Conditional Control Flow Constructs

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 endprogram 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, reminderread *, num1,num2if (abs(num1) < abs(num2)) then! abs function returns the absolute value temp = num2 num2 = num1 ! swapping the contents num1 = temp ! of num1 and num2end ifif (.not.(num2 == 0)) then!num2 contains a nonzero value quotient = num1/num2 remainder = num1 - quotient * num2 print *, quotient, remainderelse 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 ifsProblem: Compute Maximum of three numbersSolution: ?

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

11. Another Strategy • Compute the maximum of num1 and num2, say max12 • Compute the maximum of max12and 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,maxread *, num1,num2,num3if (num1 > num2) thenif (num1 > num3) then!num1 > num2,num1 > num3 max = num1else! num1 > num2 and num1 <= num3 max = num3endifelseif (num2 > num3) then !num1 <=num2 > num3 max = num2else! num1 <=num2<=num3 max = num3endifprint *, maxend 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. Ifconditions • 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: ax2 + bx + c = 0 • Can have • exactly one root • two real roots • two complex conjugate roots • Type of roots depends upon the discriminant (b2 - 4ac)

22. A Program to solve the equation program quadraticimplicit nonereal :: a, b, c, disc, x_r, x_r1, x_r2, x_im1, x_im2real, parameter :: eps = 1.0e-6read *, a, b, cif (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 zeroprint *, "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 - discprint *, “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_im1print *, "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. 0.1x2 - 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 x2 – 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)