This document describes solving a nonlinear equation to determine the time when a Saturn V rocket reaches the speed of sound using various numerical methods. Specifically, it compares the bisection method, linear interpolation method, Newton-Raphson method, Mueller's method, and the x=g(x) method. For each method, it provides the MATLAB script used to solve the equation and displays the solution time and number of iterations. The root found for all methods was approximately 70.878 seconds.

1. Islamic Azad University
Qazvin Branch
Faculty of Industrial and Mechanics , Department of Mechanical
Engineering
Subject
Compare Some Algorithms for Solving Nonlinear Equation
Thesis Advisor
Dr.Marufi
By
Parham Sagharichi Ha

2. Assignment of Numerical Analysis Parham Sagharichi Ha
parhamsagharchi@gmail.com 1
Problem
The speed v of a Saturn V rocket in vertical flight near the surface of earth can
be approximated by
𝑣 = 𝑢 ln
𝑀0
𝑀0 − 𝑚̇ 𝑡
− 𝑔𝑡
𝑢 = 2510
𝑚
𝑠
= 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑒𝑥ℎ𝑎𝑢𝑠𝑡 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑟𝑜𝑐𝑘𝑒𝑡
𝑀0 = 2.8 ∗ 106
𝑘𝑔 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑟𝑜𝑐𝑘𝑒𝑡 𝑎𝑡 𝑙𝑖𝑓𝑡𝑜𝑓𝑓
𝑚̇ = 13.3 ∗ 103
𝑘𝑔
𝑠
= 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑓𝑢𝑒𝑙 𝑐𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛
𝑔 = 9.81
𝑚
𝑠2
= 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
𝑡 = 𝑡𝑖𝑚𝑒
Determine the time when the rocket reaches the speed of sound (335 m/s).
Solution
𝑢 𝑙𝑛
𝑀0
𝑀0 − 𝑚̇ 𝑡
− 𝑔𝑡 − 𝑣 = 0
Now we want to determine time in the above equation

3. Assignment of Numerical Analysis Parham Sagharichi Ha
parhamsagharchi@gmail.com 2
Matlab
1) Bisection Method
Script :
clc
close all
clear all
%%
% Subject : Bisect Algorithm
% Author: Parham Sagharichi Ha Email :
parhamsagharchi@gmail.com
%%
%-------------------S------T------A------R------T------------
-------------%
global tolerance
tolerance = 1e-4; % for example : 1e-4 = 10^-4
u = 2510;
M0 = 2.8*10^6;
mdot = 13.3*10^3;
g = 9.81;
v = 335;
xlower = 0;
xupper = 100;
myfun = @(t)(u.*log(M0./(M0-mdot.*t))-g.*t-v);
[root,iflag] = fbisect(myfun,xlower,xupper);
switch iflag
case -2
disp('Initial range does not only contain one root')
otherwise
disp([' Root = ' num2str(root) ...
' found in ' num2str(iflag) ' iterations'])
end
%---------------F------I------N------I------S------H---------
-------------%

4. Assignment of Numerical Analysis Parham Sagharichi Ha
parhamsagharchi@gmail.com 3
Function :
function [root,iflag] = fbisect(myfun,a,b)
if a>=b
disp(' attention b>a in [a b] ')
return
end
global tolerance
x = a:0.001:b;
y = feval(myfun,x);
fa = y(1);
fb = y(end);
ymax = max(y);
ymin = min(y);
figure
plot(x,y)
grid on
hold on
plot([a a],[ymin ymax])
plot([b b],[ymin ymax])
iflag = 0;
iterations = 0 ;
while (fa*fb<0) & (b-a)>tolerance
iterations = iterations + 1;
c = (a+b)/2;
fc = feval(myfun,c);
plot([c c],[ymin ymax])
pause
if fa*fc<0
b = c; fb = fc;
elseif fa*fc>0
a = c; fa = fc;
else
iflag = 1;
root = c
return
end
end
switch iterations
case 0
iflag = -2; root = NaN;
otherwise
iflag = iterations; root = c;
end

5. Assignment of Numerical Analysis Parham Sagharichi Ha
parhamsagharchi@gmail.com 4
Result :
Root = 70.8779 found in 20 iterations
2) Linear Interpolation (False Position) Method :
Script :
clc
close all
clear all
%%
% Subject : False Postion Algorithm
% Author: Parham Sagharichi Ha Email :
parhamsagharchi@gmail.com
%%
%-------------------S------T------A------R------T------------
-------------%
global tolerance
tolerance = 1e-4; % for example : 1e-4 = 10^-4
u = 2510;
M0 = 2.8*10^6;
mdot = 13.3*10^3;
g = 9.81;
v = 335;
xlower = 0;
xupper = 100;
myfun = @(t)(u.*log(M0./(M0-mdot.*t))-g.*t-v);
[root,iflag] = finter(myfun,xlower,xupper);
switch iflag
case -2
disp('Initial range does not only contain one root')
otherwise
disp([' Root = ' num2str(root) ...
' found in ' num2str(iflag) ' iterations'])
end
%---------------F------I------N------I------S------H---------
-------------%

6. Assignment of Numerical Analysis Parham Sagharichi Ha
parhamsagharchi@gmail.com 5
Function :
function [root,iflag] = finter(myfun,a,b)
if a>=b
disp(' attention b>a in [a b] ')
return
end
global tolerance
x = a:0.001:b;
y = feval(myfun,x);
fa = y(1);
fb = y(end);
ymax = max(y);
ymin = min(y);
figure
plot(x,y)
grid on
hold on
plot([a a],[ymin ymax])
plot([b b],[ymin ymax])
iflag = 0;
iterations = 0 ;
while (fa*fb<0) & (b-a)>tolerance
iterations = iterations + 1;
c = b - (fb)*(a-b)/(fa-fb);
fc = feval(myfun,c);
plot([c c],[ymin ymax])
pause
if fa*fc<0
b = c; fb = fc;
elseif fa*fc>0
a = c; fa = fc;
else
iflag = 1;
root = c
return
end
end
switch iterations
case 0
iflag = -2; root = NaN;
otherwise

7. Assignment of Numerical Analysis Parham Sagharichi Ha
parhamsagharchi@gmail.com 6
iflag = iterations; root = c;
end
Result :
Root = 70.878 found in 24 iterations
3) Newton-Raphson Method :
Script :
clc
close all
clear all
%%
% Subject : Newton_Raphson Algorithm
% Author: Parham Sagharichi Ha Email :
parhamsagharchi@gmail.com
%%
%-------------------S------T------A------R------T------------
-------------%
format short E
tolerance = 1e-4; % for example : 1e-4 = 10^-4
xlower = 0;
xupper = 100;
xguess = 45;
if (xguess>xupper)||(xlower>xguess)
disp(' error , repate again ')
return
end
xrange = xlower:0.1:xupper;
s = size(xrange);
u = 2510;
M0 = 2.8*10^6;
mdot = 13.3*10^3;
g = 9.81;
v = 335;
syms x
myfun = u.*log(M0./(M0-mdot.*x))-g.*x-v;

8. Assignment of Numerical Analysis Parham Sagharichi Ha
parhamsagharchi@gmail.com 7
u = 2510;
M0 = 2.8*10^6;
mdot = 13.3*10^3;
g = 9.81;
v = 335;
for i = 1:s(2);
y(i) = double(subs(myfun,[x],[xrange(i)]));
end
fa = y(1);
fb = y(end);
ymax = max(y);
ymin = min(y);
figure
plot(xrange,y)
grid on
hold on
plot([xlower xlower],[ymin ymax])
plot([xupper xupper],[ymin ymax])
plot([xlower xupper],[0 0])
iflag = 0;
iterations = 1 ;
f = double(subs(myfun,[x],xguess));
myfun_prime = jacobian(myfun,x);
fprime = double(subs(myfun_prime,[x],xguess));
xn = xguess;
xnew = xn - f/fprime;
plot([xn xn],[0 f])
pause
plot([xn xnew],[f 0])
while (abs(xnew-xn)>tolerance) & (iterations<30)
iterations = iterations + 1;
xn = xnew;
f = double(subs(myfun,[x],xn));
fprime = double(subs(myfun_prime,[x],xn));
xnew = xn - f/fprime;
root = xnew;
pause
plot([xn xn],[0 f])
pause
plot([xn xnew],[f 0])
end

9. Assignment of Numerical Analysis Parham Sagharichi Ha
parhamsagharchi@gmail.com 8
switch iterations
case 30
disp(' Not root found ');
otherwise
disp([' Root = ' num2str(root) ...
' found in ' num2str(iterations) ' iterations
'])
end
%---------------F------I------N------I------S------H---------
-------------%
Result :
Root = 70.878 found in 5 iterations
4) Mueller’s Method :
Script :
clc
close all
clear all
%%
% Subject : Mueller’s Algorithm
% Author: Parham Sagharichi Ha Email :
parhamsagharchi@gmail.com
%%
%-------------------S------T------A------R------T------------
-------------%
tolerance = 1e-4; % for example : 1e-4 = 10^-4
u = 2510;
M0 = 2.8*10^6;
mdot = 13.3*10^3;
g = 9.81;
v = 335;
xlower = 0;
xupper = 100;

10. Assignment of Numerical Analysis Parham Sagharichi Ha
parhamsagharchi@gmail.com 9
xguess = 45;
if (xguess>xupper)||(xlower>xguess)
disp(' error , repate again ')
return
end
myfun = @(t)(u.*log(M0./(M0-mdot.*t))-g.*t-v);
x = [xlower xguess xupper]';%[x2 x0 x1]
xe = xlower:0.1:xupper;
ye = feval(myfun,xe);
ymax = max(ye);
ymin = min(ye);
figure
plot(xe,ye)
grid on
hold on
rline = plot([xlower xlower],[ymin ymax]);
mline = plot([xguess xguess],[ymin ymax]);
fline = plot([xupper xupper],[ymin ymax]);
pause
iterations = 0;
while (true)
iterations = iterations +1;
y = feval(myfun,x);%[f2 f0 f1]
h1 = x(3)-x(2);
h2 = x(2)-x(1);
gamma = h2/h1;
c = y(2);
a = (gamma*y(3)-y(2)*(1+gamma)+y(1))/(gamma*h1^2*(1+gamma));
b = (y(3)-y(2)-a*h1^2)/h1;
if b>0
root = x(2)-(2*c)/(b+sqrt(b^2-4*a*c));
else
root = x(2)-(2*c)/(b-sqrt(b^2-4*a*c));
end
pause
rootline = plot([root root],[ymin ymax]);
if root>x(2)
x = [x(2) root x(3)];
else
x = [x(1) root x(2)];
end
pause
delete(rootline)
delete(rline)
delete(mline)
delete(fline)

11. Assignment of Numerical Analysis Parham Sagharichi Ha
parhamsagharchi@gmail.com 10
rline = plot([x(1) x(1)],[ymin ymax]);
mline = plot([x(2) x(2)],[ymin ymax]);
fline = plot([x(3) x(3)],[ymin ymax]);
if (abs(feval(myfun,root))<(10^-8))&(iterations<30)
break
end
end
switch iterations
case 30
disp(' Not root found ');
otherwise
disp([' Root = ' num2str(root) ...
' found in ' num2str(iterations) ' iterations
'])
end
Result :
Root = 70.878 found in 5 iterations
5) 𝑥 = 𝑔(𝑥) Method :
𝑢 𝑙𝑛
𝑀0
𝑀0 − 𝑚̇ 𝑡
− 𝑔𝑡 − 𝑣 = 0
First Equation :
𝑡 =
𝑢
𝑔
𝑙𝑛
𝑀0
𝑀0 − 𝑚̇ 𝑡
−
𝑣
𝑔
Second Equation :
𝑡 =
𝑀0
𝑚̇
(
exp (
𝑔𝑡 + 𝑣
𝑢
) − 1
exp (
𝑔𝑡 + 𝑣
𝑢
)
)

12. Assignment of Numerical Analysis Parham Sagharichi Ha
parhamsagharchi@gmail.com 11
Script :
clc
close all
clear all
%%
% Subject : x=g(x) Algorithm
% Author: Parham Sagharichi Ha Email :
parhamsagharchi@gmail.com
%%
%-------------------S------T------A------R------T------------
-------------%
tolerance = 1e-4; % for example : 1e-4 = 10^-4
u = 2510;
M0 = 2.8*10^6;
mdot = 13.3*10^3;
g = 9.81;
v = 335;
xlower = 0;
xupper = 100;
xguess = 45;
if (xguess>xupper)||(xlower>xguess)
disp(' error , repate again ')
return
end
myfun1 = @(t)((u/g).*log(M0./(M0-mdot.*t))-v/g);
myfun2 = @(t)((M0/mdot).*(exp((g.*t+v)/u)-
1)./exp((g.*t+v)/u));
xold1 = xguess;
xnew1 = feval(myfun1,xold1);
iterations1 = 0;
while (abs(xnew1-xold1)>tolerance)&(iterations1<30)
iterations1 = iterations1 + 1;
xold1 = xnew1;
xnew1 = feval(myfun1,xold1);

13. Assignment of Numerical Analysis Parham Sagharichi Ha
parhamsagharchi@gmail.com 12
end
root1 = xnew1(end);
switch iterations1
case 30
disp(' Not root found ');
otherwise
disp([' Root1 = ' num2str(root1) ...
' found in ' num2str(iterations1) ' iterations1
'])
end
xold2 = xguess;
xnew2 = feval(myfun2,xold2);
iterations2 = 0;
while (abs(xnew2-xold2)>tolerance)&(iterations2<30)
iterations2 = iterations2 + 1;
xold2 = xnew2;
xnew2 = feval(myfun2,xold2);
end
root2 = xnew2(end)
switch iterations2
case 30
disp(' Not root found ');
otherwise
disp([' Root2 = ' num2str(root2) ...
' found in ' num2str(iterations2) ' iterations2
'])
end
Result :
Not root found
root2 =
7.0878e+01
Root2 = 70.8779 found in 20 iterations2
References
Kiusalaas, J. (2009) Numerical Methods in Engineering with MATLAB®