How can I use the Bisection method to converge on an arbitrary value?

I am working on coding a function to solve for the water level inside a water tank. The volume of the tank is described by:

V = pi * x^2 *((3*R-x)/3)

I have R = 4 m and I need to find the height (x) where V = 100 m^3. I chose to use the Bisection Method to find this height and use a while loop that should end when the height that gives out Volume equal to 100 m^3. Though I am having a hard time getting the loop to run more than just once and I don't know what could be going wrong. I'm new to Matlab so I might be missing something very obvious but any help would be appreciated.

Here is the code that I have managed to come up with so far:

% Define function

R = 4;

v = @(x) pi.*x.^2.*((3.*R-x)./3);

% Define variables

xL = 0;

xH = 2.*R;

vL = v(xL);

vH = v(xH);

% Define xr

xr = .5.*(xL+xH); % Bisection method

vR = v(xr);

% Define the loop's target value

tar = 100;

while v(xr) == tar

xr = .5.*(xL+xH);

% Update xH and xL

if vH>vR

xH = xr;

else

xL = xr;

end

end

root = xr;

Once again thanks for any help it is very appreciated

Answer 1

[Voss]

In fact your loop runs 0 times (because v(xr) is not equal to 100).

% Define function
R = 4;
v = @(x) pi.*x.^2.*((3.*R-x)./3);
% Define variables
xL = 0;
xH = 2.*R;
vL = v(xL);
vH = v(xH);
% Define xr
xr = .5.*(xL+xH); % Bisection method
vR = v(xr);
% Define the loop's target value
tar = 100;

disp(v(xr));

  134.0413

while v(xr) == tar
    disp('looping');
    xr = .5.*(xL+xH);
     % Update xH and xL
    if vH>vR
       xH = xr;
    else
       xL = xr;
    end
end
root = xr;

It should loop until v(xr) == tar, so: while v(xr) ~= tar, and in fact it probably should not be with exact (in)equality but using some small error, e.g., while abs(v(xr) - tar) > max_error. And you need to update your volumes as you go and use tar to tell which of xH or xL need to be updated each time.

% Define function
R = 4;
v = @(x) pi.*x.^2.*((3.*R-x)./3);
% Define variables
xL = 0;
xH = 2.*R;
vL = v(xL);
vH = v(xH);
% Define xr
xr = .5.*(xL+xH); % Bisection method
vR = v(xr);
% Define the loop's target value
tar = 100;
max_error = 1e-6;

while abs(v(xr) - tar) > max_error
    xr = .5.*(xL+xH);
    vR = v(xr); % update vR too
    fprintf(1,'looping: xr = %12.8f, vR = %12.8f\n',xr,vR);
     % Update xH and xL
    if tar < vR % use tar to determine which side of the target you're on
       xH = xr;
       vH = v(xH); % update vH too
    else
       xL = xr;
       vL = v(xL); % update vL too
    end
end

looping: xr =   4.00000000, vR = 134.04128655
looping: xr =   2.00000000, vR =  41.88790205
looping: xr =   3.00000000, vR =  84.82300165
looping: xr =   3.50000000, vR = 109.03944502
looping: xr =   3.25000000, vR =  96.78396118
looping: xr =   3.37500000, vR = 102.88102348
looping: xr =   3.31250000, vR =  99.82405544
looping: xr =   3.34375000, vR = 101.35052611
looping: xr =   3.32812500, vR = 100.58677545
looping: xr =   3.32031250, vR = 100.20528511
looping: xr =   3.31640625, vR = 100.01463751
looping: xr =   3.31445312, vR =  99.91933825
looping: xr =   3.31542969, vR =  99.96698583
looping: xr =   3.31591797, vR =  99.99081115
looping: xr =   3.31616211, vR = 100.00272420
looping: xr =   3.31604004, vR =  99.99676765
looping: xr =   3.31610107, vR =  99.99974592
looping: xr =   3.31613159, vR = 100.00123506
looping: xr =   3.31611633, vR = 100.00049049
looping: xr =   3.31610870, vR = 100.00011820
looping: xr =   3.31610489, vR =  99.99993206
looping: xr =   3.31610680, vR = 100.00002513
looping: xr =   3.31610584, vR =  99.99997859
looping: xr =   3.31610632, vR = 100.00000186
looping: xr =   3.31610608, vR =  99.99999023
looping: xr =   3.31610620, vR =  99.99999604
looping: xr =   3.31610626, vR =  99.99999895
looping: xr =   3.31610629, vR = 100.00000041

format long;
root = xr

root = 
   3.316106289625168

v(root)

ans = 
     1.000000004077111e+02

[Logan Parrish]

Thanks a ton! I was really struggling trying to wrap my head around how to get my function to approach the target value and your explanation really understand where I went wrong.