lect10a - DS CS 11002 Computer Sc Engg IIT Kharagpur $ 1...

PDS: CS 11002 Computer Sc & Engg: IIT Kharagpur 1
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From Inductive Definition to Loop
Lect 10A Goutam Biswas

PDS: CS 11002 Computer Sc & Engg: IIT Kharagpur 2
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Definition
Consider the factorial function n ! . It may be
viewed as a sequence
0! , 1! , 2! , 3! , 4! , · · · = 1 , 1 , 2 , 6 , 24 , · · · . The n
th
term of the sequence t
n
can be defined as
follows:
t
n
=







1 if n = 0 ( basis ) ,
nt
n - 1
if n > 0 .
Lect 10A Goutam Biswas

PDS: CS 11002 Computer Sc & Engg: IIT Kharagpur 3
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Converting the Definition to Code
1. Initialize a variable fact to the value of the
basis . Also initialize a variable i as the
sequence index of the next element to
compute.
2. Repeat the following steps until i exceeds
the input n .
(a) multiply fact by i ,
(b) increment i to the next index.
Lect 10A Goutam Biswas

PDS: CS 11002 Computer Sc & Engg: IIT Kharagpur 4
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A Code
fact = i = 1 ;
while(i<=n) fact *= i++ ;
Lect 10A Goutam Biswas

PDS: CS 11002 Computer Sc & Engg: IIT Kharagpur 5
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Definition
The n
th
term t
n
of the Fibonacci sequence
0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , · · · is defined inductively as
follows:
t
n
=







n if n = 0 , 1 ( basis ) ,
t
n - 1
+ t
n - 2
if n > 1 .
Lect 10A Goutam Biswas

PDS: CS 11002 Computer Sc & Engg: IIT Kharagpur 6
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Converting the Definition to Code
1. Initialize two variables f0 and f1 to zero and