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Air University Subject: Numerical Analysis and Computation Assignment No. 1

Submission Date: 9th October, 2015 Class: BEMTS V (C), BSCS V (A, B)

Question no 1 : Define the following

1. Norm of a matrix (2, )

2. Band matrices

3. Diagonal matrix

4. Scalar matrix

5. Tri diagonal matrix

6. Positive definite matrix

7. Symmetric matrices

8. Diagonally dominant system

9. Positive definite matrix

10. Ill condition system

Question no 2 : Find the fourth Taylor polynomial () for the function () = about = .

a) Find an upper bound for|() 4()|, for 0 0.4.

b) Approximate () 0.4

0 using 4()

0.4

0.

c) Approximate (0.2) using4 (0.2), and nd the error.

Question no 3: Compute the absolute error and relative error in approximations of by , with

relative error at most for each value of

a) = , = 2.718

b) = 2, = 1.414 c) = 10, = 22000

d) = 10, = 1400 e) = 8!, = 39900

f) = 9!, = 18(9/)9

Question no 4: Perform the following computations (i) exactly, (ii) using three-digit chopping

arithmetic, and (iii) using three-digit rounding arithmetic. (iv) Compute the relative errors using

tolerance .

a) 4 5 + 1

3

b) 4 5 . 1

3

c) ( 1 3 3

11 ) + 3

20

d) ( 1 3 + 3 11) 3 20

Question no 5: Let

() =

a) Find lim0

() .

b) Use four-digit rounding arithmetic to evaluate(0.1).

c) Replace each trigonometric function with its third Maclaurin polynomial, and repeat part (b).

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d) The actual value is (0.1) = 1.99899998. Find the relative error for the values obtained in

parts (b) and (c).

Question no 6: Find and norms of the vectors.

a) = (3, 4,0, 3 2)

b) = (2,1, 3,4)

c) = (, , 2 ) for a xed

positive integer k

d) = (4

+1,

2

2, 2 )

for a xed

positive integer k

Question no 7: For each of the following linear systems, obtain a solution by graphical methods, if

possible. Explain the results from a geometrical standpoint

a) 1 + 22 = 3, 1 2 = 0.

b) 1 + 22 = 3, 21 + 42 = 6.

c) 1 + 22 = 0, 21 + 42 = 0.

d) 21 + 2 = 1, 41 + 22 = 2, 1 32 = 5.

Question no 8: Use Gaussian elimination with backward substitution and two-digit rounding

arithmetic to solve the following linear systems. Do not reorder the equations. (The exact solution to

each system is = , = , = .)

a) 41 2 + 3 = 8, 21 + 52 + 23 = 3, 1 + 22 + 43 = 11.

b) 41 + 2 + 23 = 9, 21 + 42 3 = 5, 1 + 2 33 = 9.

Question no 9:Use Gaussian elimination and three-digit chopping arithmetic to solve the following

linear systems, and compare the approximations to the actual solution.

a) 0.031 + 58.92 = 59.2, [10,1].

5.311 6.102 = 47.0. b) 3.031 12.12 + 143 = 119 [0,10, 1/7].

3.031 + 12.12 73 = 120 6.111 14.22 + 213 = 139

c) 191 + 2.112 1003 + 4 = 1.12, [0.176,0.0126, 0.0206, 1.18]

14.21 0.1222 + 12.23 4 = 3.44

1002 99.93 + 4 = 2.15

15.31 + 0.1102 13.13 4 = 4.16

d) 1 2 + 23 34 = 11,

21 + 2 23 +

3

74= 0,

51 62 + 3 24 =

31 + 22 73 + 1/94 = 2 [0.788, 3.12,0.167,4.55].

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Question no 10: Given the linear system

21 62 = 3, 31 2 = 3/2.

a) Find value(s) of for which the system has no solutions.

b) Find value(s) of for which the system has an innite number of solutions.

c) Assuming a unique solution exists for a given, nd the solution.

Question no 11: Given the linear system

1 2 + 3 = 2, 1 + 22 3 = 3, 1 + 2 + 3 = 2.

a) Find value(s) of for which the system has no solutions.

b) Find value(s) of for which the system has an innite number of solutions.

c) Assuming a unique solution exists for a given , nd the solution.

Question no 12: Determine which of the following matrices are (i) symmetric, (ii) singular, (iii) strictly

diagonally dominant, (iv) positive denite.

a) [2 11 3

]

b) [2 1 00 3 21 2 4

]

c) [2 1 0

1 4 20 2 2

]

d) [

22

34

11

25

3 7 1.5 16 9 3 7

]