Chapter 6, Problem 28E

To determine

To write:

A menu-driven program to investigate the constant $\text{π}$ for the given different given options.

Answer to Problem 28E

Solution:

The script file is,

% MATLAB code to calcualte the value of pi.

%script file.

select = pivalues;

while select ˜= 4

switch select

%select the different options to get the value of pi.

case 1

machinformula

%print the result by usin Machin's formula.

case 2

leibn

%get approximate by using Leibniz.

case 3

leibgood

%approximate pi value using Leibniz untill good approximation.

end

select = pivalues;

end

% end of function

%The script file should be placed in the same folder.

The function file is,

% MATLAB code to calcualte the value of pi by different options.

%function file.

function select = pivalues

select = menu('select a option for pi', 'Machin''s formula', 'Leibniz''s formula:n-terms','Leibniz''s formula:good approximation','Exit');

%select the different options in menu.

while select == 0

disp('not valid! please select one of the options')

select = menu('select a option for pi', 'Machin''s formula', 'Leibniz''s formula:n-terms','Leibniz''s formula:good approximation','Exit');

end

end

% end of function

%The function file should be placed in the same folder.

The function file is,

% MATLAB code to calcualte the value of pi by Machinformula.

%function file.

function machinformula

formula = (4*atan(1/5)-atan(1/239));

%define the variable pivalues.

fprintf('Using the MATLAB constant, pi = %.6f\n', pi)

fprintf('the value of pi using Machin''s formula, pi = %.6f\n', 4*formula)

%print the value of pi.

end

% end of function

%The function file should be placed in the same folder.

The function file is,

% MATLAB code to calcualte the value of pi by leibniz's formula for the specific terms.

%function file.

function leibn

fprintf('Approximate value of pi by using Leibniz'' formula\n')

n = askforn;

approxvaluepi = 0;

denominator = -1;

signterm = -1;

for i = 1:n

denominator = denominator + 2;

signterm = -signterm;

approxvaluepi = approxvaluepi + signterm * (4/denominator);

end

fprintf('An approximation of pi with n = %d is %.2f\n', n, approxvaluepi)

end

function out = askforn

inputnum = input('A positive integer for n is entered: ');

num2 = int32(inputnum);

while num2 ˜= inputnum || num2 < 0

inputnum = input('Not Valid! Enter a positive integer:');

num2 = int32(inputnum);

end

out = inputnum;

end

% end of function

%The function file should be placed in the same folder.

The function file is,

% MATLAB code to calcualte the value of pi by leibniz's formula till the good apprroxiamtion is found..

%script file

function leibgood

error = 0.01;

N = 1;

S = 2;

runsum = 0;

difference = 1;

while error < difference

term = (-1)^S*4/N;

temp = runsum;

runsum = runsum + term;

difference = abs(temp-runsum);

N = N+2;

S = S+1;

end

fprintf('An approximation of pi using Leibniz ''series within %.2f is %.2f\n', error, runsum)

%print the value of pi.

end

% end of function

%The script file should be placed in the same folder.

Explanation of Solution

Machin’s formula is given as,

$\begin{array}{l}\frac{\pi }{4}=4\arctan \left(\frac{1}{5}\right)-\arctan \left(\frac{1}{239}\right)\\ \pi =3.1416\end{array}$

Leibniz’s formula is given as,

$\pi =\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}+\mathrm{...}$

The approximation till the fourth term is given as,

$\begin{array}{rll}\pi & =& \frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}\\ & =& 2.8952\end{array}$

MATLAB Code:

% MATLAB code to calcualte the value of pi.

%script file.

select = pivalues;

while select ˜= 4

switch select

%select the different options to get the value of pi.

case 1

machinformula

%print the result by usin Machin's formula.

case 2

leibn

%get approximate by using Leibniz.

case 3

leibgood

%approximate pi value using Leibniz untill good approximation.

end

select = pivalues;

end

% end of function

%The script file should be placed in the same folder.

% MATLAB code to calcualte the value of pi by different options.

%function file.

function select = pivalues

select = menu('select a option for pi', 'Machin''s formula', 'Leibniz''s formula:n-terms','Leibniz''s formula:good approximation','Exit');

%select the different options in menu.

while select == 0

disp('not valid! please select one of the options')

select = menu('select a option for pi', 'Machin''s formula', 'Leibniz''s formula:n-terms','Leibniz''s formula:good approximation','Exit');

end

end

% end of function

%The function file should be placed in the same folder.

% MATLAB code to calcualte the value of pi by Machinformula.

%function file.

function machinformula

formula = (4*atan(1/5)-atan(1/239));

%define the variable pivalues.

fprintf('Using the MATLAB constant, pi = %.6f\n', pi)

fprintf('the value of pi using Machin''s formula, pi = %.6f\n', 4*formula)

%print the value of pi.

end

% end of function

%The function file should be placed in the same folder.

% MATLAB code to calcualte the value of pi by leibniz's formula for the specific terms.

%function file.

function leibn

fprintf('Approximate value of pi by using Leibniz'' formula\n')

n = askforn;

approxvaluepi = 0;

denominator = -1;

signterm = -1;

for i = 1:n

denominator = denominator + 2;

signterm = -signterm;

approxvaluepi = approxvaluepi + signterm * (4/denominator);

end

fprintf('An approximation of pi with n = %d is %.2f\n', n, approxvaluepi)

end

function out = askforn

inputnum = input('A positive integer for n is entered: ');

num2 = int32(inputnum);

while num2 ˜= inputnum || num2 < 0

inputnum = input('Not Valid! Enter a positive integer:');

num2 = int32(inputnum);

end

out = inputnum;

end

% end of function

%The function file should be placed in the same folder.

% MATLAB code to calcualte the value of pi by leibniz's formula till the good apprroxiamtion is found..

%script file

function leibgood

error = 0.01;

N = 1;

S = 2;

runsum = 0;

difference = 1;

while error < difference

term = (-1)^S*4/N;

temp = runsum;

runsum = runsum + term;

difference = abs(temp-runsum);

N = N+2;

S = S+1;

end

fprintf('An approximation of pi using Leibniz ''series within %.2f is %.2f\n', error, runsum)

%print the value of pi.

end

% end of function

%The script file should be placed in the same folder.

Save the MATLAB script with name, main.m, and the function files with names machinformula.m, leibn.m, pivalues.m and leibgood.m in the current folder. Execute the program by typing the script name at the command window to generate result.

Result:

The results is,

Therefore, the result is stated above.