Page of 2 1
Answer only four questions [Total mark 120], [30 for each question]
1- (a) Give the definition of fixed point of a function
g
, also if
],,[)(],,[ baxgbaCg
and
for all
],,[ bax
show that
g
has at least one fixed point in
ba,
.
(b) Show that
2
sin5.0)( x
xg
has a unique fixed point on
2,0
.
(c) Use fixed-point iteration to find an approximation to the fixed point that is accurate
to within
2
10
.
2- (a) Construct a natural cubic spline that passes through the points (1, 2), (2, 3), and (3, 5).
(b) Suppose that
,6,4,3,2,1 43210 xxxxx
and
x
exf )(
. Use the Lagrange
interpolating polynomial
)(
4,2,1 xp
to approximate
)5(f
.
(c) Use the most accurate three point formulas and second derivative midpoint formula to
approximate
)0.2('f
and
)0.2(''f
using the data given in the following table; estimate
the actual and the maximum error.
x
1.8
1.9
2.0
2.1
2.2
x
xexf )(
10.889365
12.703199
14.778112
17.148957
19.855030
3- (a) Use Newton's iterative method to find the approximate solution for
,0)cos( xx
4
],2,0[ 0
xx
(Only four iterations).
(b) Use a formula of higher order accuracy to approximate
1
0
2)cos( dxex x
with
8n
.
and estimate the error.
(c) Use Composite Simpson’s rule with
4n
and
2m
to approximate.
.)2(
0.2
4.1
5.1
0.1
dxdyyxnl
4- (a) Drive Runge-Kutta method of order two.
(b) Use the Runge-Kutta method of order four with
2.0h
to obtain approximations to the
solution of the initial-value problem
,20,1
2
ttyy
,5.0)0( y
and compare
Al - Azhar University
Final Exam
Time: 3 Hours
Faculty of Science
Year 2018/2019
Date: 1/2019
Mathematics Department
Numerical analysis [M 304]
