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FORTRAN 90+: COMPLEX NUMBER DEMONSTRATION

The approach needed to program complex numbers is demonstrated in the Theory section using snippets of code. Several applications that use complex numbers can be found in the Numerical Methods section.

DEMONSTRATION: COMPLEX ROOT FINDING

In some engineering applications, either solution of algebraic equations involves roots that are complex numbers, or the algebraic equations themselves involve complex numbers (in addition to possible roots). Thus, it is necessary to understand how to program both types of problems.

Fortunately, most of the root finding methods available can be modified to include complex numbers, whether in the equation and/or in the roots. One mathematical concept to be aware of is the concept of complex conjugate pairs. That is they appear as r±i expressions.

Consider the equation x² = d. When d = 4, then the root is real and the bisection code.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!  Uses bisection method to find square root
!
!   x*x - c  = 0
!   over the interval [a,b]
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
PROGRAM sqrt
  IMPLICIT NONE
  INTEGER :: nmax, n
  REAL :: tol, d
  REAL :: a, b, c
  REAL :: fa, fc

  WRITE(*,*) ’ Input Constant d (x^2-d=0) ’
  READ(*,*) d
  WRITE(*,*) ’ Input Lower and Upper Limits’
  READ(*,*) a, b

  tol = 0.0001
  nmax = 100

  fa = f(a,d)
  c = (a + b ) /2.
  fc = f(c,d)
  n = 1

  DO WHILE (n <= nmax)
     IF((b-a)/2.0 < tol) THEN
       WRITE(*,*) ’Root’,c,’gives’,fc,’after’,n,’iterations’
       EXIT
     ENDIF
     IF(fa*fc > 0.0) THEN
        a = c
     ELSE
        b = c
     ENDIF
     c = (a + b ) /2.
     fc = f(c,d)
     n = n + 1
  ENDDO

CONTAINS

   REAL  FUNCTION F(x,n)
   REAL :: x
   REAL :: n
   f = x*x - n
   END FUNCTION F
   
END PROGRAM

Input Constant d (x^2-d=0) 
4.
  Input Lower and Upper Limits
0.,3.
 Root   1.9999695 gives  -1.2207031E-04 after 15 iterations

So after fifteen iterations, the real root 2.0 is obtained (within the significant digits of the tolerance provided - decreasing the value of the tolerance increases accuracy). Increasing to seven significant digits only requires a few more iterations. Although only one root is computed, mathematically it is known that a real square root is symmetric about zero.