If you want to find the square root of a positive REAL number in a FORTRAN program, the simplest method is to use the SQRT function. It is quite reasonable, however, to ask how it is done.

Here is a program to find a square root. It uses a method known as "Newton's Algorithm".

         PROGRAM SQROOT

         IMPLICIT NONE
         REAL, PARAMETER :: E = 0.0001
         REAL :: NUM, X1, X2, RESULT
          
         PRINT *, 'We will find the square root of a number.'
         PRINT *, 'Pick a real number.'

         READ *, NUM

         IF (NUM < 0.0) THEN
           PRINT *, 'Your number is negative:  no real root.'
         ELSE
           IF (NUM == 0.0) THEN
             RESULT = 0.0
           ELSE
             X1 = NUM
             X2 = 1.0
             DO WHILE (ABS(X1 - X2) > E)
               X1 = (X1 + X2) / 2.0
               X2 = NUM / X1
               PRINT *, 'Our current values are:  ', X1, ' and ', X2
             END DO
           END IF
           PRINT *, 'The square root of ', NUM, ' is:  ', X1
         END IF

         END PROGRAM SQROOT

A few notes

• We use IMPLICIT NONE to switch off the implicit naming convention. If we leave this out, NUM would be assumed to be an INTEGER.

• We are finding square roots to only a few digits of accuracy. This is controlled by the value of E. If we wanted to have more accuracy, it would be necessary only to change the value of E. (It might also be necessary to switch to the use of DOUBLE PRECISION variables.)

• As we are working with REAL values, we cannot find the square root of a negative number. Therefore, we check separately for this.

• The square root of 0.0 is 0.0. Although the program checks for this separately, it would work without it.

• This program prints out the values of X1 and X2 that we compute as we go through the loop. This is not strictly necessary, but it does illustrate the convergence of the X1 and X2 values.

• We are printing all the numbers here using the default format. We certainly could modify the program to use FORMAT statements and produce neater output. The default format does demonstrate that REAL numbers are not always stored as we might suppose.

• We are using the ABS function here. We could get along without it.

Output from the SQROOT program

 We will find the square root of a number.
 Pick a real number.
 Our current values are:     1.5000000      and    1.3333334    
 Our current values are:     1.4166667      and    1.4117646    
 Our current values are:     1.4142157      and    1.4142114    
 The square root of    2.0000000      is:     1.4142157    


 We will find the square root of a number.
 Pick a real number.
 Our current values are:    0.50500000      and   1.98019799E-02
 Our current values are:    0.26240098      and   3.81096117E-02
 Our current values are:    0.15025529      and   6.65533915E-02
 Our current values are:    0.10840434      and   9.22472253E-02
 Our current values are:    0.10032578      and   9.96752754E-02
 Our current values are:    0.10000053      and   9.99994650E-02
 The square root of   9.99999978E-03  is:    0.10000053    


 We will find the square root of a number.
 Pick a real number.
 The square root of    0.0000000      is:    3.21393408E-39


 We will find the square root of a number.
 Pick a real number.
 The square root of    1.0000000      is:     1.0000000    


 We will find the square root of a number.
 Pick a real number.
 Your number,   -1.0000000      is negative:  no real root.

 
 We will find the square root of a number.
 Pick a real number.
 Our current values are:     3200.5000      and    1.9996876    
 Our current values are:     1601.2499      and    3.9968777    
 Our current values are:     802.62335      and    7.9738522    
 Our current values are:     405.29861      and    15.790826    
 Our current values are:     210.54472      and    30.397343    
 Our current values are:     120.47103      and    53.124805    
 Our current values are:     86.797920      and    73.734482    
 Our current values are:     80.266205      and    79.734680    
 Our current values are:     80.000443      and    79.999557    
 Our current values are:     80.000000      and    80.000000    
 The square root of    6400.0000      is:     80.000000