numerical methods and statics

8/13/2019 Numerical Methods and statics

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MA2264 /MA1251 - NUMERICAL METHODS

UNIT-I

SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS

1. If a function f(x) = 0 is continuous in the interval (a, b) and if f (a) and f (b) are ofopposite signs. Then one of the root lies between a and b.

2. xa!ple of Alge!"#$ e%&"'#()" (i) x#$ 2x % & = 0' (ii) 2x#$ #x $ = 0.

#. xa!ple of T!")*$e)+e)'"l e%&"'#()" (i) x $ cosx = 0' (ii) xex2 = 0' (iii)012log10 =xx .

*. Reg&l" F"l*# Me',(+")()(

)()(

afbf

abfbafx

=

(+irst iteration of egula +alsi -ethod).

&. I'e!"'#e Me',(+" )(1 nn xx =+ .

. onvergence condition of iterative !ethod is 1)(


2/27

1&. D#"g()"ll D(#)")'" 8n nn !atrix 8 is said to be diagonall do!inant if theabsolute value of each leading diagonal ele!ent is greater than or e3ual to the su! of the

absolute values of the re!aining ele!ents in that row.

4iven sste! of e3uations is 1111 dzcybxa =++ ' 2222 dzcybxa =++ '#### dzcybxa =++ is a diagonal sste! is if 111 cba +

222 cab + ### bac +

1. G"&** "$(# Me',(+" If the given sste! of e3uation is diagonall do!inant then

( ) ( )( )nnn zcybda

x111

1

)1( 1 =+

( ) ( )( )nnn zcxad

by 222

2

)1( 1 =+

( ) ( )( )nnn ybxad

cz

###

#

)1( 1 =+

1/. G"&** Se#+el Me',(+" If the given sste! of e3uation is diagonall do!inant then

( ) ( )( )nnn zcybda

x 1111

)1( 1 =+

( ) ( )( )nnn zcxad

by 2

1

22

2

)1( 1 = ++

( ) ( )( )1#

1##

#

)1( 1 +++ = nnn ybxadc

z

1. 6ufficient condition for iterative !ethods (4auss 6eidel -ethod > 4auss 5acobi -ethod)

to convergence is the coefficient !atrix should be diagonall do!inant.

1. The iteration !ethod is a self correcting !ethod since the round off error is s!aller.

20. ?h 4auss 6eidel iteration is a !ethod of successive corrections@

8ns" 9ecause we replace approxi!ations b corresponding new ones as soon the latter

have been co!puted.

21. o!pare G"&** El##)"'#() Me',(+and G"&** (!+") Me',(+


3/27

22. I)e!*e ( " M"'!#7" Aet 8 be an nn nonsingular !atrix. If B is the inverse of the!atrix 8 then 8B = I (i.e.) B = I 81. ?e start with aug!ented !atrix of 8 with identit!atrix I of the sa!e order and convert 8 into the re3uired for! (i.e.) identit then the

inverse is for!ed. 78 C I : 7I C 81:.

2#. o!pare G"&** El##)"'#() Me',(+and G"&** Se#+el Me',(+.

4auss li!ination -ethod 4auss 5ordan -ethod

1. Direct -ethod

2. oefficient !atrix is

transfor!ed into uppertriangular !atrix.

#. ?e obtain the solution

b bac; substitution!ethod.

1. Direct -ethod

2. oefficient !atrix is transfor!ed

into diagonal !atrix.#. Eo need of bac; substitution

!ethod. 6ince finall this sste!

of e3uation each has onl oneun;nown.

4auss li!ination -ethod 4auss 6eidel -ethod

1. Direct -ethod2. It has the advantage that it is finite

and wor;s in theor for an non

singular set of e3uation.#. ?e obtain exact value.

1. Indirect -ethod2. It converges onl for

diagonall

do!inant.

#. 8pproxi!ate value

which is self correct!ethod.


4/27

2*. o!pare G"&** "$(#and G"&** Se#+el Me',(+*.

2&. ?h G"&** Se#+el e',(+is better !ethod than "$(#0* e',(+@8ns" 6ince the current value of the un;nowns at each stage of iteration are used in

proceeding to the next stage of iteration, the convergence in 4auss 6eidel !ethod will be

!ore rapid than in 4auss 5acobi !ethod.

UNIT-II

INTERPOLATION AND APPRO8IMATION

2. xplain briefl I)'e!(l"'#(). 8ns" Interpolation is the process of co!puting the values of a function for an value of

the independent variable within an interval for which so!e values are given.

2/. Definition of I)'e!(l"'#()and e7'!"(l"'#()9

4auss 5acobi -ethod 4auss 6eidel -ethod

1. Indirect -ethod

2. onvergence rate is slow.#. ondition for convergence is

diagonall do!inant.

1. Indirect -ethod

2. The rate of convergence of this !ethod isroughl twice that of 5acobi.

#. ondition for convergence is diagonall

do!inant.


5/27

8ns" I)'e!(l"'#()" It is the process of finding the inter!ediate values of a function fro!

a set of its values specific points given in a tabulated for!. The process of

co!puting corresponding to x 1,...2,1,0,1 =< 0 then the process is called extrapolation.

2. 6tate Ne.'()0* F(!."!+ #)'e!(l"'#()for!ula.

h

xxuwherey

n

nuuuu

yuuu

yuu

yu

yxyuh

n 00

0

#

0

2

000

F

))1()...(2)(1(...

....F#

)2)(1(

F2

)1(

F1)()=(x"8ns

=

+

+

+

++==+

2. 6tate Ne.'()0* B"$;."!+ #)'e!(l"'#()for!ula.

h

xxpwherey

n

npppp

yppp

ypp

yp

yxyph

nn

n

nnnn

=

++++

+++

++

++==+

F

))1()...(2)(1(...

....F#

)2)(1(

F2

)1(

F1)()=(x"8ns #2n

#0. rror in Ne.'()0* (!."!+"

n

n xxwherefn

nuuuuAns


6/27

=01

01

01

01

x

):g(x)7g(x

x

):f(x)7f(x

xx

= f(x) g(x).

#&. Divided difference table"

B G G 2 G #G

B0

B1

B2

B#

G0

G1

G2

G#

),(

)()(

10

01

01

xxf

xx

xfxf

=

),(

)()(

21

12

12

xxf

xx

xfxf

=

),(

)()(

#2

2#

2#

xxfxx

xfxf

=

),,(

),(),(

210

02

1021

xxxf

xx

xxfxxf

=

),,(

,(),(

#21

1#

21#2

xxxf

xx

xxfxxf

=

),,,(

,,(),,(

#210

0#

210#21

xxxxf

xx

xxxfxxxf

=

#. ?rite L"g!")g#")0* (l)(#"l (!&l".

n

nnnnnn

n

n

n

n

n

yxxxxxxxxxx

xxxxxxxxxx

yxxxxxxxx

xxxxxxxxy

xxxxxxxx

xxxxxxxxyAns

))...()()()((

))...()()()((...

...))...()()((

))...()()((

))...()()((

))...()()(("

1#210

1#210

1

1#12101

#200

0#02010

#21

+

+

+

=

#&. ?hat is the assu!ption we !a;e when Aagrange


7/27

8ns" Aet ( ))(, ii xfx , i = 0, 1, 2... n be the given (n %1) pairs of a data. The third ordercurves e!ploed to connect each pair of data points are called cubic splines. () 8

s!ooth polno!ial curve is ;nown as cubic spline.8 cubic spline function f(x) w.r.t. the points x0, x1, .....xn is a polno!ial of

degree three in each interval (xi1,xi) i = 1, 2, ...n such that )(xf , )(xf and )(xf are continuous.

#. ?rite down the for!ula of C$ Sl#)e.

( )[ ]

iiiii

iiiiiii

xxxMh

yxxh

Mh

yxxh

MxxMxxh

xyAns

+

++=

1

2

1

1

2

1

#

11

#

'

)(1

)(

1)(

1)("

and [ ] )1....(#,2,12

* 11211 =+=++ ++ niforyyyhMMM iiiiii

where - = y ()

=+++

+

++++

i

ii

i

iiiiiiiii

h

ff

h

ffahahhah 1

1

11111 )(2 where 1= iii xxh

and ii yf = ' i = 1,2,#,....

UNIT < III

NUMERICAL DIFFERENTIATION AND INTEGRATION

#. Derivatives of G based on Ne.'()0* (!."!+ #)'e!(l"'#() (!&l""

( ) ( ) ( )

( )

++++

+++++

=

........2*10010&*0&

F&

1

221*F*

12#

F#

112

F2

1

1

0

&2#*

0

*2#

0

#2

0

2

0

yuuuu

yuuuyuuyuy

hdx

dy


8/27

( ) ( )

+++++++

= ........102112212

1

11112

1

)1(1 0&2#

0

*2

0

#

0

2

22

2

yuuuyuuyuyhdx

yd

( ) ( )

+++++= ......../2*

1

#22

1

#1 0

&2

0

*

0

#

#

#

yuuyuyhdx

yd whereh

xxu 0

=

If 0xx = then u = 0

+++=

=

.....&

1

*

1

#

1

2

11 0

&

0

*

0

#

0

2

0

0

yyyyy

hdx

dy

xx


9/27

++=

=

........

(

&

12

111 0

&

0

*

0

#

0

2

22

2

0

yyyy

hdx

yd

xx

++=

=

........*

/

2

#

1 0

&

0

*

0

#

##

#

0

yyyhdx

ydxx

*0. Derivatives of G based on Ne.'()0* "$;."!+ #)'e!(l"'#() (!&l""

( ) ( ) ( )

( )

++++++

+++++++++

=

........2*10010&*0&F&

1

221*F*

12#

F#

112

F2

1

1

&2#*

*2##22

n

nnnn

ypppp

ypppyppypy

hdx

dy


10/27

( ) ( )

+++++++++++

= ........102112212

1

11112

1

)1(1

&2#*2#2

22

2

nnnn yuupyppypyhdx

yd

( ) ( )

+++++++= ......../2*

1

#22

1

#

1

&2*#

#

#

nnn yppypyhdx

yd whereh

xxp n

=

If nxx= then p = 0

+++++=

=

.....&

1

*

1

#

1

2

11

&*#2

nnnnn

xx

yyyyy

hdx

dy

n


11/27

++++=

=

........

(

&

12

111

&*#2

22

2

nnnn

xx

yyyy

hdx

yd

n

+++=

=

........*

/

2

#

#1

&*#

#

#

nnn

xx

yyyhdx

yd

n

*1. ?hat are the two tpes of errors involving in the nu!erical co!putation ofderivatives@

8ns" (i) Truncation error' (ii) ounding error (To produce exact result is rounded tothe nu!ber of digits).

*2. Define the error of approxi!ation.

8ns" The approxi!ation !a further deteriorate as the order of derivative increases.The 3uantit (r)= f(r)(x) $ H( r )n(x) is called the error of approxi!ate in the r

thorder

derivative. ?here f(x) is the given e3uation and H(x) is approxi!ate values of f(x).

*#. To find M"7#" and M#)#" of a tabulated function"

Aet = f(x)

+inddx

dyand e3uate to ero. 8nd solving for x.

+ind2

2

dx

yd' If

2

2

dx

ydat x is $ve has !axi!u! at that point x.


12/27

If2

2

dx

ydat x is %ve has !ini!u! at that point x.

?e


13/27

*. S#*()0* ()e-',#!+ !&le"

rule

#

1

( ) ( ) ( )[ ]( ) ( )( )

+

+=

+++++++=

ordinatesevenofSum

ordinatesoddremainingofSumordinateslastandtheofSumh

yyyyyyh

dxxf

st

n

x

x

n

*

21

#

....*...2#)(#1*20

0

*. S#*()0* ',!ee e#g,', !&le"

rule

#

( ) ( ) ( )[ ]....*...#

#)( #*210

0

++++++++++= yyyyyyyyyh

dxxfn

x

x

n

&0. rror in S#*()0* ()e-',#!+ !&le"

{ },.....,!ax> LLL2LLL0 yyMn

abhwhere =

=

&1. The error in S#*()0* O)e ',#!+ !&leis of (!+e! *h .

&2. ?hen does S#*()0* !&legive exact result@8ns" Holno!ials of degree 2 .

. ?hen is T!"e=(#+"l !&leapplicable@8ns" +or an intervals.

&*. ?hen is S#*()0*#

1!&leapplicable@

Mhab

E10

)( *


14/27

8ns" ?hen there even no. of intervals.

&&. ?hen is S#*()0*

#!&leapplicable@

8ns" ?hen the intervals are in !ultiples of three.

&9 R(e!g0* Me',(+" =b

a

dxxfI )(

+=#

122

IIII

&/. 6tate R(e!g0* e',(+ #)'eg!"'#() (!&l"to find the value of =b

a

dxxfI )(

using2

handh .

&. 6tate T.( P(#)' G"&**#") Q&"+!"'&!e (!&l":

8ns"

+

==

#

1

#

1)(

1

1

ffdxxfI

&. 6tate T,!ee P(#)' G"&**#") Q&"+!"'&!e (!&l":

8ns"

( )

++

==

&

#

&0

&

#

&)(

1

1

fffdxxfI

0. I) G"&**#")0* Q&"+!"'&!e" If the li!it is fro! a to b then we shall appl a suitable

change of variable to bring the integration fro! 1 to 1

( ) ( )

dtab

dxabtab

xPut2

'2

=++= 9

=

+=

hh

hh

h II

II

Ih

hIAns2

2

2

*#

1

#2,"


15/27

1. 6tate T!"e=(+#"l (!&l" (! D(&le I)'eg!"l*:

( )dxdyyxfId

c

b

a= ,

( )

( )

( )

+

+=

nodeseriortheatfofvaluestheofSum

boundarytheonnodesremainingtheatfofvaluestheofSum

cornersfourtheatfofvaluesofSumhk

I

int*

2*

2. 6tate S#*()0* !&le (! D(&le I)'eg!"l*:

( )dxdyyxfId

c

b

a

= ,

( )( )( )( )

( )( )( )

++

+

+

+

+

=

matrixtheofroweventheonpositionseventheatfofvaluestheofSum

positionsoddtheatfofvaluestheofSum

matrixtheofrowoddtheon

positionseventheatfofvaluestheofSum

positionsoddtheatfofvaluestheofSum

boundarytheonpositionseventheatfofvaluestheofSum

boundarytheonpositionsoddtheatfofvaluestheofSum

cornersfourtheatfofvaluesofSum

hkI

1(

*

*

2


16/27

#. ?h is Trapeoidal rule so called@

8ns" 9ecause it approxi!ates the integral b the su! of the areas of n trapeoids.

*. o!pare Trapeoidal rule and 6i!pson


17/27

8 !ethod that uses values fro! !ore than one preceding step is called

!ultistep !ethod.

/. 6tate T"l(! *e!#e* (!&l""

0L

0

*LLL

0

#LL0

2L01 .......

F*F#F2F1)( xxhwherey

hy

hy

hy

hyxy v =++++==

. ?hat are the !erits and de!erits of Talor


18/27

( )

( )#*

2#

12

1

,2

,

2

2,

2

,

kyhxhfk

ky

hxhfk

ky

hxhfk

yxhfk

nn

nn

nn

nn

++=

++=

++=

=

/2. 6tate F(&!', O!+e! R&)ge-@&''" Me',(+ (!&l""(for +irst order 6i!ultaneous

differential e3uations)

Aet ( ) ( )zyxfdx

dzandzyxf

dx

dy,,,, 21 ==

yyy nn +=+1 > zzz nn +=+1

?here ( )*#21 22

1kkkky +++= and ?here ( )*#21 22

1llllz +++=

( )

( )#1*

21#

112

11

,

2,2

2,

2

,

kyhxhfk

ky

hxhfk

ky

hxhfk

yxhfk

nn

nn

nn

nn

++=

++=

++=

=

( )

( )#2*

22#

122

21

,

2,2

2,

2

,

kyhxhfl

ky

hxhfl

ky

hxhfl

yxhfl

nn

nn

nn

nn

++=

++=

++=

=

/#. 6tate F(&!', O!+e! R&)ge-@&''" Me',(+ (!&l""(for second order differential

e3uations)

Aet ( ) ( )zyxfdx

dz

dx

ydandzzyxf

dx

dy,,,, 22

2

1 ====

yyy nn +=+1

?here ( )*#21 22

1kkkky +++= and ?here ( )*#21 22

1llllz +++=


19/27

( )

( )#1*

21#

112

11

,2

,

2

2,

2

,

kyhxhfk

ky

hxhfk

ky

hxhfk

yxhfk

nn

nn

nn

nn

++=

++=

++=

=

( )

( )#2*

22#

122

21

,2

,

2

2,

2

,

kyhxhfl

ky

hxhfl

ky

hxhfl

yxhfl

nn

nn

nn

nn

++=

++=

++=

=

/*. uler


20/27

/. 8da!


21/27

&. ?h ungePutta for!ula is called fourth order@

8ns" The fourth order ungePutta !ethod agree with Talor series solution up to

the ter!s of h*. Nence it is called fourth order .P. !ethod.

. E!!(!" 7M#l)e0* P!e+#$'(!

)(*&

1* )&(1

& = nyhE where 1# + nn xx

M#l)e0* C(!!e$'(!

)(01 )&(1&

= nyhE where 11 + nn xx

/. R(&)+ ( e!!(!: ?hen we are wor;ing with deci!al nu!bers. ?e approxi!ate thedeci!als to the re3uired degree of accurac. The error due to

these approxi!ations is called round off error.

T!&)$"'#() e!!(!: The error caused b using approxi!ate for!ula in co!putationsis ;nown as truncation error.

. E!!(!: A+"0* P!e+#$'(!

( ) 1#)&(&

/20

2&1+ = nn xxwhereyhE

E!!(!: A+"0* C(!!e$'(!

( ) 12)&(&

/20

1+

= nn xxwhereyhE

. o!pare the -ilne


22/27

2. It does not have the sa!e

instabilit proble!s asthe -ilne !ethod.

2. 9ut is about as efficient.

#. 8 !odification of8da! AND PARTIAL

DIFFERENTIAL EQUATIONS

0. Define B(&)+"! "l&e !(le. 8ns" ?hen the differential e3uation is to be solved satisfing the conditions

specified at the end points of an interval the proble! is called boundar valueproble!.

1. Define D#e!e)$e Q&('#e)'*

8ns" 8 difference 3uotient is the 3uotient obtained b dividing the differencebetween two values of a function b the difference between two corresponding

values of the independent variable. ( )11L

2

1+ = iii yy

hy

2. F#)#'e D#e!e)$e Me',(+*"

( )11L

2

1+ = iii yy

hy

( )112LL

21

+ += iiii yyyh

y

#. lassification of Hartial Differential 3uations of the 6econd rder

0,,,,2

22

2

2

=

+

+

+

y

u

x

uuyxf

y

uC

yx

u

x

uA

(i) If 0*2 AC then the e3uation is Nperbolic.


23/27

(iii) If 0*2 = AC then the e3uation is Harabolic.

*. Be)+e!-S$,#+'0* D#e!e)$e E%&"'#()" (xplicit -ethod)2

22

x

u

t

u

=

1, +!iu

!iu ,1 !iu , !iu ,1+

()

!iu ,1 !iu , !iu ,1+

1, +!iu

2,1,,11, )21(ahk"hereuuuu !i!i!i!i =++= ++ and 21

=a

Be)+e!-S$,#+'0* D#e!e)$e E%&"'#()

If then2

1=

( )!i!i!i uuu ,1,11, 21 ++ +=

This is valid onl if2

2h

ak=


24/27

&. C!");-N#$,(l*()0* D#e!e)$e E%&"'#()" 7I!plicit -ethod:2

22

x

u

t

u

=

( ) ( ) !i!i!i!i!i!i

uuuuuu,1,1,1,1,11,1

1212 +++++ +=++

?here 2ah

k= and 2

1

=a

C!");-N#$,(l*()0* D#e!e)$e E%&"'#()?hen 1= (i.e.) 2ahk= the ran;Eicholson


25/27

100. 6tate the explicit sche!e for!ula for the solution of the wave e3uation.

8ns" The for!ula to solve nu!ericall the wave e3uation ttxx uua =2

is

( ) ( ) ,,1,122

,

22

1, 12 ++ ++= !i!i!i!i!i uuuauau

?hena

hkei

h

k

a=== ).(

12

2

2

2

The wave e3uation is

1, !iu

and2

0)0,( 0,10,11, + +== iiit uuuxu !iu ,1 !iu , !iu ,1+

1, +!iu

101. Define the local truncation error.

8ns" =E( )

+

+ ++i

ii

ii

iii yh

yyfy

h

yyy

2

2 112

11

102. 6tate finite difference approxi!ation for2

2

dxyd and state the order of truncation

error.

8ns" =iy ( )

2

11 2

h

yyy iii + + and the order of truncation error is (h2).

10#. +orward +inite difference for!ula"

h

yxuyhxuyxux

),(),()( 00000,0

+=

k

yxukyxuyxuy

),(),()( 00000,0

+=

Truncation error is ( ) hxxwhereyuh

xx + 000,2

,,1,11, ++ +=

!i!i!i!i uuuu


26/27

10*. 9ac;ward +inite difference for!ula"h

yhxuyxuyxux

),(),()( 00000,0

=

k

kyxuyxuyxuy

),(),()( 00000,0

=

Truncation error is ( ) hxxwhereyuh

xx + 000,2

10&. 6econd order finite difference for!ulae"

( ) ( ) ( )2

0000000000

0,0

,,2,),(),()(

h

yhxuyxuyhxu

h

yxuyhxuyxu xxxx

++=

+=

( ) ( ) ( )2

0000000000

0,0

,,2,),(),()(

k

kyxuyxukyxu

k

yxukyxuyxu

yy

yy

++=

+=

Truncation error is ( ) hxhxwhereyuh

xxxx + 0002

,12

10. Ea!e at least two nu!erical !ethods that are used to solve one di!ensional

diffusion e3uation.

8ns' (i) 9ender6ch!idt -ethod (ii) ran;Eicholson -ethod.

10/. 6tate standard five point for!ula for solving uxx% u = 0.

8ns"

10. 6tate diagonal five point for!ula for solving uxx% u = 0.

8ns"10. ?rite down one di!ensional wave e3uation and its boundar conditions.

8ns"2

22

2

2

x

u

t

u

=

9oundar conditions are

(i) u(0,t) = 0(ii) u(l, t) = 0 0t

(iii) u(x,0) = f(x), 0QxQl

(iv) ut(x,0) = 0, 0QxQl

110. 6tate the explicit for!ula for the one di!ensional wave e3uation with

m

#aand

h

kwherea === 222 01 .

8ns"

!i!i!i!i!i uuuuu ,1,1,,1,1 *=+++ ++

!i!i!i!i uuuu 1,11,11,11,1 *=+++

++++

,,1,11, ++ +=

!i!i!i!i uuuu


27/27

111. ?rite down the finite difference for! of the e3uation ),(2 yxfu= 8ns"

112. ?rite Aaplace

numerical methods and statics

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