Reference no: EM133189673

BCA-305 Numerical Analysis

Part - A

Q:1 Attempt all :
i. Define relation between operators.
ii. Make forward difference table.

iii. Given (x0 , y0),(x1, y1) (x2, y2 ) Lagrange's interpolation formula is .....
iv. Extrapolation is defined as ......
v. Define Numerical Integration.
vi. The eigen value value of a Δ matrix are .....

vii. Eigen value of the matrix are

viii. The 4^th divide difference for x0 , x1, x2, x3 is....
ix. The ntli divide difference of a polynomial of degree n is :
a. a) zero b) constant c) a variable d) none of these
z. Define Interpolation.
xi. Write down String's formula.
xii. Newton's backward interpolation formula is
xiii. The 4th divide difference for x1, X2, x3, is....
xiv. If the Eigen values of a matrix A are -4 , 3, 1 then the dominant eigen value of A is .
xv. Write the formula for Bessel's equation.

Part - B

Attempt any five:
i. The following data given f(x) , the indicated HP and x, the speed in knots developed by a ship.

X	8	10	12	14	16
Y	1000	1900	3250	5400	8950

Find y when x = 9 , using Newton's forward interpolation formula.

ii. Given the values

x	5	7	11	13	17
F(x)	150	392	1452	2366	5202

Evaluate f(9) ,using Lagrange's formula.

iii. Find I(2.5) using Gauss's backward interpolation formula from the following data :

X	1	2	3	4
F(x)	1	8	27	64

iv. Prove the relations :
i) hD = log (1 + Δ) = - log (1 + ∇) ii) μ = 1/2(E^1/2 - E^-1/2)
v. Find derivatives using Newton's backward difference formula
vi. Obtain by Power Method ,the numerically dominant Eigen values and Eigen vectors of the matrix :

 

vii. Derive Newton's cote's Quardrature formula.

viii. ' Derive Trapezoidal formula and solve ₀∫⁶dx/1+x²

Part - C

Attempt any two :

i. Solve by Newton's forward Interpolation formula find the value off(1.6) ,if

X	1	1.4	1.8	2.2
F(x)	3.49	4.82	5.96	6.5

And also derive the central difference table upto 5th differences.
ii. Derive Newton's divided difference formula. Also find the value of f(8) from the following table :

(x)	4	5	7	10	11	13
F(x)	48	100	294	900	1210	2028

iii. Find by Taylor's series method , (a) The value ofy at x = 0.1 and x = 0.2 to 5 places of decimals from dy/dx = x²y -1, y(0) = 1

(b) given dy/dx = y-x/y+x with initial condition y = 1 at x = 0 : find y for x = 0 by Euler's method.

iv. Derive the formula for Newton's Forward and Backward Interpolation.