Programming Portion 1. Create a MATLAB function nroot(x,n) that computes the nth

Question

Programming Portion 1. Create a MATLAB function nroot(x,n) that computes the nth roots of r. Use a for or a while loop. Your loop should break after 100 itera- tions, or when r" is correct to 12 significant figures Create an additional function pow(x,n) that computes rn where n is a positive integer using either a loop or MATLAB's array structure. Your function nroot(x,n) and pow(x,n) must not make use of the 'A' operator. Use the "Modified Babylonian Method' derived in class. 2. Create a MATLAB function "pi0' that uses the following infinite prod- uct. Use the nroot() function from problem 1. The function pi0 will not have any input and should return an approx imation to pi with 16 significant figures. Diary File Exercises 1. Set the tolerance in your nroot0 function to 10 15 Use the tic, and toc commands to compare the speeds of your nroot 0 function and MAT- LAB's nthroot() function when computing 2. Do 100 comparisons

Explanation / Answer

Programming Portion Ans

1) The nroot() function

function [y ] = nroot( x,n )
% Function to compute n th root of x
% Using Babylonian Algorithm
y = 1;
   for i=1:100
       y = y -y/n+x/(n*pow(y,n-1));
   end
end

The pow() function

function [ y ] = pow( x,n)
% Function to compute y =x^n
y = 1;
for i=1:n
    y = y*x;
end
end

2) The PI() function

function [s] = PI( )
% Function to find the value of pi using infinite product
s= 1;a=1/2;
for i =1:100
    rt = nroot(a,2);
    s = s*rt;
    a = 1/2+(1/2)*rt;
end
s = 2/s;
end

TESTING FUNCTIONS

>> fprintf('Actual value=%0.16f PI() function value%0.16f',pi,PI());
Actual value=3.1415926535897931 PI() function value3.1415926535897931

Diary File exercies

1) The modified nroot() function with tolerence 10^-15

function [y ] = nroot( x,n )
% Function to compute n th root of x
% Using Babylonian Algorithm
y = 1; yp = -1;
   while abs(y-yp)>10^-15
       yp =y;
       y = y -y/n+x/(n*pow(y,n-1));
   end
end

Testing of nroot() and nthroot() functions

T = zeros(100,2);
for i =1:100
    tic;
    c = nroot(2,2);
    T(i,1) = toc;
    tic;
    c = nthroot(2,2);
    T(i,2) = toc;
end
Avg = sum(T)/100;
fprintf('Average time for nroot =%f ',Avg(1));
fprintf('Average time for nthroot =%f ', Avg(2));

OUTPUT

Average time for nroot =0.000010
Average time for nthroot =0.000089

2)

function [y ] = nroot( x,n )
% Function to compute n th root of x
% Using Babylonian Algorithm
y = 10^100; yp = -1; ct =0;
   while abs(y-yp)>10^-16
       yp =y;
       y = y -y/n+x/(n*pow(y,n-1));
       ct =ct+1;
   end
   fprintf('The Function took %d iterations',ct);
end

OUTPUT

>> nroot(2,2)
The Function took 337 iterations
ans =

    1.4142

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