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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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