBO_RESULT`, respectively.

RECURSIVE FUNCTION FACTORIAL(N) RESULT (FAC_RESULT)
IMPLICIT NONE
INTEGER, INTENT(IN) :: N
INTEGER :: FAC_RESULT
IF ( N <=1 ) THEN
FAC_RESULT = 1
ELSE
FAC_RESULT = N * FACTORIAL(N-1)
END IF
END FUNCTION FACTORIAL
RECURSIVE FUNCTION FIBONACCI(N) RESULT (FIBO_RESULT)
IMPLICIT NONE
INTEGER, INTENT(IN) :: N
INTEGER :: FIBO_RESULT
IF ( N <= 2 ) THEN
FIBO_RESULT = 1
ELSE
FIBO_RESULT = FIBONACCI(N-1) + FIBONACCI(N-2)
END IF
END FUNCTION FIBONACCI

The reason that the above calculation of the Fibonacci-numbers is
so inefficient is that each call with a certain value of N generates two
calls for the routine, which in its turn generates four calls, and so
on. Old values (or calls) are not re-used.
On page 222 Brainerd,
Goldberg and Adams (1990) demonstrate
an interesting use of recursive technique for the calculation of an
exponential function of a matrix. They give the immediate (straight
forward) expression, with the successive multiplication with a matrix,
as well as a recursive variant, which can pick out the suitable squares to
optimize the calculation. Recursion is also excellent to
code an adaptive algorithm, see exercise (9.2) below.

Another very important usage of the `RESULT ` property and the output
variable is with array valued functions. It is very easy to specify an
output variable so that it can store all the values of such a
function. Actually, it is the combination of recursive functions and
array valued functions that have forced the committee to introduce the
`RESULT ` property.

We note that not only functions, but subroutines too, can be recursive.

## Exercises

(9.1) Write a routine for the calculation of Tribonacci numbers.
These are formed like the Fibonacci-numbers, but you start
with three numbers (all 1 at the start). At each step you then add
the last three numbers to get the next one. Run
and calculate `TRIBONACCI(15)`. Note that the calculation time
increases very quickly with the argument.

Solution.
(9.2) Write an adaptive routine for quadrature, i.e. calculation of
a definite integral on a certain interval.

Solution.

Last modified: 6 April 1998

`boein@nsc.liu.se`