EAB 2113 NUMERICAL METHOD
Question 1
Develop an M-file to implement the bisection method. Using this program solve the following problem.
The velocity of falling parachutist is given as
v(t ) 
gm
(1  e  ( c / m )t ) .
c
Where v (t ) = velocity of parachutist = 40 m / s ,
g = gravitational constant = 9.8 m / s 2 ,
m = the mass of the parachutist = 68.1 kg .
Find the drag coefficient, c at the time t  10 seconds using the initial bracket of the root as [13,
16] and iterate until  a  0.001 %.
% Data obtain from the question
f=inline('((9.81*68.1)/x)*(1-exp(-(x/68.1)*10))-40','x');
xl=13;
xu=16;
es=0.001;
%computation
xr=xl; %initiation
while(1) %since we dont know how many interation will take place
xrold=xr; %keep the previous xr
xr=(xl+xu)/2; %formula for bisection method
if xr~=0, ea=abs((xr-xrold)/xr)*100; end %calculate ea if xr not real
answer
test=f(xl)*f(xr);
if test<0
xu=xr;
elseif test>0
xl=xr;
else ea=0;
end
if ea<=es, break,end %end loop if the situation is satisfied
end
format long;
disp('the root for this equation is : ')
disp(xr)
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EAB 2113 NUMERICAL METHOD
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EAB 2113 NUMERICAL METHOD
Question 2
Develop an M-file to implement the false position method. solve the following problem.
The velocity of falling parachutist is given as
v(t ) 
gm
(1  e  ( c / m )t ) .
c
Where v (t ) = velocity of parachutist = 40 m / s ,
g = gravitational constant = 9.8 m / s 2 ,
m = the mass of the parachutist = 68.1 kg .
Find the drag coefficient, c at the time t  10 seconds using the initial bracket of the root as [13,
16] and iterate until  a  0.001 %.
f=inline('((9.81*68.1)/x)*(1-exp (-(x/68.1)*10))-40','x');
xl=13;
xu=16;
es=0.001;
xr=xl
while (1)
xrold=xr;
xr=xu-(((f(xu)*(xl-xu))/f(xl)-f(xu)));
if xr~=0, ea=abs((xr-xrold)/xr)*100; end
test=f(xl)*f(xr);
if test<0
xu=xr;
elseif test>0
xl=xr;
else test>0
ea=0;
end
if ea<=es, break, end
end
format long;
disp ('the root for this equation is : ')
disp(xr)
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EAB 2113 NUMERICAL METHOD
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EAB 2113 NUMERICAL METHOD
Question 3
Locate the root of f ( x)  2 sin( x)  x
x0  2 .
(a)
Using the MATLAB function fzero with an initial guess of
(b)
Using Newton-Raphson method by writing a function M-file. Use an initial guess of
x0  0.5 and iterate until  a  0.001 %.
a)
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EAB 2113 NUMERICAL METHOD
function root=newraph1(f,df,xr,es)
%f=inline('2*sin(sqrt(x))-x')
%df=inline('-cos(sqrt(x))/sqrt(x)-1')
while(1)
xr_old=xr;
xr=xr-(f(xr)/df(xr));
if xr~=0,ea=abs((xr-xr_old)/xr)*100;
end
if ea<=es,break,end
end
fprintf('\n\nthe root for this question is %2.6f\n',xr);
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EAB 2113 NUMERICAL METHOD
Question 4
Develop an M-file to implement the modified secant method. Using this program determine the
loest positive root of f ( x)  8 sin( x) e  x  1 with an initial guess of x0  0.3 and   0.01.
Iterate until  a  0.000001% .
f=inline('8*sin(x)*exp(-x)-1','x')
Ea=1;
xo=0.3;
fprintf('The initial guess is %.2f',xo)
while(Ea>0.000001)
xold=xo;
xo=xo-(0.01*xo*f(xo))/(f(xo+0.01+xo)-f(xo));
Ea=((abs(xo-xold))/xo)*100;
end
fprintf('\nThe lowest positive root of function is %10.6f with approximate
error %7.4f',xo,Ea)
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EAB 2113 NUMERICAL METHOD
Question 5
Find the solution of the following set of linear algebraic equations
x  2 y  3z  1
3x  3 y  4 z  1
2 x  3 y  3z  2
a) Using the left division
b) Using Gaussian elimination
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EAB 2113 NUMERICAL METHOD
c) Using LU decomposition
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EAB 2113 NUMERICAL METHOD
Question 6
Develop a function M-file Tridiag.m to solve the following tridiagonal system with the Thomas
algorithm.
 f1
e
 2









g1
f2
g2
e3
f3
g3
.
.
.
.
.
.
.
.
.
e n 1
f n 1
en
  x1   r1 
  x  r 
  2   2 
  x3   r3 
 
 

 .   . 
  .   . 
 
 

  .   . 
g n 1   x n 1  rn 1 
 
 

f n   x n   rn 
Thomas Algorithm:
(i)
Decomposition:
ek 
ek
and
f k 1
f k  f k  ek . g k 1 , where k  2,3,4,    , n .
(ii)
Forward substitution:
(iii)
Back substitution:
rk  rk  ek . rk 1 , where k  2,3,4,    , n .
xn 
and xk 
rn
fn
(rk  g k . xk 1 )
, where k  n  1, n  2,  ,2,1 .
fk
Using your program, solve the following tridiagonal system.
 0.020875
 2.01475

 0.020875

2.01475
 0.020875



 0.020875
2.01475
 0.020875


 0.020875
2.01475 

 x1   4.175 
x  

 2 = 0 
 x3   0 
  

 x 4  2.0875
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EAB 2113 NUMERICAL METHOD
function x=Tri(e,f,g,r)
%e=input('e= ');
%f=input('f= ');
%g=input('g= ');
%r=input('r= ');
e=[0 -0.020875 -0.020875 -0.020875];
f=[2.01475 2.01475 2.01475 2.01475];
g=[-0.020875 -0.020875 -0.020875 0];
r=[4.175 0 0 2.0875];
n=length(f);
for k=2:n
factor=e(k)/f(k-1);
f(k)=f(k)-factor*g(k-1);
r(k)=r(k)-factor*r(k-1);
fprintf('factor=%2.4f\t f(k)=%2.4f\t
end
x(n)=r(n)/f(n);
for k=n-1:-1:1
x(k)=(r(k)-g(k)*x(k+1))/f(k);
fprintf('x(k)=%2.4f\n',x(k))
end
r(k)=%2.4f\n',factor,f(k),r(k))
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EAB 2113 NUMERICAL METHOD
Question 7
Q7.
Develop a MATLAB script file to determine the solution of the following system of
linear equations using the Gauss-Seidel iteration method by performing first seven
iterations.
9 x1  2 x 2  3x3  2 x 4  54.5
2 x1  8 x 2  2 x3  3x 4  14
 3x1  2 x 2  11x3  4 x 4  12.5
 2 x1  3x 2  2 x3  10 x 4  21
i=1;x1=0;x2=0;x3=0;x4=0;
disp(' i
x1
x2
x3
x4')
fprintf('%2.0f %8.5f %8.5f %8.5f %8.5f\n',i,x1,x2,x3,x4)
for i=2:8
x1=(54.5-(-2*x2+3*x3+2*x4))/9;
x2=(-14-(2*x1-2*x3+3*x4))/8;
x3=(12.5-(-3*x1+2*x2-4*x4))/11;
x4=(-21-(-2*x1+3*x2+2*x3))/10;
fprintf('%2.0f %8.5f %8.5f %8.5f %8.5f\n',i,x1,x2,x3,x4)
end
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EAB 2113 NUMERICAL METHOD
Question 8
.
Develop a script M-file to estimate f ( 2.75) using Lagrange interpolating polynomials of
order 1, 2 and 3 for the following data.
x
f(x)
0
0
1
0.5
2
0.8
3
0.9
4
0.941176
5
0.961538
For each estimate find the true percent relative error if the try function is given by
f ( x) 
x2
.
(1  x 2 )
function fint=LagrangeINT(x,y,xint)
n=length(x);
for i=1:n
L(i)=1;
for j=1:n
if j~=i
L(i)=L(i)*(xint-x(j))/(x(i)-x(j));
end
end
end
f=(xint^2)/(1+xint^2);
et=abs((f-sum(y.*L))/f)/100;
fprintf('\nx=%.8f\n',sum(y.*L));
fprintf('Error=%6f%%\n',et);
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EAB 2113 NUMERICAL METHOD
Question 9
.
The force on a sailboat mast can be represented by the following function:
H
 z   2.5 z / H
F   200
dz
e
0
7 z
where z  the elevation above the deck and H  the height of the mast. Compute F for the
case where H  30 using
(i)
the M-file for Trapezoidal rule with the step size h  0.1 .
the MATLAB trapz function.
function evaluate = trapezoidal1(f,H,h)
%f=inline('200*(z/(7+z))*exp(-25*z/30)')
%y(for the trapz function)=200*(z./(7+z)).*exp(-25.*z./30)
a=0;
while(1)
n=(H-a)/h;
t0=f(a);
t1=0;
for i=1:n-1
t1=t1+f(a+h*i);
end
t2=f(H);
value=h/2*(t0+2*t1+t2);
break
end
fprintf('The Trapezoidal Integral: %2.8f\n',value)
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EAB 2113 NUMERICAL METHOD
Question 10
Develop an M-file to implement Simpson’s 1/3 rule. Using your program solve the following
problem.
The velocity of falling parachutist is given as
v(t ) 
gm
(1  e ( c / m ) t ) .
c
Where v (t ) = velocity of parachutist,
g = gravitational constant = 9.8 m / s 2 ,
m = mass of the parachutist = 45 kg ,
c = the drag coefficient = 32.5 kg / s .
If the distance, d, traveled by the parachutist is given by
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d   v(t )dt ,
0
find the distance using Simpson’s 1/3 rule for the segments 10, 20, 50, and 100.
function distance=simpson(V,a,b,n)
h=(b-a)/n;
t(1)=a;
for i=2:n+1
t(i)=t(i-1)+h;
end
P=0;
for i=2:2:n
P=P+V(t(i));
end
T=0;
for i=3:2:n-1
T=T+V(t(i));
end
d=(h/3)*(V(a)+(4*P)+(2*T)+V(b));
fprintf('\nThe distance travelled by the parachutist=%9.6f\n',d)
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EAB 2113 NUMERICAL METHOD
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EAB 2113 NUMERICAL METHOD
Question 11
Develop an M-file for Euler’s method to solve a first order ordinary differential equation (ODE).
i(t)
E
The current around the circuit at time t is governed by the following differential equation
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di
 2i  3e  2t ,
dt
i(0)  2 .
Using your program, solve the above initial value problem over the interval from t  0 to 2 with
the step size h  0.1 .
function [t,i]= Eulode(didt,tspan,i0,h)
%input:
%didt= name of the function f(t,i)
%tspan= [ti,tf] where ti and tf= initial and final valus of independent
%variable
%y0= initial value of the dependent variable
%h=step size
%output:
%[t,i] where t= vector of the independent variable
%
i= vector of the solution for the dependent variable
tx=tspan(1);
ty=tspan(2);
t=(tx:h:ty)';
n=length(t);
i=i0*ones(n,1);
for x=1:n-1
i(x+1)=i(x)+feval(didt,t(x),i(x))*h;
end
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EAB 2113 NUMERICAL METHOD
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EAB 2113 NUMERICAL METHOD
Question 12
Develop an M-file for Fourth-Order Runge-Kutta method to solve a first order ordinary
differential equation (ODE).
Using your program solve the following initial value problem over the interval from x  0 to 2
with the step size h  0.2 .
dy

dx
x2  y ,
y (0)  0.8 .
function [x,y]=rk40de(dydx,xspan,y0,h)
xi=xspan(1);
xf=xspan(2);
x=(xi:h:xf)';
n=length(x);
y=y0*ones(n,1);
for i=1:n-1
k1=feval(dydx,x(i),y(i));
k2=feval(dydx,x(i)+(1/2)*h,y(i)+(1/2)*k1*h);
k3=feval(dydx,x(i)+(1/2)*h,y(i)+(1/2)*k2*h);
k4=feval(dydx,x(i)+h,y(i)+k3*h);
y(i+1)=y(i)+(1/6)*(k1+2*k2+2*k3+k4)*h;
end
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EAB 2113 NUMERICAL METHOD
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