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BCS-054 : COMPUTER ORIENTED NUMERICAL TECHNIQUES

(1)Attempt to solve the following system of linear equations using the Gauss elimination method :

3x1 + 2x2 + x3 = 3

2x1 +x2 +x3 = 0

6x1 + 2x2 + 4x3 = 6

Does the solution exist ? If yes, how many

(2)Give the formula for next approximation of values of x1, x2 and x3 using Gauss-Jacobi

iterative method for solving the following system of linear equations :

a11x1 a12x2 a13x3 = bl

a21x1 a22x2 a23x3 = b2

a31x1 + a32x2 + a33x3 = b3

(3) Solve the following NP using Euler's method :

y` = fft, y) =1 + y; given y(0) = 1            

Find the solution on [0, 0.8] with h = 0-2.

(4) Using Maclaurin's series expansion, find the value of (1 - x) -1, at x = 0, by taking

the first three terms and find truncation error.

(5) Discuss the relative merits and demerits of direct methods over iterative methods for

solving a given system of linear equations

(6) Using 8-decimal digit floating-point representation (with four digits for

mantissa, two for exponent and one each for sign of exponent and mantissa),

represent the following numbers in normalized floating point form (use

chopping, if required) :

(i) 89543

(ii) — 89.766

(iii) 0.0007345

(7)Write the following system of equations in matrix form

— 9x — 8y = — 4

3x + 4y = — 17

(8) Show one iteration of solving the following system of linear equations using any

iterative method. You may assume

x = y = 0 as the initial estimate :

— 6x + 8y = — 2

4x + 7y = — 11

(9) Find an interval in which the following

equation has a root : 2

x2 + 9x + 20 = 0

(10)State the following two formulae for interpolation :

(i) Newton's backward difference formula

(ii) Bessel's formula

(11)From the Newton's backward difference formula asked in part k(i)

above, derive the formula for finding the derivative of a function at x = xo.

(12)Explain each of the following concepts with a suitable example :

(i) Order of a differential equation

(ii) Initial Value Problem

(iii) Degree of a differential equation

(iv) Non-linear . differential equation

(13)For each of the three numbers of find relative error in its normalized

 floating point representation ?

(14) Using an 8-decimal digit floating point representation (4 digits for mantissa, 2 for

exponent and one each for sign of exponent and sign for mantissa), represent the

following numbers in normalized floating point form (using chopping, if required) :

(i) 87426

(ii) - 94.27

- 0.000346

(15) What is underflow ? Explain it with an example of multiplication in which

underflow occurs.

(16) Write the following system of linear equations in matrix form :

6x + 8y = 10

- 5x + 3y = 11

(17) State the following two formulae for (equal interval) interpolation :

(i) Newton's Backward Difference Formula

(ii) Newton's Forward Difference Formula

(18) Let min. and max. represent respectivelyminimum and maximum

 positive real numbers representable by some floating

point number system. Can every real number between max. and min. be

representable by such a number system ? Explain the reason for your answer

(19) For each of the following numbers, find the floating point representation, if

possible normalized, using chopping, if required.

(i) 3/11

(ii) 74.0365

Further, find the absolute error, if any, in

each case.

(20) Solve the following system of equations, using partial pivoting Gaussian

elimination method (compute upto two places of decimal only) :

4x1 - 5x2 + 6x3 = 24

3x1 - 7x2 + 2x3 = 17

5x1 + 2x2 - 4x3 = - 21

(21) For solving the following system of linear equations

a11 x1 + a12 X2 4- a13 = b1 ,

a21 + a22 X2 + a23 x3 = b2 and

a31 x1 + a32 x2 + a33 x3 = b3

with an * 0 * a22 and a33 * 0, by iterative

Gauss-Jacobi Method, with initial approximations as x 1 = 1 = x2 = x3,

find the values of next approximations of xl , x2 and x3.

(22) For the table given above, find Newton's forward differences interpolating

polynomial and find the value f(1.7) using the polynomial.

(23) If, f(x) represents the distance covered by a

particle in x units of time, estimate the velocity and acceleration of the particle at

x = 1-5.

(24) Solve the following 117P using Euler's method :

y' = fix, y) = x + y, given y(0) = 1.

(25) Using 8-decimal digit floating point representation (4 digits for mantissa, 2 for

exponent and one each for sign of exponent and mantissa), represent the following

numbers in normalised floating point form :

(i) 89.36

(ii) — 0.00004375

(iii) 87604

(use chopping, if required)

(26) Find the sum of two floating numbers

x1-- .5307 x 104 and x2 = .4252 x 103

(27) Write the following system of linear equations in matrix form :

5x — 9y =14

2x + 5y =11

(28) State the following two formulae for interpolation :

(i) Newton's Forward difference formula

(ii) Stirling's formula

(29) Define each of the concepts with suitable  example.

(i) Degree and order of a differential equation

(ii) Initial Value Problem

(30) Explain the advantages of normalized  floating point number over un-normalized

numbers.

(31) For solving a system of linear equations :

an x1+ a12 x2 + a13 x3 = bi;

a21 x1 + a22 x2 + a23 x3 = b2 and

a31 x1 + a32 x2 + a33 x3 = b3, by iterative

Gauss-Jacobi Method, with initial

approximations, x1 = 0 = x2 = x3, give formulas for next approximations of x1, x2

and x3.

(32) What are the advantages of iterative  methods over direct methods for solving a

system of linear equations.

(33) We are given the Initial Value Problem (IVP)  y'= 1 — 2xy, y(0.2) = 0.1948 with h= 0.2,

using Euler's Method, find y(0.4). The independent variable is x.

(34) Find the product of the two numbers a and b given above

(35) Write the following system of linear equations in matrix form

8x + lly = 19

12x + 5y = 17

(36) Write briefly the steps of bisection method to find out the roots of an equation ?

(37) What are the relative advantages of iterative methods over direct methods

 for solving a system of linear equations ?

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