NUMERICAL ANALYSIS PAPER

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NUMERICAL ANALYSIS ME 406 5th Sem May 2k5

NUMERICAL ANALYSIS ME 406 5th Sem May 2k5

Max marks 60

Note: Section A is compulsory. Attempt any Four questions from section B and two from Section C.

Section A Marks 2 each

1.(a) With a=0, b=1, the following function changes sign in (a, b), what point does the bisection method locate? Is this point a zero of f(x)?

(b) Find the function whose first difference is 9x2+11x+5.

(c) What are the major drawbacks of the Lagrange’s form of interpolation?

(d) Prove that :

(e) Show that the matrix

is Invertible, but that A cannot be written as the product of a lower triangular matrix with an upper triangular matrix.

(f) Calculate the number of additions and number of multiplications necessary to multiply an n x n matrix with an n-vector.

(g) If third differences are constant, prove that:

(h) Prove that:

i) Why higher order Newton-Cole’s formula for numerical integration are not commonly used?(j) Convert the following second order initial value problem into a system of first order initial value problem.

ty’’-y+4t3y=0.˜

Y(1) = 1, y’ (1) = 2.

Section B Marks 5 each

2. Show that bisection method always converges and its order of convergence is one.

3. Solve he equations:

x1+ x2+ x3=6

3x1+ (3+ e)x2+ 4x3=20

2x1+ x2+3 x3=13

using Gauss elimination method, where e is small such that 1+ e2 ˜ 1.

4 (a) Define the operators d and m and prove that d(f(x)g(x)] = mf(x)dg(x) + mg(x)df(x). where d = delta and m is mu

(b) Use Newton’s formula for interpolation to find the number of deaths at 40-50 and 50-55 if the following are the number of deaths on for successive ten year age groups:

Age group Deaths

25-35 13229

35-45 18139

45-55 24225

55-65 314965.

Use stirling formula to find the first derivative of the function y 2ex x – 1 tabulated below at the point x = 0.6

X y

0.4 1.5836494

0.5 1.7974426

0.6 2.0442376

0.7 2.3275054

0.8 2.6510818

Compare with the true value which is 2.044238

6. Derive Simpson’s 1/3 formula for numerical integration and show that its local truncation error is of the order h3.

Section C Marks 10 each

7. (a) Use Picard’s method to approximate y where x 0.1, x = 0.2, given that y = 0 when x = 0, dy/dx =x+y. Compare the results with exact value.

(b) Find the three term Taylor series solution for the third order initial value problemW”’ + WW” = 0.W(0) = 0, W’ = (0), W”(0) =1.Find the bound on the error for t ? [0, 0.2]

8. By considering the limit of the three-point Lagrange interpolating polynomial relative to

9 (a) Factorize the following matrix into LU decomposition using direct factorization

with uii =1 for all i:

(b) The equation ex – 4x2 = 0 has a root between x = 4 and x = 5. Show that we cannot find this root using fixed point interaction with natural iteration function x = ½ ex/2

Find root of this equation correct to three decimal places selecting an appropriate iteration function.