[4]
QUESTION 1
^ ^
Write down the iteration formulae of the T a y l o r series method of order two for the following
initial-value problem ( I V P ) . T h e n perform one iteration w i t h step length ti - (J.6.
u
' ( i ) = ucos(3t + n ) ,
ii(4) = 5
._3(5)5in(i7)-fr^^67)-^^'V)7).
U:
2
QUESTION 2
(61
Perform one iteration of the improved E u l e r method w i t h h = 0.5 to approximate the
solution of the following I V P . R o u n d to 3 decimals.
x'{t)
=
tx
+ COSZ,
y'{t)
^ e -
ty,
z'{t)
=
^tii)
r
xsm{y
x{2)
=
3
y(2) = 4
+ z),
z(2)
=
5
^cos{s)2.5(knil)
3
3
^a>s(5Jis)_
QUESTION 3
F i n d the L U decomposition of the m a t r i x
7
A =
49
14
2
-5
4
--37
0 '
4
-8
33
1
0
b -5
if
-1
6
-19
-17
I
0
0
(9
7 •
0
1
0
0
-z
0
(9
1
0
0
0
-z
- 3 7 35_
* -
(7
0
7
-!•
1
0
c
C
5
6
0
1
0
C
®
0
0
I
@
0
7
® I
1
0
1 0
-3
7
Z
1
i
-10
-
2
0
—
1
0
3
)
-15
7
-1
0
3
0
0
J
0
(7
-5
0
2
0 0 ~ 1
0
0
0
0
1
0
-
i
0
4
5
^z^il
- [-t)k\
Ki ^ ll-h - ill
QUESTION 4
Consider the system of equations
" -11
4
-5
151
2
8
6
7 '
-4
12
X
y
z
=
" 34 "
45
56
4.1 W h a t can we deduce from a relevant theorem about the convergence of an iteration
sequence generated by the Gauss-Seidel method for the system?
answer.
G i v e a reason for your
•
sijsh/n
and
(2)
we (jm cki^ya^e noihm^ -^^^^
4.2 Perform one iteration of the Gauss-Seidel method w i t h the starting values xa = 7,
= 8, 2o = 9 to approximate the solution of the system,
(3)
2,
-
2.^^^
5
QUESTION 5
Perform one iteration of Newton's method w i t h the starting values VQ
approximate the solution of the following system. R o u n d to 3 decimals.
[51
2, Wo = S to
v"^ + Inw = A
w s'mv = b
-2.2722V
^ 1
1
2 .Z 7 Z _
6
QUESTION 6
Consider the following system of equations in fixed point form:
[4]
X = e'"^ cosy + V2
y = \n{x^ +y'^) + 3.
Assume that the system has a lixed point p i n
= {{x,y)
: 1 < x < 2, 4 < y < 5}.
Prove by means of a relevant theorem that the fixed point iteration sequence {x^^^) w i t h
starting point
= (xQ,yo) will converge to p if x^^^ is close enough to p. (Do N O T
perform any iterations.)
2
T
•\uiSi;
2-
~
0)^ ^ ( v /
~
O.UiS
or e(
^0 ll7(y,^jL=m.4^.>^^}^^'^^^^^'
So ]]7(f)h
io f
^ '
fsmcef^^)^
; f ^^"^ ; i c}osc emucjh io
7
f.
'^M^'^-