Assignment II
Semester: Spring 2024
Faculty
:
Faculty of Science
Department
: Math & Computer Science
Division/ program: Math/CS
Course Name
: Numerical Methods
Course Code: Math 348
Note: Use maximum precision that your calculator provides in all calculations and provide
the final answer in as many numbers of significant figures as the question specified.
.
1. Approximate f(8.4) using Lagrange interpolating polynomial of degree three. Provide
your answer in eight significant figures.
x
f(x)
a) 17.87833;
8.1
17.61549
8.3
17.56492
b) 17.87716;
8.6
18.50515
8.7
18.2091
c) 17.810000;
d) 17.60859;
2. Three values of the Gamma functions (x) are given in the table below. Use Lagrange’s
second order interpolating polynomial to estimate (1.01). Provide your answer in four
significant figures.
x
(x)
a) 0.9943;
1.00
1.0000
b) 0.9835;
1.02
0.9888
1.04
0.9784
c) 0.9935;
d) 0.9843
3. Use Newton third order interpolating polynomial to estimate f(9.2). Provide your
answer in four significant figures.
pg. 1
Dr. N. Yassine
X
f(x)
a) 2.291;
8.0
2.079
9.0
2.197
b) 2.519;
9.5
2.251
c) 2.219;
11.0
2.398
d) 2.215; e) none of the above.
4. Use multiple applications of Simpson’s 1/3 rule with n = 6 to approximate the
3 / 8
integral
 tan x dx .
0
a) 0.960547;
b) 0.961055;
c) 0.961944;
d) 0.998816
2
5. Estimate the integral  e − x dx using a multiple application of the trapezoidal rule with
1
5 segments.
2
6.
Apply multiple applications of Simpson’s 1/3 rule with n = 6 to evaluate  sin( x 2 )dx .
0
7. Given the following data points, f(1) = 1.5, f(2) = 2.5, f(3) = 4.5, use a second order
LaGrange interpolating polynomial to estimate the value of f(2.1).
a) 2.655
b) 2.820
c) 2.995
d) 3.180
e) 3.375
8. Calculate the integral of the function f(x) = sin(x)/x from 0 to 5 using a multiple
application of the trapezoidal rule with 5 segments. Note that f(0) is 1.
a) 1.8383
b) 1.4453
c) 1.7520
d) 1.5996
e) 1.5581
9. Determine the missing entries in the table.
X
f (X )
0.0
F [ X 0, X 1]
0.4
0.7
pg. 2
1st DD
2nd DD
50/7
10
6
Dr. N. Yassine
10. Find the ratio of the coefficients b3/b2 in Newton’s third order interpolating polynomial
for the following data:
x
1
2
3
4
y
1.9
3
6
10
Note: b3 is the coefficient of (x-1)(x-2)(x-3) and b2 is the coefficient of (x-1)(x-2).
a) -0.15789
11.
b) -0.055556 c) -0.20513
e) 0.33333
Fit quadratic splines to estimate f(2.2). Provide your answer in three significant figures.
X
0
3.0
6.0
9.0
f(x)
1.0
2.2
3.1
5.9
a) 2.22;
pg. 3
d) -0.23232
b) 1.85;
c) 2.25;
d) 1.88;
e) none of the above
Dr. N. Yassine
FORMULAE
Polynomial interpolation
f ( x) = b0 + b1 ( x − x0 ) + b2 ( x − x0 )( x − x1 ) +  + bn ( x − x0 )( x − x1 )  ( x − x n −1 )
b0 = f ( x0 )
b1 = f [ x1 , x0 ]
bn = f [ x n , x n −1 , , x1 , x0 ]
f [ xi , x j ] =
f ( xi ) − f ( x j )
xi − x j
f [ x n , x n −1 ,  , x1 , x0 ] =
f [ x n , x n −1 ,  , x1 ] − f [ x n −1 , , x1 , x0 ]
x n − x0
n
f n ( x) =  Li ( x) f ( xi )
i =0
n
Li ( x) = 
j =0
j i
pg. 4
x − xj
xi − x j
Dr. N. Yassine
Integration
h
 f (a) + f (b)
2
n −1

h
b−a
=  f ( x 0 ) + 2  f ( xi ) + f ( x n )  h =
2 
n

i =1
h
=  f ( x0 ) + 4 f ( x1 ) + f ( x n )
3
n −1
n−2

h
=  f ( x 0 ) + 4  f ( xi ) + 2  f ( x j ) + f ( x n ) 
3

i =1,3,5,...
j = 2, 4,6,...


3h
b−a
=  f ( x0 ) + 3 f ( x1 ) + 3 f ( x 2 ) + f ( x3 )
h=
8
3
I (h1 ) − I (h2 )
= I (h2 ) +
2
 h1 
1 −  
 h2 
I=
I
I
I
I
I
n
4
1
16
1
64
1
I m − Il , I = I m − Il , I =
I m − I l ,...
3
3
15
15
63
63
x
c
2
0.5773502692 1.000000000000
I=
h=
b−a
n
− 0.5773502692 1.000000000000
3
0.7745966692 0.555555555556
0.0000000000 0.888888888889
− 0.7745966692 0.555555555556
b
 (b − a )t + b + a  b − a 

dt
2
2 


−1
1
 f ( x)dx =  f 
a
pg. 5
Dr. N. Yassine