Nested Loops, and Miscellaneous Loop Techniques

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Nested Loops, and Miscellaneous Loop Techniques

• Venkatesh Ramamoorthy
• 16-March-2005

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Exercise Theorem of Pythagoras

• If a, b and c are the sides of a right-angled
  triangle, with c the length of the hypotenuse,
  then
• c2 a2 b2
• Some examples are
• a 3, b 4, c 5
• a 5, b 12, c 13
• a 15, b 8, c 17
• a 20, b 21, c 29

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Pythagorean Triplets

• The triplet (a, b, c) is called a Pythagorean
  Triplet
• Examples
• (3,4,5)
• (5,12,13)
• (15,8,17)
• (20,21,29)

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Goal

• To enumerate the first several Pythagorean
  Triplets until a 100, b 100 and c 100
• a 3, b 4, c 5
• a 4, b 3, c 5
• a 5, b 12, c 13
• a 6, b 8, c 10
• a 7, b 24, c 25
• Etc, etc
• a 100,b 100, c 25

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How?

• Run through all the values of a from 1 through
  100
• For each of those values of a, run through all
  the values of b from 1 through 100
• For each of those values of b, run through all
  the values of c from 1 through 100
• For all the above combinations of a, b and c,
  check if the following equation is satisfied
• a2 b2 c2

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Remember the 3-digit odometer?

• Hundreds Tens Ones
• 0 0 0
• 0 0 1
• 0 0 2
• 0 0 9
• 0 1 0
• 0 1 1
• 0 1 2
• .. .. ..
• 0 9 0
• 0 9 1
• .. .. ..
• 0 9 9
• 1 0 0
• 1 0 1
• .. .. ..
• 9 9 9

In this odometer, the rightmost digit varies most
frequently, followed by the middle, while the
leftmost digit varies least frequently
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Trace

• a b c Is a2b2c2?
• 1 1 1 No
• 1 1 2 No
• 1 1 3 No
• .. .. .. ..
• 1 1 100 No
• 1 2 1 No
• 1 2 2 No
• 1 2 3 No
• .. .. .. ..
• 1 2 100 No
• 1 3 1 No
• .. .. .. ..
• 1 3 100 No
• .. .. .. ..
• 1 100 100 No
• 2 1 1 No
• 2 1 2 No
• .. .. .. No

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Pseudocode Nested Loops!!

1. for a 1 to 100 in steps of 1, repeat
2. for b 1 to 100 in steps of 1, repeat
3. for c 1 to 100 in steps of 1, repeat
4. if (a2 b2 equals c2) then
5. Display a, b, c
6. end-if
7. end-for
8. end-for
9. end-for

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More on Nesting Nested IFs

• if (condition-1)
• if (condition-2)
• if (condition-3)
• statement-set-1
• else
• statement-set-2
• else
• statement-set-3

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More on NestingNested while-loops

• while (condition-1)
• statement-1
• while (condition-2)
• statement-2
• for (index1 indexltn index)
• statement-set-1

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Exercise

• Modify the Pythagorean Triplet program to print
  the first 100 triplets
• Increment a counter whenever a triplet is found
  and displayed.
• If the counter exceeds 100, abnormally terminate
  the loop.
• Remember the break statement?

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Displaying decimal values

• Use stdfixed
• To display float and double decimal numbers in
  fixed-point format
• This is opposed to a scientific format
• Use stdsetprecision(d)
• To display float and double decimal numbers
  correct to d decimal places
• Note that these only display the number inside
  the variable in the specified manner
• The variable still contains the original!

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Set the floating-point number format

• By using a separate cout, this can be done
• cout ltlt fixed ltlt setprecision(2)
• You can also set to display in d decimal places,
  where d is an input
• Ensure that d has been separately validated
• cout ltlt fixed ltlt setprecision(d)

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The header files / namespaces to use for such
displaying

• using stdfixed
• include ltiomanipgt
• using stdsetprecision
• Note the above order in which these statements
  are used!

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Example

• What does the following program segment do?
• using namespace stdfixed
• include ltiomanipgt
• using namespace stdsetprecision
• double result
• result 2
• cout ltlt fixed ltlt setprecision(2)
• cout ltlt result ltlt \n

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Example modified

• Now I change result to 3.1415926
• What does the following program segment do?
• using namespace stdfixed
• include ltiomanipgt
• using namespace stdsetprecision
• double result
• result 3.1415926
• cout ltlt fixed ltlt setprecision(2)
• cout ltlt result ltlt \n

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Another modification

• Now I change the precision from 2 to 4!
• What does the following program segment do?
• using namespace stdfixed
• include ltiomanipgt
• using namespace stdsetprecision
• double result
• result 3.1415926
• cout ltlt fixed ltlt setprecision(4)
• cout ltlt result ltlt \n

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Example modified

• Now I change the content of result!
• What does the following program segment do?
• using namespace stdfixed
• include ltiomanipgt
• using namespace stdsetprecision
• double result
• result 3.14
• cout ltlt fixed ltlt setprecision(4)
• cout ltlt result ltlt \n

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An interesting problem

• What does the following program segment do?
• using namespace stdfixed
• include ltiomanipgt
• using namespace stdsetprecision
• double term1, term2, sum
• cout ltlt fixed ltlt setprecision(2)
• term1 14.275
• cout ltlt term1 ltlt term1 ltlt \n
• term2 18.675
• cout ltlt term2 ltlt term2 ltlt \n
• result term1 term2
• cout ltlt result ltlt \n

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Loops Summation of Infinite Series

• Using the infinite series below, determine the
  value of ex for an input number x correct to d
  decimal places, where d is another input

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Problem Analysis

• Are we repeatedly going to compute xn and the
  factorial of n, and use them in each term?
• Certainly not!
• Heres where the following technique is very
  useful

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Problem analysis

• To sum such series,
• Always try to relate the previous term with the
  current term
• Or, always try to relate the current term with
  the next term
• In other words, try to relate two consecutive
  terms

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Current and Next terms

• Current term

• Next term

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The relation

• Therefore,

• In other words,

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The loop

• Repeatedly keep on accumulating the current term
  into a variable sum
• Then, determine the next term from the current
  term
• By multiplying the current term with x/(n1)
• Increment n by 1

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When will the loop terminate?

• When the current term becomes zero!
• Why?
• What should be the value of the current term for
  the first-time?

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The Pseudo-code

1. Input x and d
2. Set current_term 1, sum 0
3. Set n 1
4. While (current_term is not equal to zero),
   repeat
5. sum sum current_term
6. current_term (current_term x) / (n 1)
7. n n 1
8. End-while
9. Display sum
10. Stop