UTP CM Matlab Assignment and Test Questions - May 2015 Sem

8/20/2019 UTP CM Matlab Assignment and Test Questions - May 2015 Sem

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FCM2043/FDM2043 - COMPUTATIONAL METHODS

JANUARY 2015 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()( )( t mcec

gmt v −−= .

!here )(t v " velocity of parachutist " sm #$ % g " gravitational constant " &

'.( sm %m " the mass of the parachutist " kg 1.' .

*ind the drag coefficient% c at the time 1$=t seconds using the initial brac+et of

the root as ,1% 1 and iterate until $$1.$≤aε /.

Q2. Develop an M-file to implement the falseposition method. Using this program

solve the following problem.


)1()( )( t mcec

gmt v −−= .

!here )(t v " velocity of parachutist " sm #$ % g " gravitational constant " &

'.( sm %m

" the mass of the parachutist "kg 1.'

.*ind the drag coefficient% c at the time 1$=t seconds using the initial brac+et of

the root as ,1% 1 and iterate until $$1.$≤aε /.

Q3. 0ocate the root of x x x f −= )sin(&)(

(a) Using the MT02 function fzero with an initial guess of

&$ = x .

(b) Using 3ewton-4aphson method by writing a function M-file. Use an

initial guess of 5.$$ = x and iterate until $$1.$≤aε /.

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

determine the loest positive root of 1)sin(')( −= − xe x x f with an initial guess

of -.$$ = x and $1.$=δ . 6terate until /$$$$$1.$=aε .

Q5. *ind the solution of the following set of linear algebraic e7uations



&--&

1#--

1-&

=++

=++

=++

z y x

z y x

z y x

(a) Using the left-division 8.

(b) Using 9aussian elimination.

(c) Using the 0U decomposition.

,:int; <ince ,LUx " % let ,Ux " !% so that ,L! " . 3ow% first solve for ! and then for x.

Q"# 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

f e

g f e

g f e

g f e

g f

1

-

&

1

1

-

&

1

111

---

&&&

11

.

.

.

.

.

.

...

......

Thomas lgorithm;(i) Decomposition;

1−

=k

k k

f

ee and

1. −−= k k k k g e f f % where nk %%#%-%& −−−−−−= .(ii) *orward substitution;

1. −−= k k k k r er r % where nk %%#%-%& −−−−−−= .(iii) 2ac+ substitution;

n

n

n f

r x =

andk

k k k k

f

x g r x

).( 1+−= % where 1%&%%&%1 −−−−−−= nnk .

Using your program% solve the following tridiagonal system.

−−−

−−

−

$1#=5.&$&$'=5.$

$&$'=5.$$1#=5.&$&$'=5.$$&$'=5.$$1#=5.&$&$'=5.$

$&$'=5.$$1#=5.&

×

#

&

1

x

x x

x

"

$'=5.&

$$

1=5.#

Q$. Develop a MT02 script file to determine the solution of the following system

of linear e7uations using the 9auss-<eidel iteration method by performing first

seven iterations.



&11$&-&

5.1&#11&-

1#-&'&

5.5#&-&(

#-&1

#-&1

#-&1

#-&1

−=+++−

=−++−

−=+−+

=++−

x x x x

x x x x

x x x x

x x x x

Q%. Develop a script M-file to estimate )=5.&( f using 0agrange interpolating

polynomials of order 1% & and for the following data.

x $ 1 & # 5

f ( x) $ $.5 $.' $. $.#11= $.15'

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

)1()( &

&

x x x f += .

Q&. The force on a sailboat mast can be represented by the following function;

∫ −

+=

H H z dz e

z

z F

$

5.&

=&$$

where = z the elevation above the dec+ and = H the height of the mast.>ompute F for the case where -$= H using

(i) the M-file for Trape?oidal rule with the step si?e 1.$=h .

the MATLAB trapz function.

Q10. Develop an M-file to implement <impson@s 1 rule. Using your program solvethe following problem.


)1()( )( t mcec

gmt v −−= .

!here )(t v " velocity of parachutist% g " gravitational constant "

&'.( sm %

m " mass of the parachutist " kg #5 %

c " the drag coefficient " skg 5.-& .6f the distance% d% traveled by the parachutist is given by

∫ =

$

)( dt t vd %

find the distance using <impson@s 1 rule for the segments 1$% &$% 5$% and 1$$.



Q11. Develop an M-file for Auler@s method to solve a first order ordinary differentiale7uation (BDA).

The current around the circuit at time t is governed by the following differentiale7uation

t eidt

di &-&- −+= % &)$( =i .

Using your program% solve the above initial value problem over the interval from$=t to & with the step si?e 1.$=h .

Q12. Develop an M-file for *ourth-Brder 4unge-Cutta method to solve a first order

ordinary differential e7uation (BDA).Using your program solve the following initial value problem over the interval

from $= x to & with the step si?e &.$=h .

'.$)$(%&

=+= y y xdx

dy.

i(t )

E

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UTP CM Matlab Assignment and Test Questions - May 2015 Sem