11th lecture

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Elementary Programming using Mathcad (INF-20303)

• 11th lecture

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Recursion
For example in the series
we use the operator ! factorial.
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In the algorithm for the series we have used
Along this line
So factorial(5) calls factorial(4),
factorial(4) calls factorial(3),
factorial(3) calls factorial(2), etc.
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Fibonacci
1, 1, 2, 3, 5, 8, 13, 21, ...
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Applying recursion
fibonacci(n)
This algorithm expresses clearly that a term
equals the sum of the two preceding terms.
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Bisection
No tests for roots on the specified interval!
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• Layout of a recursive function
• no recursive calls needed (stopping case), see
  factorial(1)
• one or more cases where a recursive call is
  needed

Make sure the function stops!

• Advantage formulation is simple and clear
• Disadvantage inefficient
• calling a function takes time a.o. transport of
  arguments
• more calls are needed, see fibonacci

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Towers of Hanoi
Bring disks from pole A to pole C.
A
B
C
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• n1 A to C
• n 2 A to B. A to C. B to C
• n 3 Use step n 2 to move the first two disks
  to B while
• using C as temporary storage.
• A to B. Use step n 2 to move these
  two disks
• from B to C using A as temporary
  storage
• n-2 ..
• n-1 Use step n-2 to move n-2 disks from A to B
  (C temporary).
• Move A to C.
• Use step n-2 to move the n-2 disks from
  B to C (A temporary)
• n Use step n-1 to move n-1 disks from A to B
  (C temporary)
• Move A to C.
• Use step n-1 to move the n-1 disks from
  B to C (A temporary)

Demo
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• n1 Move A to C
• Use step n-1 to move n-1 disks from A to B (C
  temporary)
• Move A to C
• Use step n-1 to move the n-1 disks from
  B to C (A temporary)

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Answers section 19.1, assignment 1 Make a vector
with the digits of a given integer value.

• For example 1325, how we can find the digits?
• divide 1325 by 1000 gives 1
• then 1325 - 1000 325
• divide 325 by 100 gives 3
• then 325 - 300 25
• divide 25 by 10 gives 2
• then 25 - 20 5
• divide 5 by 1 gives 5

Needed number of digits of the value?
We can find the number, however .
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• A better approach
• determine remainder on division by 10, gives 5
• calculate result of division 132510 132.5
• truncate 132.5 with function floor, gives 132
• determine remainder on division by 10, gives 2
• calculate result of division 13210 13.2
• truncate 13.2 with function floor, gives 13
• determine remainder on division by 10, gives
  3
• calculate result of division 1310 1.3
• truncate 1.3 with function floor, gives 1

5
1
2
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Digits appear in reverse order.
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• Core of the algorithm is
• determine remainder on division by 10
• calculate result of division by 10
• truncate result of division.

x is digit, insert in vector
Before the loop!
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Repeat until N lt 10
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This works well in Mathcad for N 0, because
Mathcad initializes v 0.
Better is N 0, however infinite loop!
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So solves that problem! (in other programming
languages)
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With error message if given number (N) is
negative
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Answer section 19.1, assignment 2 Bubble sort.
Phase 1
Phase 2
Phase 3
Ready
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With 5 elements Repeat 5 - 1 4 times. In
general n elements Repeat n-1 times.
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• Suppose the algorithm is arrived at element j
• of vector v
• The algorithm looks backwards, a.o. this means
• algorithm starts at element
• test whether the element should be exchanged

in v n elements
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For one pass, we need to repeat
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In principle, repeat this last(v) times
(maximum). Vector with n elements last(v) n-1
number of needed repetitions.
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• After phase 1 the largest element is in last
  position, a.s.o.
• so, every phase we have to visit one element
  less

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Alternative
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Sometimes less number of phases, see the example.
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We do not need to know the exact number of swaps
Sufficient is
again is a so-called boolean variable!
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What is the problem here?
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k 4
k 1, again 1
20 10 7 50 3
50 20 10 7 3
20 50 10 7 3
20 10 50 7 3
7 3 10 20 50
3 7 10 20 50
20 10 7 3 50
k 3, again 1
k 0, again 1
20 10 7 3 50
10 20 7 3 50
10 7 20 3 50
10 7 3 20 50
Conclusion one more phase. However, we are
ready but the algorithm does not know!
k 2, again 1
10 7 3 20 50
7 10 3 20 50
7 3 10 20 50
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k 0
j 1 if v1 lt v0 j 0 if v0 lt v1
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Solution
Maximally last(v) phases!
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So bubble4 operates ok!
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• Sometimes a variable with the values
• 0 (false or not true) or
• 1 (true)
• Assume you want adult to become 1 if age 18
• adult to become 0
  if age lt 18
• adult is a so-called boolean variable
• You can accomplish that in two ways

Or