..
'
Mathematics
Department
King Saud University
M254
?S
Time : 120 minutes
Midterm Exam
Second term
2004/2005
Answer the following questions
·/ 1. Give a brief description of the bisection met~od for approximating the roots of a
.
.
(
gtven equatwn.
I
a. Using the bisection method find a 2nd approximate value of 1i 4_r ·
b. State the error formula and compute an estimated error bound of the value
obtained in part(a).
..J
,r2. Show that if g (x) is continuous on some interval I = [a, b] and a s g(x) s b for all
x
E
I , then g (x) has at least one fixed point in I.
3. Consider the equation In x - 2 = 0 on the interval [7 ,8].
a. U se Newton method to findx 1 , starting with x 0 = 7.5.
~·
Show that the convergence is quadratic.
I
4. Approximate Jx 2 e-.rdx using trapezium method with n =5 ..
0
/a.
·A.
Compute an upper error bound
Find n if the maximum error is not to exceeds 0.0001.
5. State the error formula in the process of approximating a functionf(x) by an
interpolating polynomial Pn (x) .
~ Prove that in the linear case the error fo rmula is:
!J(x) -
2
p 1(x)l s h M ,IJ"(x)!s M
~
8
b. Show that the interpolating polynomial is unique.
.J
1
i
6. Construct the divided difference table for the function f (x) = x + 3 using the nodes
0,2 ,5,6 . .
a. Compute an approximate value of J(O. 7)
__.b. Compute an error bound and compare with actual error.
~
1
Good
Luck
Join your Cy berclass : omar.ourwebclass .com
I
·~
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.
King Saud University:
Math. Dept.
1st Se1nester 1425-26 H MidteriTl Exan1.
Quest ion I:
Math-254
Tin1e: 2 Hours
(4+4+4)
-- -
(a) Find an approximation to .j3 correct to within
method.
10- 3
-*+3 ·P .
by using the bisection
-:..
a,
=
(b) Show that g(x)
2-x has a unique fixed point in
1], Use the fixed point
iteration method to approximate the fixed point accurate to within 10- 3 .
Estimate the number of iterations to achieve this accuracy (verify) .
(c) Let g(x) be continuous on [a,b] and be such that
a< x
< b =>
a ~
g(x)
~ b
Prove that g ( x) has at least one fixed point in [a, b].
Question II:
(4+4+4) _.
~
I
(a) Write down a suitable iterative method that converges quadratically to the root
a = 1 of the equation f(x) = (x- 1) 2 ln(x) .
~?
rr--1 (b) Find the positive value of
ll-;
/j---~
Q
that makes the coefficient of
~ L/-.....:........ · Lagrange polynomial P2(x) equals 1.
X
in quadratic
I
(c) Consider f(x)
the error .
L
-L
= ex
2
and x
= 0,0 .25_,0.5\1:
.)(
Compute .E:(0 .75) and estimate
L
::=.
--z_:x
><-
e_
(3+4+4) -;.
Question III:
(a ) Let f(x) = x 2 +ex and x 0
difference f[xo, x1, xo) .
(b) Iff E C 2[xo, x1] and h
1
= 0, x 1 = 1.
= x1 h
Xl
f(x) dx = -[f(xo)
2
xo
Find the value of the second divided
xo . Then show that
+ f(xl)]-
h3
-f"(ry),
12
(c) Find the step size h so that the absolute value of the error for th e Simpson's
1
r
rule is less than 10.
5
when it is used to approximate the integral [
vO
2
ex dx .
,#
7
..
C
~'/- ~- I t:~
t.
~o
/
\
Math. Dept~· - Math-254
King Saud University:
1427 H Midterm Exam.
Summer Semester
Time: 120 mins.
Maximum Marks = 40
(5 marks)
Problem 1:
Let g(x) be a continuous function on [a, b], and suppose that g satisfies the property
implies that
Show that the equation x
vJ
a ::; g(x) ::; b
~c~l
\!2:_ -~ " ~A 2 -~ .c:.
J <3:-J .:J<--""'1 --::. 9cc..\,-o)
= g(x ) has at least On(:l solution a in [a, b).
v_/_?
Problem 2:
Show that the Newton method for finding reciprocals by solving .!_ - c
X
n >0
= 0 results in
><""~ 1 ::
'1<.71
wi ll converge to t he solution a
(2)
:::. 1-\--o,.
\';:L- L0 \
I
--=:l_
= 0 . 5(x~ - 3) ,
= 3 of the non linear equation x
2
+ ~ -t _::_
-
for all
2x - 3
= c114
r,~..:f_ Cov-- V<:....-f£_.L
Xn+l
n 2: 0
= 0, but the other one will not.
= cx;:; 3 ,
and this sequence fail in finding the fourth root of the positive numb er c.
)'(_tt
~ <--
-
r
----=<..
-
~
\1\A .............
-r c.
4--
\?(0-=-
Problem 5: ·.
Solve the following nonlinear system
0.1x 2
+ 0.1:./
0. 1x + O.lx!;
u~i .1g Newton's method , starting with (x 0 ,y0 )T
=
- 0.8
=
- 0.8
= (0.5,0.5)T,
~-s
(6 marks)
find (x 1 ,yi)T.
Problem 6:
(5 marks)
Use Ga uss elimination method with partial pivoting to find a uE!_que solut}Q!l to the followin g system
Problem 7 :
Find the LU decomposition (with
u;i
x + 2y + 3z
6
2x + 2:y + 3z
· 3x + 3y + 3z
7
= 1) of the matrix A
~
:: ·x ""' ( 'L -=-~r c_x-'"'
can be written as
__ C?
\..-
(6 marks)
~
Problem 4:
Show that the equation x
~ ~'
'>( ~
~I)
/(""--+'-----.-<."X.'-' t-- <=- )(.
;;.
Xn+I
~~)
j>< 't.-<..
~ ::.:~- ---· -
';
~
= v'2xn + 3,
t-t 'X
~<..x)
->(
Prnblem 3:
Sh ow th at one of the following iterative schemes
Xn+l
X -
(~
the iteration
• .... ~ t:._-;:_ t...,J
Also show that this iteration converges quadratically.
-==----·
(1)
(6 marks)
-&-·
9
, (6 marks)
of the linear system in the Proble m 6.
---,-----------~'------------------~ · . ,.
.
------
King Saud University:
Math. Dept.
Math-254
First Semester
1426 H
First Midterm Exam.
Maximum Marks = 20
Time: 90 mins.
Ql: (~Find the second approximation to the (5) 113 using the bisection method .
(b'YShow that the Newton method for fmding reciprocals by solving
~
- c = 0
X
results in the iteration
Xn+ l
= Xn(2- ex~),
n ~ O
Also show that this iteration converges quadratically.
(c) Show that the equation
X
= c115
can be written as
n >O
and this sequence fail in finding the fifth root of the positive number c.
q:z;
(a) Let g(x) be a continuous function on [a, b], and suppose that g satisfi es
the property
a<x <b
implies that a ~ g(x) ~ b
Then the equation x
(~hich
(a)
= g(x)
has at least one solution a in [a, b).
of the following rearrangements
Xn+l
=
eXn 2
1
,
(b)
Xn+l
= exn
-
Xn-
1,
(c) Xn+l
= ln( 2X + 1)
11
of the equation e:i: - 2x - 1 = 0 has a unique fixed-point in [1 , 2) ? ll?e the S\}it~e sequence to find th~ second am:;:;imation us:g fixed-point method, taki_::g
xo = 1.5
(c)/ Compute the number of iterations needed to get the accuracy within 10- 6 ,
using part (b).
Q3: (a) The equation e 2 x = 2x + 1 has a rogt a = 0. Using x 0 = 0.2 , compute
first approximation to the zero by using a suitable iterative method .
(6) If x = a is a root of multiplicity m of f( x) = 0, then find the rate of convergence of the Newton's method.
f~
.
( eyUse the quadratic Lagrange interpolation formula to find the required unknown
coefficients from A, B, C arrd D using
.
.
.
p(0.5) = Aj(0.2) + Bf(0.8) + Cj(1) + Df(2.5)
I
l _I --
'1\ \-
- 'i
L..\
King Saud University:
Math. Dept.
Math-254
first Semester
1427-28 H First Midterm Exam.
Maximum Marks = 20
Time: 90 mins.
Ql: (a) Use Bisecti on met hod to com pute the first two approx imate values for
-1'25 . How many bisections n (or ittv ... ~ ions) are needed to ob tain accuracy to
~
within 10- 4 ?
--
(b) If g E C 2 [o., b] and a~ x ~ b implies that a~ g( :r) < b, then prove that
g has at least one fixed point a E [a, b].
Q2: (a) The cubic eq uation x 3 = 2 can b e written in the form :r: = g( :1:) in two
ways, namely
:1;n+l
= 91 (xn) =X~+ Xn- 2, ~ Xr·+l
= 92(xn) =
2 + 5x - x
;
I
JIU~t>
j i<J;;.-'
3
~
\'
~
n. n 2: 0
W hi ch sequence converges to 2 113 and why ? Use the convergent sequence to find
firs t approximation by using x 0 = 1.2.
. - .. - - ·.. --
CD
(b) Show that the iterative scheme
_ (3A - x~ ~
A
,
2
Xn+I -
co nverges to
JA
j ll-1_) :
A > 0,
n 2: 0
K -::::
z..
5 ( '. 'L ) + )
~ <P-
and its rate of convergence is quadratic.
'
~ ----·--------·------·---- - - - - -- - - - - - -
Q3: (a) The nonlinear equation
has a root. a = 1 in [0.5, 1.5]. Use the quadratic iterative method to fin d the fir s t.
approximation of it using xo = 0.5.
(b) Find the first approximation for the nonlinear sys tem
x2 + y2 =
ex+ y
=
4
1
....-----.....using)~ ewton's methili:l} starting w·,t, initia l approximation
(:1:o, Yo)T
= (1 , -
1.5f.
CA)
=
o
"' f
.._A)
C
\\ Kins Saud University:
Math. Dept.
Math-254
S ec ~ 1d Semester
1427-28 H First .v1idterm Exam.J .M< :imum Marks = 20
Time: 90 mins.
/
Question 1:·
· . (4
6
~)f)::_~ _\.0
(a) The x 6
-
+ 3) .
&"C:..IC )--
..t..../
X...,=
'¥..._- --...--
y
10 ~ O'has a root in [1.4, 1.5]. Show that Newton 's formula for this fun ction is
Use it to find first approximate, taking x 0
of the given sequence is quadratic.
= 1.5.
Also show that the order of convergence
~~-
(b) Develop an iterative procedure for evaluating the reciprocal of a number N by using the
secant method . Then use it to find firs t apj1roximation of the reciprocal of fi . taking the
initial approximations of say x 0 = 0 and :rr -= 0.1.
•
Question 2:
(3
+ 3)
(a) If x = a be root of f( x) = 0, and f'(a) = J"( o:) = f 111 (o:)
rate of convergence of Newton's method.
= 0 but
j (4)(o:)
=f. 0. Find the
(b) The nonlinear equation
2x
= e2x-1
has a root~ = 0.5 in [0, 1] . Use the quadratic iterative method to find the first approximation of it using xo = 0.1.
--------- --------------
Que :tion 3:
(4
+ 3)
I
(c._ Find the first approximation for the nonlinear system
\
I
using Newton 's method , taking (xo ,Yo)T = (1, 1)r .
\
I
I
I
I
I
(b) Use Gaussian elimination to find the relation between a 1 and a 2 so that the following
linear system A x = b is consistent:
(-:
King Sau d University:
Math. Dept.
M-254 Semester I
Final Exam. Time : 90 min .
Max. Marks: 20
Q~:~--.~-:~ -~=-----
~7
I
v '(a) Show th <-1t the equati on 1 - 2x
+ cosx
1428-1429 H
)p>-~(~~-;)
e·-rid
= 0 has a root in [0, 1]. How manv itera-
tious of the bisectioil method are required to estimate this root to an accuracy of~ x 10- 3 ?
C'[a, b] with a :S g(:t:) :S b for all :r E [a, b]. If g is cont inuously clifferentiHble
on [a, b] with max lg'(:r)l = k < 1. Show that g has a unique fixed point. o· E [a, b].
v · (b) Let g E
:rE[Ct,bj
Question 2:
(a) Show that
(4 + 3)
Cl'
= 1 is the root for the equation , .
2
:r. ln x = (2x - 1) 1nx
!1\)\. .
~
~--- Use
quadrG1;!ic convergent iterati-ve method to find the first approximation of o- starting
witl/~()_-5) · _ :.) l
~~
~ ~
,·f.!:>
.• .
1\
f'\
/-#b) \Vrit e down New tou.:s formula tha t can be used to approximate th e_5:~~e root of a
posJt. Jve numb er A, and find Its order of convergence
/57
(4 + :3)
Question 3:
(a) Find the first approximation for the nonlinear system
3:r2
3xy
2
-
:r
3
y2
1
~/
using Newton 's method, starting with initial approximation (x 0 ,y0 )T
= (1, 1f.
,.- 7,{b) :- i11d all Ya.lues of 0: so that the foiiO\ving 1;near system
,::. ;;:.
2:r- y + 3.::
5
+ 29 + 2.::
-2x + Ct.y + 3..::
4:r
6
1
has an infinite number of solutions using simple Gauss elimination method.
King Saud University:
Math. Dept .
Su1nmer Se1nester Midterm Exan'l .
M-254
1427-1428 H
Qu est ions :
(7 +6+5+ 7 +6+ 2+ 2)
(1.) Which of the following iterations
(x~ +ex" - 2)
(ii) Xn+l = )3xn- ex,.+ 2, n = 0, 1, ...
,
3
is most suitable to approximate the root of the equation x 2 - 3x + ex - 2 = 0 in
t.he interval [- 1, 0] ? Starting with xo = 0, find the first approximation x 1 of the
root. Also, compu te the error bound for the approximation.
i)
(
Xn+ l =
N~ o
(2.) Show that the equation x - - has a root
X
JN.
Use the Newton's method for
th is equation to find second approximation using N = 9 and :ro = 2.5
)3.) Show that the iterative scheme
(3a- x~ ) xn
, a > 0, n ;:::: 0
2a
and its rate of convergence is quadratic.
Xn+ l
converges to
fo
=
( 4.) Find th e first approximation for the nonlinear system
/
+y
4x 3
')
x-y
=
6
=
1
using Newt on's method, st arti ng with initial approximation
(xo, Yo)T = (1 , 1)r.
~5 - )
·
Use Gauss-elimination method to find the solution of the linear syst em
Ax = b , where
A = [ ~~! ] , b = [ ~ ]'
4 9 16
}~-)
11
~on~tru~ the LU decomposition of the matrix
usmg Iii - 1 .
..
,.
.
;
~
• • ·· •
• ~;-.
• r t
A given in par t (5) by
.'
• -
·
..:....:.· ~ ·:•M'"•:•·-_,-:_ - .
- _f
'
.
.: . - "---;-;·!i~"O-;.C »~' ...:---•- <
~> ) i JX7I)i LEii~Cl ·the -~le terminant of~the. matr~~ -il 'gkveb. ;i)~: p-arC[,5)fi)y,Fsii1g; ; LU_
d ecompos i ~ion by lii
·~·
~·.'·
=
1.
i
~-
;;~·u,nq H)~if,f,_-, 11 'i)y
1\
= i
·
~3
King Saud University:
First Semester
1429-30 H
Maximum Marks = 20
Math. Dept.
Math-254
II Midterm Exam.
Time: - 90 mins
. ..
,/·,.···
I
Question 1:
(4
+ 4 + 2)
-~-.:.
Cc psider the linear system Ax = b , where
(
4 2 1)
A= - -~
:
(a) {lve the given system by using
t;omy/sition of A using I;;
I
I
~ ~
·------··
,
~imple
~
Gauss-Elimination method.
~ 1.
Also,~
/
(dUs~Jacobi method to find second approximate solution x(Jt of the linear system
/
/
~~using the initial approximation x(o)
=
[1, 1, 1]T. Also, compute the error bound .
' j
(!J If condition number of the matrix A
A/
the matrix A.
is 19/5, then find [=- norm of the inverse of
·
·
Question 2:
(2
(a) Let ::4 be a nonsingular matrix and
stem Ax .'= h. Prove "that
wh~re
x be
+
4
+
4)
an approximat e solution to the linear
r is the residual vector for the system with respect to
x.
Find the value of a if the constant t erm in p 2 (x) is 5. Also, find approximation of
f(2 .5).
.
-
.
.
- ·
~-
f (x) = xe(x+l), with points x 0 = 0, x 1 = 0.5, x 2 = 1.5, x 3 = 2, x 4 __: 3, and
x 5 = 3.5. Compute the error bound for tile.. approxim-a tion ore2 - using -Lagrange
method of degree 5.
.
.
·
(c) Let
King Saud University:
Math. Dept.
Math-254
2nd Semester
1427-28 H
Second Midterm Exam.
Max. Marks = 20
Time: 90 Mins.
Question 1:
(3
+ 3)
(4
+ 3)
(a) Consider the matrix
(i) Use LU-decomposition (lii = l) to factorize A into LU.
(ii) Use part {i) to solve the system Ax= [1, 1, -4jT.
{b) Let J(x) = x 3 and
a=/= 1 is a real number.
Find the value of o so that f[o , l , 1) = l.
Question 2:
(a) Consider the linear system
4xl
X1
(i) Starting with
x<o)
= [0, 0,
(ii) Find a bound for the error
+
+
+
X2
4x2
X2
-
XJ
+
4x 3
= 1
= .: 2
3
)
Of, use Gauss-Seidel iterative method to find x(l).
llx - x< 5 ) II when x< 5 ) is the fifth approximation by Gauss~Seidel method.
(b) Find the linear splines which interpolate the following data:
(0.2, 0.18), (0.3, 0.26), (0.5, 0.41)
What is its value at x
= 0.39?
(3
Question 3:
(a) Let f(x)
= xlnx be given at x 0 = 2, x 1 = 2.5,x2 = 3, and
XJ
+ 4)
= 4.
(i) Construct the-divided difference table for f(x) with respect to these points.
(ii) Use Newton's interpolation formula to find the best approximation of f(3 .5) by a quadratic
polynomial, and find the error bound for this approximation.
(b) Suppose :X= [0.1, 2)T is an approximate solution for the linear system Ax = b, where
A= (
~ , ~-01
Fmd a bound for the relat1ve error
llx - xll
llxll .
.
.
) '
.,
Math . Dept.
Math-254
King Saud University :
Semester I
1426-1427 H
II MidTerm Examination
Max. Marks 20
Time: 90 min.
(3 + 4)
Question 1:
(a) Use th e New ton p olynomial of degree 2 that approximate the fun ction ln( 6_2) by
taking the fun ction f (x ) = ln(x + 2) and points from x = 7, 4, 3, 8 and 5.
(b) Determine t he value of step size h and subintervals n to approxim ate t he in tegral
2
1
{ - dx to within 10- s, then compute d e approximation, using the Sim pson 's rule.
lo -r + 4
.
-----------------------(3 + 4)
~u e stion 2:
(a) Let x 0 = a, x 1 = b and h = b - a, then show that
1
b
b - a [ f(a)
f( x) dx :::::::-
2
a
+ f(a + h) ]
(b) Let f (x) = x 2 ln x. Then
(i) Compute the approximate value of !"(2) , taking h = 0.1 using the nu merical
differentiation formul a .
bound for your apj..:
0: 1 Compute t he error
'C imation.
Question 3:
(3
+ 3)
(a) Use a suitabl e interpolati_ng polynomial and its error term to deri ve t he d iff~at i o n
formula
..
- ·
! '( 1 ) =
x
2
f( x l + h) - f( x l - h) _ h fm( )
2h
6
rJ '
rJ E
(x1 - h, x 1 + h)
(b) Show t ha t the foll owing matrix is nonsingular using the Gauss eliminat ion metho d:
(~ ~
-:- 1 ' 1
3 1
~ r ~)
-5
3
7 - 2
Also, solve t he lin ear system A x = [10 , 31, - 2, 18jT .
---------....----·
King Saud University:
Math. Dept.
Math-254
1st Semester 1427-28 H Second Midterm Exam.
Time: 90 Mins.
_,
Max: Marks
20
--~ \ ()_\..-v -\>\---Question 1: Cons ider a lin~ar s_vs t rm Ax = b , wh Prc
A=
~
1
- ]
1
(
()
())
5
3 -3
-
3
-2
;.)
0 0
'
(a) Usc G auss clirninat.iml m eth o d to solvr thi s
r --~.~co~
"";L ' \
L_;,
s~r s t. c m .
A ;'t~~:!~oJ L
J
C''
@ b ) Factorize A into LC hy Doolittk 's met hod (/;; = 1) . Sh ow t hat tht· mat rix A is
noosni~ular.
c-_,.-'ii-\b
II\ I
+o
A ~ rta=~
L~ ~e\u
~ --
~ ~
~ J Supp(_ls<' v:~· want to solv~, tl lf' foll ow in~
sv s t c m will pvr hc tt.<'r solut 10 11 and wh y 1
0
,
~
:;'
\
e ' o\ U ::.c,, I~
e<~, c 1 L e~,
sys tems b~· Gauss elimin a ti o n met ho d . whi c
(D o no t. solv(' th e sys te m s) .
A ,
L- u
r o- o od \ - l_oJ\
1.{-t~
u,._ u,\
u u
o u
o
1. 'L tJ
0
1 0
0
u
(ii )
OA 003x 1 - L 502:r2
0. 0003x 1 + L203:c2
'ti
2_;)01
1.206
Question 2: C ons id e r th e lin ear sys t e m Ax = b , \·vh er c
-5 1
A = ( :) ~
0)
-~
,
(a) Find tlw Jacobi it.(•rati o n matrix T, 1 an d s bow th at
- \ J ::: \::)\. (L+U)
IIT1 11 <
1.
=
(b) Use .Jacobi m e thod to find firs t app rox imat.(' solution x(l ) o f th e linear sv st.('rn by
usin g x( 0l = [0. 0, OJT _ Also. co mpute th e e rro r b o und !lx - x (Io)jj . :::: un\"" \~ -"-.,) l'lr
~
\ - \\\~
~ Cornput<' t.h('
Question
IJliiiJhN
of st('ps JW(·ded t o g <•t. t.lw a ccmacv within 1W 5 _
1pposp :r 0 • .T 1 ar<' t wo di s tin c t. numbers in t. h P. inte r va l [a, h] wi t h
.. ) E C 2 [u. , b]- Find a b o und for th e e rror in lin e<tr interpo la ti o n .
\
~ (b) Given f(O) = 1, f(2) = 3, f(3)
interpolatiou formula t.o approximate
/
=
4 <t nd f( S)
f (4 _;) ) _
-···· -
(c) Considn th e points
:r
+ 2 1n(:r +
formula) _
6- Use th e quadrati c Lagran ge
------------
a nd t IH' fun ction .f (-:) =
2), <·D rnpuU• an ('rror >0111H fo r t.h C' a pprox1ma t.i o u o f j( 2) (hy Lagr a uge
:r. 0 =
\~\
\\I
s:c
~\e-('1.)\
1
~~ ~
[
(i )
L2-: 'ioos -'.soL~
0_0003:rl + L203.r:.!
L20G
() _.1003:rl - L S02T 2 = 2.501
1
t '1
1.1
.
'1-J \
.,
K . ng Saud University:
Math. Dept.
Math-254
1st Semester 1426-27 H Final Exain.
Time: 3 Hours
(5
Question 1:
+ 6. + 6)
J.a)
Show that the equation xP- N = 0 can be written as x = Nx 1 - P, and that the associated
iterative process Xn+l = N x~-p, n ~ 0 will not be successful in finding the pth root of the positive
number N, where p = [2, 3, .. .].
lf/J{b) What is the multiplicity of the
(l'_ method that converges qu adratically
\when xo
= 0.5.
root a == 0 of the equ a tion x 2 = 2 - 2 cos x. Use a numerical
to compute the first approxim ation round to four decim al places
·
(c) Let f(x) = (x+2)ln(x+2) and points: x 0 = O,x 1 = l,x 2 = 2 and x 3 = 3. Use the quadratic
. _ Lagrange_interpolation.formula based on the points. to. approxima te-j(2.8).-- Also, compute the~
ernr and an error bound for your approximation .·
----
Question 2:
~b)
(5
Use qu adratic interpolation which interpolates f( x)
merical integration rule for approximating the integral
.·;:t';· :.·,~::-
:t
1
+ 5 + 6)
points xo, x 1 an d x 2 , derive a suitable nu-
f (x)
dx .
;_. :, (b) Let;.f{x) = .x + ln(x +2}.;.Use a three point formula to approximate·j'(2) :using:<j(2.1) and 1(1.9).
Find actual error and error bound for your approximation .
(c) Use Taylor's method of order two on the initial value problem to approximate y(l.O l) .
Ques.tion 3:(5
+ 6 + 6)
Consider the following matrices
A=
c:
2
10
1
i).
-4
- 0.22
0.03
A-' = (
- 0.06
- 0.05
-0.11
- 0.04
-0. 07 )
-0. 04
,
- 0.28
II
(a) Find the LU-decomposition of A .
t--{ ;: ,Ji' . F;
h=(,. , ~n
P-;r :. fVJ
-
A
'I
(b) Show that Jacobi method conve~ges for the~iven linear system. If the first approxim ate solution of
the given linear system by the Jacobi method is x(l) = [0 .6, - 2.7, - If, by using iJ?1tial approximation
x(o) = [0, 0, O]T, then compute number of. steps needed
to get accuracy within 10-\''
.
(c) Compute an upper bound for the relative error in solving the given linear system .
...
K:ng Saud University: Math. :rlept. Final Exam. M-254
F ~st Semester 1427-28 H Ma:r Marks = 50 Time: 3 Hours
~uestion 1:
(4 + 5 + 4 + 4)
(a) Let g E C[a , b] with g(x) E [a, b] for all x E [a, b] and g'( x) is continuous fun ction on
(a , b) such that Jg'(x) J ~ k < 1 for all x E (a, b). Show that
Jo. - :.r nl ~
k"
1
n >1
_ k Jx1 - :coJ ,
(b) Use ~ewto n 's method to find t he second approxim at ion to th e root of
x
4
-
3x 2 + -Lr
-
1 = 0. using xo = 0. 5.
(c) If the iteratiw scheme
= a:1·~
x·n+I
b
+-- 5,
~-"
n 2 0 converges too. = l. th en find
5
the Yalu Ps of o e1ncl b so that th e conwrgence is quadratic:.
")(,~ ~ -\- 'o(d) Find thr .L1cohi<lll matrix and it s inn' rse from th e follo\\'ing nonlinear syst em \\'ith
x 0 = 1 and Yo = - 1:
T
2
+ 4y-') =
9
2
\ 8y - 14x = - -.15
and
------·------------------·--- -- ·----
Question 2:
(a) Lrt f(:t)
=
3r
-3_, cm d
"'~
(-:1
T
+3+
-1 + 5)
= 0. l. 2. .3, and -1. Construct the cliYickd cliffC'n'll et' tabl e and
then use it to int erpolate f( 3.5) usin g th e :\ewton diYid ed difference quadrati c: formu la.
{b) If f( x) = 1: + ln :r , th en find the Yalue of the second di,·ided difference f[ 2, 3, 3).
(c) Consider a linear system Ax = b , wh ere
A =
(3 2)
2 50
b =
'
(1)
1
.
l'sin g J acobi nwt lwei and Gnuss-Seidelme:h :..lCI. \\'hic:h OIH' \Yill conYerge fc1ster aud '"h~-"
(d' If approximate solution of the system i' ·)art (c) is [0.1 , 0.4f, th en find upper bound
fo · t.he relatiw error in solving the given li11 r sys tem in part (c)_
Question 3:
(4 + 5 + 4 + -'1)
(a) Find step-size h. so that the absolut e Yalu e of th e error for composite Trap ezoid al
·l d:r
rule is less than 5 x 10- .J when it is used to approxim ate
- _
j
2
1'
(b) L'se a suitable int erpola t.in g polynomial and its error term to deri n' the differentiation
formula
2
- !( x i - h) _ h
! '( XI. ) = f(xi + h) 2h
6
(c) Given th e function j(1') =
'r/ '
+ 1 a t th e distinct points /
2.2 2.5 and 2.8. l ·se a
1 1
su itable numerical form ula to compute th e approxim ation of ~ a t :t~
. -.
1pute
'(__. S
the actual error and an error bound for ,von approxim at ion.
( (. 1 Use Tay lor's second-order method to .
roximate th e n1lue of y(l.O- :
___
2.x y'
1:
In J '
f "'( )
+ y = 2x 2 ,
y( 1) = 0.25 ,
n
=2
Math 254
Final exam, 1st semester 1425 H
Department of Mathematics
King Saud University
Time: 3 -Hours
.
~
1. Use the fixed point iteration method to find a root of the equatiOn e
1.982
=X
near x 0
= 1.300
(make only 4-iterations).
2. Use quadratic ¥lynomial
interpolation
to approximate f(1.5) given that
X,
'j.._
X: -3
1
3
f(x):
2
1.5
3.5
5
~then estimate the error if lf(3)Cx)I~M, for all xE[-3,3].
Prove that there is only one polynomial of degree < 2 that satisfies
P2 (xJ = YP i = 0, 1, 2.
4. Use an efficient method to numerically approximate the integral
x+1
2
15
0 . 5 sxs 1
--,
.
Jf(x)dx where f( x ) =
1
0.5
-
\
, given that h
. ,
1S
X
=
0.25.
S 1.5
X
5. Given the two systems
5x1 + 7x2 = 0.7
5£1 + 7£2 = 0.69
7x1 + 10x2 =1.0
7£, +10£2 =1.01,
bound the relative error in x . Is the system ill-conditioned? Why?
6. Use Euler's method with h = 0.2 to find y(0.4), given that
dy
dx
~
2
=X
-y,
and
y(O)=l.
Find Newton iteration formula us.ed to approximate the-value of
±'
b > 0.
Then show that the proper value of the initial guess x 0 that makes
Rei(xn+l) =a- xn+l
a
n ~~
0 must satisfy 0 < x 0 <
2
b
.
8. Use the LU-Decomposition method (fii = 1) to solve the system Ax= b,
where A =
_/l·
l~ ~ =~l ;~ J~]·
lo
b
1#=-f
lo
2 -2 " .
Find the Jacobi iteration matrix M for the matrix A in question number 8.
For any system Ax= b, do the iterations will converge to the exact solution
or not? Why?.
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