1. Conditional Control Flow Constructs

2. Sequential Control Flow Execution order follows the textual order straight line flow Many simple problems can not be solved only with such sequential flow Here is an example Problem: Computation of a maximum of two numbers

3. Solution Program max2 Implicit none integer num1,num2, max read *, num1,num2 if (num1 > num2) then !control comes here if num1 > num2 max = num1 !control jumps to the end of the if statement (endif) else !control comes here if num1 < = num2 max = num2 endif print *, max end program max2

4. Another Problem Input two numbers Compute the quotient and remainder of the bigger number divided by the smaller number The numbers are input in unknown order

5. Solution Program Quo_Rem1 Implicit none integer num1,num2, quotient, remainder read *, num1,num2 if (num1 > num2) then quotient = num1/num2 ! / is integer division remainder = num1 – (quotient * num2) else quotient = num2/num1 remainder = num2 – (quotient * num1) endif print *, quotient, remainder end program Quo_Rem1

6. Problems with the solution What happens if num1 or num2 is negative? –Problem Specification needs to be clarified What if one of the numbers is 0? –Divide by zero leads to overflow –Should be avoided –need to test it before dividing

7. Program Quo_Rem2 integer:: num1,num2, temp, quotient, reminder read *, num1,num2 if (abs(num1) < abs(num2)) then ! abs function returns the absolute value temp = num2 num2 = num1 ! swapping the contents num1 = temp ! of num1 and num2 end if if (.not.(num2 == 0)) then !num2 contains a nonzero value quotient = num1/num2 remainder = num1 - quotient * num2 print *, quotient, remainder else print *,"cannot divide! one of the numbers is zero" end if

8. The program Quo_Rem2 The body of if-then-else is simpler The else clause is missing Complex conditions can appear in the if-condition Successive if-statements allowed

9. Another Problem %Nesting of ifs Problem: Compute Maximum of three numbers Solution: ?

10. A Strategy Let num1,num2,num3 store the three numbers. 1.Compute the maximum of num1 and num2. Name it max12 2.Compute the maximum of num2 and num3. Name it max23 3.Compute the maximum of max12 and max23, which is the maximum of num1.num2,num3

11. Another Strategy 1.Compute the maximum of num1 and num2, say max12 2.Compute the maximum of max12 and num3, which is the required maximum? Which is the better strategy? The latter - less number of steps Now we are ready to write the program

12. Program max3 integer num1,num2,num3,max read *, num1,num2,num3 if (num1 > num2) then if (num1 > num3) then !num1 > num2,num1 > num3 max = num1 else ! num1 > num2 and num1 num3) then !num1 num3 max = num2 else ! num1 <=num2<=num3 max = num3 endif print *, max end program max2

13. Nested If statements If statements in the then clause or else clause elseif construct Arbitrary series of nesting permitted elseif and else corresponds to the innermost if for which the endif is yet to come if and endif are like left and right brackets.

14. Further Observations else clause can be missing endif can be dropped if there is only one statement long series of ifs can be confusing (at most 20 levels allowed) indentation improves readability use indentation and comments

15. If conditions Control flow branches in conditional statements Branching decided by evaluating the conditions conditions are expressions of a new type called LOGICAL Examples: (x > 0), (z == 1),.NOT. (num == 2) All these involve relational operators: >,== Relational operators are defined over many data types to compare values

16. Relations over Arithmetic Given e1,e2 expressions over integer (or real), –e1 == e2 ( equality ) –e1 < e2 ( less than ) –e1 <= e2 ( less than or equal ) –e1 > e2 ( greater than ) –e1 >= e2 ( greater than or equal ) –e1 /= e2 ( not equal )

17. LOGICAL VARIABLES Fortran 90 has a built in LOGICAL DATA TYPE Expressions involving relational operators are of type LOGICAL Variables can be declared to have type LOGICAL Declarations: LOGICAL :: found, goodness Logical variables assume just two values –.TRUE.,.FALSE. They can be used in if conditions, eg. – if (found) then stop –if (goodness) then x = 0

18. LOGICAL OPERATORS Operators over logical type values They are.not.,.and.,.or.,.eqv.,.neqv..not. p is.true. iff p is.false. p.and. q is.true. iff both p and q are.false. p.or. q is.true. iff one of (or both) p,q.true. p.eqv. q is.true. iff both p and q has the same truth value p.neqv. q is.true. iff both p and q have different truth values

19. Operator precedence A general logical expression may include arithmetic, relational and logical operators operator precedence defined to specify order of evaluation arithmetic operators are evaluated first followed by relational operators logical operators are evaluated last precedence amongst logical operators.not.,.and.,.or.,.eqv. and.neqv.

20. If condition The condition in an `if' statement is a logical expressions, eg. 1. if ((a>=b).eqv. x) then... -a,b arithmetic variables, x logical variable 2. if (((rate*prin)> 100).AND..NOT. (closed)) then...

21. Quadratic Equation Solving Roots of quadratic equation: ax 2 + bx + c = 0 Can have –exactly one root –two real roots –two complex conjugate roots Type of roots depends upon the discriminant (b 2 - 4ac)

22. A Program to solve the equation program quadratic implicit none real :: a, b, c, disc, x_r, x_r1, x_r2, x_im1, x_im2 real, parameter :: eps = 1.0e-6 read *, a, b, c if (a == 0.0) then ! a is 0, not a quadratic equation print *, "equation is not quadratic" else disc = b*b – 4.0*a*c x_r = -b/(2.0*a)

23. if ( abs(disc) < eps ) then ! discriminant nearly zero print *, "double real root", x_r elseif ( disc > 0 ) then ! two distinct real roots disc = sqrt(disc)/(2.0*a) x_r1 = x_r + disc ! disc temporary variable x_r2 = x_r - disc print *, “two real roots”, x_r1, “ and”, x_r2 else ! disc is negative, complex conjugate roots x_im1 = sqrt(-disc)/(2.0*a) x_im2 = - x_im1 print *, "complex conjugate roots", x_r, "+", & x_im1, "i and", x_r, "-", x_im2, "i" endif endif end program quadratic

24. Comparison of Real Numbers instead of disc == 0.0 we have checked abs(disc) < eps as condition for double root misleading results occur otherwise, eg. 0.1x 2 - 0.3 x + 0.225 = 0 has 1.5 as double root errors in representation give disc > 0.0 roots obtained are 1.5004526 and 1.4995474

25. Comparing Real Numbers double roots undesirable in many applications two roots close to each other may be treated as double if abs(disc) is small, roots are close a parameter eps (epsilon) is usually used for comparing reals two reals are treated as equal if absolute value of difference is < eps

26. Quadratic Equations numerical problems in solving quadratic equations by this method if two roots differ by orders of magnitude, smaller root cannot be found accurately x 2 – 1000.001x + 1.0 = 0 two real roots 1.0000000e +03, 1.0070801e -03 what happens if actual roots are 1.0e +4 and 1.0 e -4 ?

27. Strategies Strategies are high level descriptions of computations They are intuitive and understandable to humans Easier to write, analyze, explore and change compared to programs Develop strategies first Only when a strategy is finalized, write the programs Precise statement of strategies is called Algorithm

28. Algorithm Is a sequence of steps Each step is precise and unambiguous to people Each step is a high level instruction, that can be carried out by mechanically Each step can be translated into `low level' programs Is at a much higher level than HLL itself

29. Analyzing algorithms Before writing the program, analyze and choose the efficient algorithm Metrics for algorithm –Number of steps –Complexity of steps number of primitive operations (like addition, multiplication, comparison)