Chapter6 1 Recursion Version3.0 Question

7/29/2019 Chapter6 1 Recursion Version3.0 Question

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Principles of Programming - NI2005 1

Simple Recursion

Recursion is where a function calls itself.Concept of recursive function:

A recursive function is called to solve a problem

The function only knows how to solve the simplest

case of the problem. When the simplest case isgiven as an input, the function will immediately

return with an answer.

However, if a more complex input is given, a

recursive function will divide the problem into 2pieces: a part that it knows how to solve and

another part that it does not know how to solve.


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Simple Recursion cont

The part that it does not know how to solve alwaysresembles the original problem, but of a slightly simplerversion.

Therefore, the function calls itself to solve this simplerpiece of problem that it does now know how to solve.

This is called the recursion step.The recursion step is done until the problem convergesto become the simplest case.

This simplest case will be solved by the function whichwill then return the answer to the previous copy of the

function.

The sequence ofreturns will then go all the way upuntil the original call of the function finally return theresult.


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Simple Recursion cont

Any problem that can be solved recursively canalso be solved iteratively (using loop).

Recursive functions are slow and takes a lot ofmemory space compared to iterative functions

So why bother with recursion? There are 2reasons:

Recursion approach more naturally resembles the

problem and therefore the program is easier to

understand and debug.

Iterative solution might not be apparent.


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Example 1 - Factorial

Analysis:n!= n * (n-1) * (n-2) * (n-3) * (n-4) * 1

n! could be rewritten as n * (n-1)!

Example:5! = 5 * 4 * 3 * 2 * 1 = 5 * (4)!, where n = 5

Fact: 0! Or 1! is equal to 1

Therefore, we could divide this problem into

two stages for n!:Simplest case: if (n


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#include double fact(double);

void main(void){

double n, result;

printf("Please enter the value of n:");scanf("%lf",&n);

result = fact(n);

printf(" %.f! = %.2f\n\n",n,result);}


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double fact(double n){

if (n


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Tracing a problem - factorial

Fact (4)

Fact (4)

= 4 * Fact (3)Fact (3)

= 3 * Fact (2)

Fact (2)

= 2 * Fact (1)

Fact (2)

2 * 1

Fact (3)

3 * 2

Fact (1)

= return 1

Fact (4)

4 * 6

double fact(double n){

if (n


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Example 2 : xy

In this example, we want to calculate x to the powerof y (i.e. xy)

If we analyze the formula for xy, we could see that xycould be written as (x being multiplied to itself, ytimes).

An example is 24, which can be written as24 = 2 x 2 x 2 x 2 (in this case, x = 2, y = 4)

24 could also be rewritten as

24 = 21x 23 where 21 = 2 (i.e the number itself)

Therefore, we could divide the problem into twostage:

Simplest case: when y = 1, the answer is x

Recursive case, we need to solve for x * x(y-1)


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Example 2 : xy

#include double XpowY(double,int);

void main(void){

double power, x;int y;

printf("Enter the value of x and y:");scanf("%lf%d",&x,&y);

power = XpowY(x,y);

printf("%.2f to the power of %d is %.2f\n\n",x,y,power);}


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Example 2 : xy

double XpowY(double x, int y){

if (y ==1 )

return 1;elsereturn x * XpowY(x, y-1);

}

Sample Output:

Enter the value of x and y: 2

3

2.0 to the power of 3 is 8.00


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Tracing a problem - XpowY


?


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Summary

Recursive definitionExample of recursive code

Factorial

X power of yTracing technique applied in recursive


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