Matlab Assignment Answers

8/11/2019 Matlab Assignment Answers

1/25

EAB 2113 NUMERICAL METHOD

Question 1

Develop an M-file to implement the i!e"tion metho#$ U!in% thi! p&o%&am !olve the

follo'in% p&olem$

The velo"it( of fallin% pa&a"h)ti!t i! %iven a!

)1()( )/( tmc

ec

gmtv =

$

*he&e

)(tv

+ velo"it( of pa&a"h)ti!t +

sm /40

,

g+ %&avitational "on!tant +

2/8.9 sm

,

m+ the ma!! of the pa&a"h)ti!t +

kg1.68

$

in# the #&a% "oeffi"ient, cat the time

10=t

!e"on#! )!in% the initial &a".et

of the &oot a! /13, 10 an# ite&ate )ntil

001.0a$

1

% Data obtain from the questionf=inline('((9.81*68.1)/x)*(1-ex(-(x/68.1)*1!))-"!'#'x')$xl=1$xu=16$

es=!.!!1$%&omutationxr=xl$ %initiationhile(1) %sin&e e ont no ho man interation ill tae la&e

xrol=xr$ %ee the re+ious xr xr=(xl,xu)/$ %formula for bise&tion metho

ifxr=!# ea=abs((xr-xrol)/xr)*1!!$ en%&al&ulate ea if xr not realanser

test=f(xl)*f(xr)$

iftest! xu=xr$ elseiftest0! xl=xr$ elseea=!$ en

ifea=es# brea#en%en loo if the situation is satisfieenformat lon$is('the root for this equation is 2 ')is(xr)


2/25


2


3/25


Question 2

Develop an M-file to implement the fal!e po!ition metho#$ !olve the follo'in%p&olem$


)1()( )/( tmc

ec

gmtv =

$

*he&e

)(tv

+ velo"it( of pa&a"h)ti!t +

sm /40

,


2/8.9 sm

,

3


4/25


m+ the ma!! of the pa&a"h)ti!t +

kg1.68

$

in# the #&a% "oeffi"ient, cat the time

10=t

!e"on#! )!in% the initial &a".et

of the &oot a! /13, 10 an# ite&ate )ntil

001.0a$

f=inline('((9.81*68.1)/x)*(1-ex (-(x/68.1)*1!))-"!'#'x')$xl=1$xu=16$es=!.!!1$

xr=xlhile(1)

xrol=xr$ xr=xu-(((f(xu)*(xl-xu))/f(xl)-f(xu)))$

ifxr=!# ea=abs((xr-xrol)/xr)*1!!$ en test=f(xl)*f(xr)$ iftest! xu=xr$ elseiftest0! xl=xr$ elsetest0! ea=!$ enifea=es# brea# enenformat lon$is ('the root for this equation is 2 ')is(xr)


5/25


4


6/25


Question 3

Lo"ate the &oot of

xxxf = )sin(2)(

5a6 U!in% the MATLAB f)n"tion fzero'ith an initial %)e!! of

20 =x

$

56 U!in% Ne'ton-Raph!on metho# ( '&itin% a f)n"tion M-file$ U!e an

initial %)e!! of

5.00 =x

an# ite&ate )ntil

001.0a$

a6

0


7/25


7

fun&tionroot=nerah1(f#f#xr#es)%f=inline('*sin(sqrt(x))-x')%f=inline('-&os(sqrt(x))/sqrt(x)-1')hile(1) xr3ol=xr$ xr=xr-(f(xr)/f(xr))$ ifxr=!#ea=abs((xr-xr3ol)/xr)*1!!$ en

ifea=es#brea#enenfrintf('4n4nthe root for this question is %.6f4n'#xr)$


8/25


Question 4

Develop an M-file to implement the mo#ifie# !e"ant metho#$ U!in% thi!

p&o%&am #ete&mine the loe!t po!itive &oot of

1)sin(8)( = xexxf

'ith an

initial %)e!! of

3.00 =x

an#

01.0=

$ Ite&ate )ntil

%000001.0=a$

8

f=inline('8*sin(x)*ex(-x)-1'#'x')5a=1$xo=!.$frintf('he initial uess is %.f'#xo)hile(5a0!.!!!!!1) xol=xo$ xo=xo-(!.!1*xo*f(xo))/(f(xo,!.!1,xo)-f(xo))$

5a=((abs(xo-xol))/xo)*1!!$enfrintf('4nhe loest ositi+e root of fun&tion is %1!.6f ith aroximateerror %7."f'#xo#5a)


9/25


Question 5

in# the !ol)tion of the follo'in% !et of linea& al%e&ai" e9)ation!

:


10/25


2332

1433

132

=++

=++=++

zyx

zyx

zyx

a6 U!in% the left #ivi!ion

6 U!in% ;a)!!ian elimination

1


11/25


"6 U!in% LU #e"ompo!ition

11


12/25


Question 6

Develop a f)n"tion M-file T&i#ia%$m to !olve the follo'in% t&i#ia%onal !(!tem'ith the Thoma! al%o&ithm$

=

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

.

.

.

.

.

.

...

...

...

Thoma! Al%o&ithm=

5i6 De"ompo!ition=

12


13/25


1

=k

kk

f

ee

an#

1. = kkkk geff, 'he&e

nk ,,4,3,2 =$

5ii6 o&'a !)!tit)tion=

1. = kkkk rerr, 'he&e

nk ,,4,3,2 =$

5iii6 Ba". !)!tit)tion=

n

nn

f

rx =

an#

k

kkkk

f

xgrx

).( 1+=

, 'he&e

1,2,,2,1 = nnk$

U!in% (o)& p&o%&am, !olve the follo'in% t&i#ia%onal !(!tem$

01475.2020875.0

020875.001475.2020875.0

020875.001475.2020875.0

020875.001475.2

4

3

2

1

x

x

x

x

+

0875.2

0

0

175.4

13

fun&tionx=ri(e#f##r)%e=inut('e= ')$%f=inut('f= ')$%=inut('= ')$%r=inut('r= ')$e=! -!.!!87 -!.!!87 -!.!!87:$f=.!1"7 .!1"7 .!1"7 .!1"7:$=-!.!!87 -!.!!87 -!.!!87 !:$

r=".17 ! ! .!87:$n=lenth(f)$for=2n fa&tor=e()/f(-1)$ f()=f()-fa&tor*(-1)$

r()=r()-fa&tor*r(-1)$frintf('fa&tor=%."f4t f()=%."f4t r()=%."f4n'#fa&tor#f()#r())enx(n)=r(n)/f(n)$for=n-12-121 x()=(r()-()*x(,1))/f()$ frintf('x()=%."f4n'#x())en


14/25


1


15/25


Question 7

Q7. Develop a MATLA s!"ip# $ile #o %e#e"&ine #'e sol#ion o$ #'e $olloin* s+s#e& o$

linea" ea#ions sin* #'e -assei%el i#e"a#ion &e#'o% + pe"$o"&in* $i"s# seveni#e"a#ions.

2110232

5.1241123

143282

5.542329

4321

4321

4321

4321

=+++

=++

=++

=++

xxxx

xxxx

xxxx

xxxx

14

i=1$x1=!$x=!$x=!$x"=!$is(' i x1 x x x"')frintf('%.!f %8.f %8.f %8.f %8.f4n'#i#x1#x#x#x")fori=28 x1=(".-(-*x,*x,*x"))/9$ x=(-1"-(*x1-*x,*x"))/8$ x=(1.-(-*x1,*x-"*x"))/11$ x"=(-1-(-*x1,*x,*x))/1!$ frintf('%.!f %8.f %8.f %8.f %8.f4n'#i#x1#x#x#x")en


16/25


Question 8

. Develop a s!"ip# M$ile #o es#i&a#e

)75.2(f

sin* La*"an*e in#e"pola#in* pol+no&ials o$

o"%e" 1, 2 an% 3 $o" #'e $olloin* %a#a.

x 0 1 2 3 4 5

f(x)

0 0.5 0.8 0.9 0.941176 0.961538

o& ea"h e!timate fin# the t&)e pe&"ent &elative e&&o& if the t&( f)n"tion i!

%iven (

)1()(

2

2

x

xxf

+=

$

10

fun&tionfint=;arane=12n if>=i ;(i)=;(i)*(xint-x(>))/(x(i)-x(>))$ en enenf=(xint?)/(1,xint?)$et=abs((f-sum(.*;))/f)/1!!$frintf('4nx=%.8f4n'#sum(.*;))$frintf('5rror=%6f%%4n'#et)$


17/25


Question 9

$ The fo&"e on a !ailoat ma!t "an e &ep&e!ente# ( the follo'in% f)n"tion=

+=

HHz dze

z

zF

0

/5.2

7200

'he&e=z

the elevation aove the #e". an#

=Hthe hei%ht of the ma!t$

Comp)te F fo& the "a!e 'he&e

30=H

)!in%

17


18/25


5i6 the M-file fo& T&ape>oi#al &)le 'ith the !tep !i>e

1.0=h

$

the MATLABtrapzf)n"tion$

Question 10

18

fun&tione+aluate = trae@oial1(f#A#h)%f=inline('!!*(@/(7,@))*ex(-*@/!)')%(for the tra@ fun&tion)=!!*(@./(7,@)).*ex(-.*@./!)a=!$hile(1)n=(A-a)/h$t!=f(a)$t1=!$fori=12n-1 t1=t1,f(a,h*i)$ent=f(A)$+alue=h/*(t!,*t1,t)$breaenfrintf('he rae@oial


19/25


Develop an M-file to implement ?imp!on@! 13 &)le$ U!in% (o)& p&o%&am !olvethe follo'in% p&olem$


)1()( )/( tmcec

gmtv =

$

*he&e

)(tv

+ velo"it( of pa&a"h)ti!t,


2/8.9 sm

,

m+ ma!! of the pa&a"h)ti!t +

kg45

,

c+ the #&a% "oeffi"ient +

skg/5.32

$

If the #i!tan"e, #, t&avele# ( the pa&a"h)ti!t i! %iven (

=6

0

)( dttvd

,

fin# the #i!tan"e )!in% ?imp!on@! 13 &)le fo& the !e%ment! 1


20/25


2


21/25

i5t6E


Question 11

Develop an M-file fo& E)le&@! metho# to !olve a fi&!t oe& oina&( #iffe&entiale9)ation 5ODE6$

The ")&&ent a&o)n# the "i&")it at time

t

i! %ove&ne# ( the follo'in%

#iffe&ential e9)ation

tei

dt

di 2323 +=

,

2)0( =i$

U!in% (o)& p&o%&am, !olve the aove initial val)e p&olem ove& the inte&val

f&om

0=t

to 2 'ith the !tep !i>e

1.0=h

$

21


22/25


22

fun&tiont#i:= 5uloe(it#tsan#i!#h)%inut2%it= name of the fun&tion f(t#i)%tsan= ti#tf: here ti an tf= initial an final +alus of ineenent%+ariable%!= initial +alue of the eenent +ariable%h=ste si@e%outut2%t#i: here t= +e&tor of the ineenent +ariable% i= +e&tor of the solution for the eenent +ariabletx=tsan(1)$t=tsan()$t=(tx2h2t)'$n=lenth(t)$

i=i!*ones(n#1)$forx=12n-1 i(x,1)=i(x),fe+al(it#t(x)#i(x))*h$en


23/25


23


24/25


Question 12

Develop an M-file fo& o)&th-Oe& R)n%e-)tta metho# to !olve a fi&!t oe&

oina&( #iffe&ential e9)ation 5ODE6$

U!in% (o)& p&o%&am !olve the follo'in% initial val)e p&olem ove& the inte&val

f&om

0=x

to 2 'ith the !tep !i>e

2.0=h

$

8.0)0(,2 =+= yyxdx

dy

$

2

fun&tionx#:=r"!e(x#xsan#!#h)xi=xsan(1)$

xf=xsan()$x=(xi2h2xf)'$n=lenth(x)$=!*ones(n#1)$fori=12n-1 1=fe+al(x#x(i)#(i))$ =fe+al(x#x(i),(1/)*h#(i),(1/)*1*h)$ =fe+al(x#x(i),(1/)*h#(i),(1/)**h)$ "=fe+al(x#x(i),h#(i),*h)$ (i,1)=(i),(1/6)*(1,*,*,")*h$en


25/25


24

Documents

Matlab Assignment Answers