Computer Science 111 Fundamentals of Computer Programming I

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Computer Science 111Fundamentals of Computer
Programming I

• Case StudyFibonacci Numbers

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Fibonacci Numbers

• Leonardo Fibonacci
• Italian mathematician
• 1202 presented problem about population of
  rabbits (oversimplified)
• Each pair of rabbits produces new pair each month
• Rabbits become fertile 1 month after birth
• Rabbits never die
• If we begin with a single pair of rabbits, how
  many pairs will we have after n months for
  specified value of n?

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Some comments

• It is often important to oversimplify a problem
  in the beginning in order to get a good start on
  the problem.
• Modeling real world problems is one of the most
  important uses of computers
• Faster than real experiments
• Less expensive
• Modeling usually requires many iterations in
  order to reach the appropriate level of
  complexity.

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More comments

• It turns out that the study of the number of
  pairs of rabbits over successive months leads to
  a series of numbers that occur often in nature.
• There has been a large amount of research done on
  the numbers and their importance in nature.

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Rabbit family tree

• Note Each month all pairs from previous month
  survive.
• The number of new pairs born is same as number of
  pairs two months back
• 1,1,2,3,5,8,13,

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Fibonacci Numbers

• The Fibonacci Numbers are defined by
• The first two numbers are 1 and 1.
• Each subsequent number is the sum of the
  preceding 2 numbers.
• 1,1,2,3,5,8,13,21,34,55,etc.

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Request and Analysis

• Request Write program that gives the nth
  Fibonacci number where n is a positive number.
• Analysis The user will input the positive number
  n and the program will output the nth Fibonacci
  number.
• Fibonacci number 10 is 55
• Enter a positive number indicating which
  Fibonacci number you wish to see 10

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Design

• Design get input n fib 1 (for
  first 2 numbers) twoBack 1 oneBack
  1 count 3 while count lt n fib
  oneBack twoBack twoBack oneBack oneBack
  fib count count1 output Fib

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Implementation - Variables

• KeyboardReader reader new KeyboardReader()Scre
  enWriter writer new ScreenWriter()
• int n // which Fibonacci
  numberint fib // the current Fib
  numberint oneBack // the previous Fib
  numberint twoBack // the Fib number two
  previousint count // where are we in
  sequence

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Implementation run - input

• n reader.readInt(Enter a positive number
  indicating which Fibonacci number you wish to
  see )while (nlt0) n reader.readInt(Positi
  ve, please )

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Implementation run initialization and loop

• fib 1twoBack 1oneBack 1count 3
• while (count lt n) fib oneBack
  twoBack twoBack oneBack oneBack
  fib count count 1

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Implementation run - output

• writer.print(Fibonacci number )writer.print(n)
  writer.print( is )writer.println(fib)

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Testing
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Maintenance

• Request Have the program print out each
  Fibonacci number as it goes forgot to tell you
  that.
• Analysis Easy enough to just put the output
  inside the loop. But be careful not to print
  anyone twice. And dont skip any.
• Design Just the output output 1
  if ngt1 print another 1 while
  countltn output fib

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Maintenance

• Request Oh, Im getting into this stuff now. Id
  like to see the ratios of consecutive Fib
  numbers. Only care about n 3 or more.
• Analysis Easy enough to just print the ratios
  inside the loop. Whoops, dont want to use int
  division though.
• Design Just the ratio while
  countltn output fib/oneBack
• Implementation writer.print(fib1.0/oneBack)

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Testing
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Testing

• Well have to use doubles to go further

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A Great Web Site on Fib numbers

• http//www.ee.surrey.ac.uk/Personal/R.Knott/Fibona
  cci/fib.html

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We're not talking U.S. inspected, we're talking
prime here, boys!