BCS 54 Computer Oriented Numerical Techniques

Question 1.
(a) Explain each of the following concepts, along with at least one suitable example for each:
(i) Fixed-point number representation (ii) round-off error (iii) representation of zero as floating point number (iv) significant digits in a decimal number representation (v) normalized representation of a floating point number (vi) overflow
(b) Explain with suitable example that in computer arithmatics ( i.e., numbers represented in computer, with +, −, *, / as implemented in a computer) the multiplication operation( *) may

some computer numbers a, b and c
(c) Find out to how many decimal places the value 22/ 7 is accurate as an approximation of 3.14159265, where the latter is value of ╥, calculated up to 8 places after decimal ?
(d) Calculate a bound for the truncation error in approximating f(x) = sin x by sin (x) = x − x 3 / (fact 3) + x5 / (fact 5), where −1 =< x =< 1 and (fact n) denotes factorial of n
(e) Obtain Approximate the value of (3.7)—1, using first three terms of Taylor’s series expansion.

Question 2.
(a) Solve the system of equations
4 x1+ x2+ 2X3 = 16
2×1 + 5×2 + 3×3 = 19
3×1+ 2×2 − x3 = 12
using Gauss elimination method with partial pivoting.
(b) Perform four iterations (rounded to four decimal places) using
(i) Jacobi Method and
(ii) Gauss-Seidel method ,
for the following system of equations.

Which method gives better approximation to the exact solution?

Question 3.
(a) Determine the smallest roots of the following equation:
f(x) = x2cos (x) + sin (x) =0 to three significant digits using
(i) Regula-falsi method (ii) Newton Raphson method (iii) Bisection method (iv) Secant method

Question 4.
(a) Explain what is the role of interpolation in solving numerical problems?
(b) Express Δ3 f1 as a backward difference.
(c) Express Δ3 f1 as a central difference.
(d) For the following data develop difference table and find forward differences and backward differences

Question 5.
(a) By decinnial census, the population of a town was given below.
Year (x) : 1971 1981 1991 2001 2011
Population (y): 112 132 158 189 226 (in thousands)
(i) Using Stirling’s central difference formula, estimate the population for the year 2006
(ii) Using Newton’s forward formula, estimate the population for the year 1992.
(iii) Using Newton’s backward formula, estimate the population for the year 1980.

b) If values of the function f: x  y are given as f(1) = −32, f(4) = 08, f(5) = 52, f(7) = 167 find the Lagrange’s interpolation polynomial of f(x). Also, find f(3).
Question 6.
(a) Find the values of the first and second derivatives of f(x) at x = 76 from the following table. Use 0(h2) forward difference method. Also, find Truncation Error (TE) and actual errors.

Question 7.
(a) Compute the value of the integral

Rectangular Rule (ii) Trapezoidal Rule and then (iii) Simpson’s 1/3 Rule