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FCM2043/EBB2143 - COMPUTATIONAL METHODSMAY 2013 SEMESTER

MATLAB ASSIGNMENT/TEST QUESTIONS

Q1. Develop an M-file to implement the bisection method. Using this program solve

the following problem.

The velocity of falling parachutist is given as

)1()()/( tmce

c

gmtv = .

Where )(tv = velocity of parachutist = sm /40 ,g= gravitational constant =

2/8.9 sm ,m = the mass of the parachutist = kg1.68 .

Find the drag coefficient, c at the time 10=t seconds using the initial bracket ofthe root as [13, 16] and iterate until 001.0a %.

Q2. Develop an M-file to implement the falseposition method. Using this programsolve the following problem.

The velocity of falling parachutist is given as

)1()()/( tmce

c

gmtv = .

Where )(tv = velocity of parachutist = sm /40 ,g= gravitational constant = 2/8.9 sm ,

m = the mass of the parachutist = kg1.68 .Find the drag coefficient, c at the time 10=t seconds using the initial bracket ofthe root as [13, 16] and iterate until 001.0a %.

Q3. Locate the root of xxxf = )sin(2)(

(a) Using the MATLAB function fzero with an initial guess of20 =x .

(b) Using Newton-Raphson method by writing a function M-file. Use an

initial guess of 5.00 =x and iterate until 001.0a %.

Q4. Develop an M-file to implement the modified secant method. Using this program

determine the loest positive root of 1)sin(8)( = xexxf with an initial guess

of 3.00 =x and 01.0= . Iterate until %000001.0=a .

Q5. Find the solution of the following set of linear algebraic equations

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2332

1433

132

=++

=++

=++

zyx

zyx

zyx

(a) Using the left-division \.

(b) Using Gaussian elimination.

(c) Using the LU decomposition.

[Hint: Since [LU]x = b, let [U]x = y, so that [L]y = b. Now, first solve fory and then forx.]

Q6. Develop a function M-file Tridiag.m to solve the following tridiagonal systemwith the Thomas algorithm.

=

n

n

n

n

nn

nnn

r

r

r

r

r

x

x

x

x

x

fe

gfe

gfe

gfe

gf

1

3

2

1

1

3

2

1

111

333

222

11

.

.

.

.

.

.

...

......

Thomas Algorithm:(i) Decomposition:

1

=k

k

kf

ee and

1. = kkkk geff , where nk ,,4,3,2 = .(ii) Forward substitution:

1. = kkkk rerr , where nk ,,4,3,2 = .(iii) Back substitution:

n

nn

f

rx =

andk

kkkk

f

xgrx

).( 1+= , where 1,2,,2,1 = nnk .

Using your program, solve the following tridiagonal system.

01475.2020875.0

020875.001475.2020875.0020875.001475.2020875.0

020875.001475.2

4

3

2

1

x

xx

x

=

0875.2

00

175.4

Q7. Develop a MATLAB script file to determine the solution of the following systemof linear equations using the Gauss-Seidel iteration method by performing first

seven iterations.

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2110232

5.1241123

143282

5.542329

4321

4321

4321

4321

=+++

=++

=++

=++

xxxx

xxxx

xxxx

xxxx

Q8. Develop a script M-file to estimate )75.2(f using Lagrange interpolating

polynomials of order 1, 2 and 3 for the following data.

x 0 1 2 3 4 5

f(x

)

0 0.5 0.8 0.9 0.941176 0.961538

For each estimate find the true percent relative error if the try function is given by

)1()(

2

2

x

xxf

+= .

Q9. The force on a sailboat mast can be represented by the following function:

+=

HHzdze

z

zF

0

/5.2

7200

where =z the elevation above the deck and =H the height of the mast.Compute F for the case where 30=H using

(i) the M-file for Trapezoidal rule with the step size 1.0=h .

the MATLAB trapz function.

Q10. Develop an M-file to implement Simpsons 1/3 rule. Using your program solvethe following problem.

The velocity of falling parachutist is given as

)1()( )/( tmcec

gmtv = .

Where )(tv = velocity of parachutist,g= gravitational constant =

2/8.9 sm ,

m = mass of the parachutist = kg45 ,c = the drag coefficient = skg/5.32 .If the distance, d, traveled by the parachutist is given by

=6

0

)( dttvd ,

find the distance using Simpsons 1/3 rule for the segments 10, 20, 50, and 100.

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Q11. Develop an M-file for Eulers method to solve a first order ordinary differentialequation (ODE).

The current around the circuit at time t is governed by the following differentialequation

teidt

di 2323 += , 2)0( =i .

Using your program, solve the above initial value problem over the interval from0=t to 2 with the step size 1.0=h .

Q12. 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 0=x to 2 with the step size 2.0=h .

8.0)0(,2

=+= yyxdx

dy.

i(t)

E