Question

Numerical methods question, I need help with the script

For each problem, hardcopy of step-by-step solution (hand calculation) for part (a), Matlab/Octave script, and final answer for part (b).

Answer 1

ANSWER:

(here pi = 3.14)

Volume of cylinder V = pi*r*r*h =0.5 litre =0.0005 m-3 ...(1)

Surface Area of cylinder S = 2*pi*r*r + 2*pi*r*h.......(2)

Essentially we are minimising the surface area of the cylinder

Eliminating h,

from (1) we have h = 0.0005 /p*r*r

Using this in (2) we have S = 2*pi*r*r + 2*pi*r*(0.005/pi*r*r) = 2*pi*r*r + 0.001/r

1) to minimize we can solve for derivative of zero d/dr(2*pi*r*r + 0.001/r) = 0 gives r=0.00795775 m

2) to minimze using golden section search use MATLAB,Mathematica,Maple etc

Code for Matlab below

file f function y=f(r)

% modify this - write your own function!

y=2*pi*r*r + 0.001/r;

file:golden.m

figure; hold on;

a=0; % start of interval

b=2; % end of interval

epsilon=0.000001; % accuracy value

iter= 50; % maximum number of iterations

tau=double((sqrt(5)-1)/2); % golden proportion coefficient, around 0.618

k=0; % number of iterations

x1=a+(1-tau)*(b-a); % computing x values

x2=a+tau*(b-a);

f_x1=f(x1); % computing values in x points

f_x2=f(x2);

plot(x1,f_x1,'rx') % plotting x

plot(x2,f_x2,'rx')

while ((abs(b-a)>epsilon) && (k<iter))

k=k+1;

if(f_x1<f_x2)

b=x2;

x2=x1;

x1=a+(1-tau)*(b-a);

f_x1=f(x1);

f_x2=f(x2);

plot(x1,f_x1,'rx');

else

a=x1;

x1=x2;

x2=a+tau*(b-a);

f_x1=f(x1);

f_x2=f(x2);

plot(x2,f_x2,'rx')

end

k=k+1;

end

% chooses minimum point

if(f_x1<f_x2)

sprintf('x_min=%f', x1)

sprintf('f(x_min)=%f ', f_x1)

plot(x1,f_x1,'ro')

else

sprintf('x_min=%f', x2)

sprintf('f(x_min)=%f ', f_x2)

plot(x2,f_x2,'ro')

end