do while-enddo loops

In the preceding section we learned that the statement inside the do-enddo loop is being executed while the controlling parameter lies betwen the specified initial and final value. Remember that the decision about the execution of the statements inside the do-enddo construction is made before each individual loop.

A similar construction which is frequently used in Fortran codes is the do while-enddo loop. The body of the loop is executed provided that the logical expression written behind while is true. At the end of each cycle the program returns back to the do while header and again checks the validity of the logical expression. This process occurs until the condition becomes false and the program continues below enddo. The standard do while-enddo loop looks as follows:

      do while (logical expression)
         :
         statements
         :
      enddo

The do while-enddo loops are mainly used once you want to ensure that a particular loop will be executed at least once. In numerical mathematics, this frequently occurs in the algorithms for finding roots of functions where you always start with an initial guess and converge towards the configuration corresponding to zero functional value. The whole algorithm exits once you achieve a sufficient accuracy, i.e. once the functional value lies within a given $\varepsilon$-neighborhood of zero. To demonstrate the use of these do while-enddo loops, look at the following simple problem.

Problem: Write an algorithm which finds one root of function $f(x)=x^3+4x^2-1$ using a bisection method. Search the interval $0\leq{}x\leq{}1$ which definitely contains a root and obtain the solution with accuracy $\varepsilon\leq{}10^{-6}$.

Solution:

      eps = 1.0e-6
      xleft = 0.0
      xright = 1.0
      fmid = eps+1.0   ! execute the first cycle 

      do while (ABS(fmid) .gt. eps)
         xmid = (xleft+xright)/2.0

         fleft = xleft**3 + 4*xleft**2 - 1
         fmid = xmid**3 + 4*xmid**2 - 1
         fright = xright**3 + 4*xright**2 - 1

         if (fleft*fmid .le. 0) then
            xright = xmid
         else
            xleft = xmid
         endif
      enddo

      write(*,'("Root is x = ",F8.6)') xmid

Initially, we set the requested accuracy in eps and specify the interval in which we are going to seek the solution. The first run of the do while-enddo loop calculates the position of the middle point and determines the functional values for the boundary of the interval and this middle point. The root can then lie either in the interval (xleft,xmid), provided that fleft and fmid are of opposite signs, or in the interval (xmid,xright) if fmid and fright are of opposite signs. This new interval is then bisected and the loop repeats until the functional value corresponding to the middle point xmid lies in the eps-neighborhood of zero. When this happens, the execution of the do while-enddo loop is stopped and we get the result: Root is x = 0.472834.

The algorithm above calculates the functional values at xleft and xright at the beginning of each cycle, although this is not necessary in subsequent cycles of the loop. Try to understand the algorithm in more detail and improve the program to avoid these unnecessary operations. Remember that the functional value at xmid has to be calculated in each cycle.