FACULTY OF SCIENCE AND TECHNOLOGY SEMESTER MAY / 2012 SBMA4603 NUMERICAL METHODS Name : Matric num. : PPK :
FACULTY OF SCIENCE AND TECHNOLOGY
SEMESTER MAY / 2012
SBMA4603
NUMERICAL METHODS
Name :
Matric num. :
PPK :
QUESTION 1
f(x) has a root of order 9 at x=p.
From Lemma 2.1 (page54)
f(x) can be expressed as f(x) = (x-p)m. h(x) ; h(x)≠0
When m=9;
So that, f(x) = (x-p)9. h(x)
Compute f’(p) :
From f(x)= (x-p)9. H(x)
We use a product rule of derivatives
f’(x) = u’. v + v’. u
= 9(x-p)8(1). h(x)+ h’(x).(x-p)9
= 9(x-p)8. h(x) + h’(x).(x-p)9
f’(p) = 9(p-p)8. h(x) + h’(x).(p-p)9
= 9(0)8. h(x) + h’(x).(0)9
= 0.
Compute f”(p) :
From f’(x) = 9(x-p)8. h(x) + h’(x).(x-p)9
f”(x) = 9(8)(x-p)7(1). h(x) + h’(x).9(x-p)8 + h”(x).(x-p)9+9(x-p)8(1).h(x)
= 9(8)(x-p)7. h(x) + 2[9(x-p)8 . h’(x)] + h”(x).(x-p)9
f”(p) = 9(8)(p-p)7. h(x) + 2[9(p-p)8 . h’(x)] + h”(x).(p-p)9
= 9(8)(0)7. h(x) + 2[9(0)8 . h’(x)] + h”(x).(0)9
= 0.
Compute f (9) (p) :
From f”(x) = 9(8)(x-p)7. h(x) + 2[9(x-p)8 . h’(x)] + h”(x).(x-p)9
f(9)(x) = 9(8)(7)(6)(5)(4)(3)(2)(1) (x-p)0. h(x) + 9[9(8)(7)(6)(5)(4)(3)(2) (x-p)1. h8(x)] +
h9(x).(x-p)9
f(9)(p) = 362880h(x) + 3265920(x-p)1. h8(x) + h9(x).(x-p)9
= 362880h(x) + 3265920(p-p)1. h8(x) + h9(x).(p-p)9
= 362880h(x) + 3265920(0)1. h8(x) + h9(x).(0)9
= 362880h(x) + 0 + 0
= 362880h(x)
QUESTION 1 (a)
We know that g(x) = ax3 + bx2 + cx + d.
The order of roots for g(x) = x3 + 3x2 - 4;
We found that a=1, b=3, c=0, d= - 4.
From calculation by calculator scientific, the roots for equation g(x) = x3 + 3x2 – 4 are:
x1=1 , x2= -2
when g(x)=0; g(x) = x3 + 3x2 – 4 = 0
g’(x) = 3x2 + 6x = 0
g”(x) = 6x + 6 = 0
when root x=1; g(1) = (1)3 + 3(1)2 – 4 = 0
g’(1) = 3(1)2 + 6(1) = 9 ; g’(1) ≠0
when g’(1) ≠0 ;
thus, the order = 1
The root x=1 , the order = 1 (x-1)1 = (x-1)
When root x=-2; g(-2) = (-2)3 + 3(-2)2 – 4 = 0
g’(-2) = 3(-2)2 + 6(-2) = 0
g”(-2) = 6(-2) + 6 = -6 ; g”(-2) ≠0
when g”(-2) ≠0 ;
thus, the order = 2
QUESTION 1 (b)
The order of the roots for h(x) = x3 - 9x2 + 27x - 27
We found that a=1, b= -9, c=27, d= -27
From calculation by calculator scientific, the roots for equation h(x) = x3 - 9x2 + 27x – 27 is:
x=3.
When h(x)=0; h(x) = x3 - 9x2 + 27x – 27 = 0
h’(x) = 3x2 - 18x + 27 = 0
h”(x) = 6x – 18 = 0
when root x=3; h(3) = (3)3 – 9(3)2 + 27(3)– 27 = 0
h’(3) = 3(3)2 – 18(3) + 27 = 0
h”(3) = 6(3) – 18 = 0
thus, the order = 3
The root x=3, the order = 3 (x-3)3
Thus, x3 - 9x2 + 27x – 27 = (x-3)3
QUESTION 2 (a)
The given approximations; and
≈ 0.01101 × 2-1
+ ≈ 0.01010 × 2-1
≈ 0.10111 × 2-1
≈ 0.01010 × 2-1
- ≈ 0.001101 × 2-1
≈ 0.000101 × 2-1
≈ 0.10111 × 2-1
+ ≈ 0.000101 × 2-1
≈ 0.110011 × 2-1
≈ 0.110011 × 2-1
- ≈ 0.001101 × 2-1
≈ 0.001110 × 2-1
Thus, the approximation for fraction 1/3 is 0.001110 × 2-1
QUESTION 2 (b)
First calculate :
And we are already found = 0.10111 × 2-1
≈ 0.10111 × 2-1
- ≈ 0.001101 × 2-1
≈ 0.100001 × 2-1
Then, calculate
≈ 0.100001 × 2-1
+ ≈ 0.100001 × 2-1
≈ 1.000010 × 2-1
Thus, the approximation for fraction 8/15 is 1.000010 × 2-1
QUESTION 2 (c)
≈ 0.01101 × 2-2 ≈ 0.001101× 2-1
+ ≈ 0.01101 × 2-1 ≈ 0.01101 × 2-1
≈ 0.100111 × 2-1
≈ 0.100111 × 2-1
- ≈ 1.010× 2-3 ≈ 0.01010 × 2-1
≈ 0.010011 × 2-1
Thus, the approximation for fraction 1 /10 + 1/5 - 1/6 is 0.010011 × 2-1
QUESTION 3 (a)
An infinite geometric series can be written as:
cr0 + cr1 + cr2 + …
cr0 = c =
Hence, we found c = . So that, we can find the value for r1.
So, by replacement c = into cr1 :
cr1 =
(1/4) r1 =
r1 = ÷
= × 4
QUESTION 3 (b)
(www.mathsisfun.com/binary-decimal-hexadecimal-converter.html)
The decimal fraction 8/7 is = 1.14285714285714
By converting the decimal fraction to the binary;
1.14285714285714 = 1.001 001 001 001 001 001 001 001 001 001 01
The binary representation of the decimal fraction 8/7 is
= 1.001 001 001 001 001 001 001 001 001 001 01
QUESTION 4 (a)
Thus, linear equation in matrix form is
QUESTION 4 (b)
Express the coefficient matrix of above system as a product of an upper and lower triangular
matrices.
Let the diagonal ;
References
1. Assoc Prof Dr Ishak Hashim(2011). SBMA4603 Numerical Methods. Open University
Malaysia. Meteor Doc. Sdn. Bhd. Selangor Darul Ehsan.
2. www.mathsisfun.com/binary-decimal-hexadecimal-converter.html
3. www.easycalculation.com
4. www.easysurf/fracton.htm