The Institution of Engineers,India 2005 A.M.I.E.T.E Electronics & Communication Engineering NUMERICAL ANALYSIS & COMPUTER PROGRAMMING - Question Paper

Code: A-07 Subject: NUMERICAL ANALYSIS & COMPUTER PROGRAMMING
Time: three Hours Max. Marks: 100
• ques. one is compulsory and carries 16 marks.
• ans any 3 ques. every from Part I and Part II.

Q.1 select the accurate or best option in the following: (2x8)
a. An approximation to is written as . If in , then the fault of approximation is bounded by
(A) . (B) . (C) . (D) .

b. Newton-Raphson method is applied to obtain where M > 0. Then, the method can be written as where R is

(A) . (B) . (C) . (D) .

c. The system of equations a is real, is to be solved iteratively using Jacobi method. The method converges for

(A) . (B) . (C) all a. (D) .

d. describe . The Lagrange fundamental polynomial at x = 0, based on these points, is provided by
(A) 4. (B) . (C) 4h. (D) 6.

e. The polynomial that fits the data

X x 0.1 0.3 0.5 0.7 0.9
f (x) 2.7 3.1 3.5 3.9 4.3

is provided by

(A) two + 2.5 x. (B) 2.5 + two x. (C) x + 2.6. (D) 3.5 – two x.
f. The integral is evaluated by the N+1 point trapezoidal rule. The roundoff fault in every value of is bounded by . Then, the bound on the total roundoff fault is provided by

(A) . (B) . (C) . (D) .

g. What is the output of the subsequent program?
# include “xyz.c”
main( )
{
printf(“%d”, a);
}
--------- File xyz.c --------
int a;
a =25;
(A) fault. (B) 10. (C) 25. (D) infinite loop of printing.

h. The lowest squares polynomial approximation of degree 1 to on [0, 1] is

(A) . (B) . (C) . (D) .


PART I : ans any 3 ques.. every ques. carries 14 marks.

Q.2 a. Using the decomposition method for symmetric matrices, obtain the inverse of the matrix . (7)

b. The system of equations Ax = b, where A is as described in ques. 2(a), is provided. Components of the right hand vector b were measured with an error, whose magnitude is less than or equal to . Derive the fault bounds for the components of the solution vector x. (You may use the inverse, , found in ques. 2(a)). (7)

Q.3 a. An iterative method for finding , N > 0 is written as . obtain the expression for the leading term of the fault. Hence, obtain the order of the method. (7)

b. In every paper in a degree examination, the final award of marks (named Finalavg) for any subject has the weightage of 30% for assignment (named Asavg) and 70% for the final exam (named Exavg). The roll numbers and the marks found by the students in the order of assignment and exam marks, in every subject (there are 5 subjects called paper one to paper 5) are stored (one line for roll number and 1 line per subject) in a file called “Submarks”. The learner passes the exam if the “Finalavg” in every paper is greater or equal to 50. There are 1000 students (named Roll [1] to Roll [1000]). Write a C program to evaluate the students, with the output stored in a file called “Result”. This file should contain (a) Roll number, (b) Total marks in every paper in a separate line, (c) passed or failed against every line. (7)

Q.4 a. The negative root of smallest magnitude of the formula is to be found. (i) obtain an interval of length 1, which contains the root. (ii) Perform 2 iterations of the bisection method. (iii) Taking the end points of the last interval (obtained by the bisection method) as initial approximations, perform 2 iterations of the secant method. (iv) Taking the mid point of the last interval (obtained by the bisection method) as the initial approximation, perform 2 iterations of the Newton-Raphson method. (9)

b. obtain the Choleski decomposition of matrix . (5)

Q.5 a. Solve the system of equations

using Gauss-Seidel method, with the initial approximations taken as . Perform 3 iterations. (5)

b. For the issue in Q.5 (a), write the Gauss-Seidal method in matrix form. Hence, obtain the rate of convergence of the method. (9)

Q.6 a. The system of equations

has a solution near . find the Newton’s method for solving this system. Iterate once using the provided initial approximations. (8)

b. A simple root of the formula f (x) = 0 is to be determined by the bisection method. Write a C program to obtain this root. Input (i) end points of the interval (a, b) in which the root lies, (ii) maximum number of iterations n, (iii) fault tolerance “eps”. Output, (i) number of iterations taken, (ii) root, (iii) f (root), (iv) “Iterations are not sufficient”, if convergence is not attained. (6)


PART II : ans any 3 ques.. every ques. carries 14 marks.
Q.7 a. The subsequent table of values represents a polynomial of degree It is provided that there is an fault in 1 of the tabular values of f (x) near the end of the table. Locate the fault and accurate this value. (7)
x 0 0.1 0.2 0.3 0.4
f(x) 2.00 2.11 2.28 2.39 2.56

b. A table of values, with uniform mesh, is to be constructed for the function on (1, 3). If linear interpolation is to be used, obtain the maximum value of the step length that can be used so that . (7)
Q.8 A mathematical model of a process in an experiment is taken as . A data of N points , i = 1, 2, ……, N is provided. If the parameters a, b and c are to be determined by the method of lowest squares, obtain the normal equations. Use these equations to obtain a, b, c for the data (keep four decimal accuracy).
x 1 2 4 5
f(x) 1.1 1.5 1.56 1.57



Solve the resulting equations by Gauss elimination. obtain the lowest squares fault. (4+10)
Q.9 a. Use the formula to calculate from the provided table of values with step lengths h = 0.4, 0.2 and 0.1.

x 0.2 0.3 0.4 0.6 1.0
f(x) 1.5048 2.0243 2.5768 3.8888 8.5
If the fault is of the form , find the Richardson’s extrapolation estimates for the case when h is decreased by a factor two every time. Apply these formulas to obtain a better estimate of . (8)

b. Determine the abscissas in the Gauss-Hermite integration formula such that the formula is of as high order as possible. It is provided that . (6)
Q.10 a. Evaluate the integral , using Simpson’s rule with step lengths h = 0.25 and h = 0.125. obtain a better approximation to the value of the integral using Romberg integration. Compare with the exact solution (Keep five decimal accuracy). (7)
b. The integral is to be evaluated by the Gauss-Legendre two point or the three point rules or . Write a C program to evaluate the integral when User should select the two point or three point formula (Use switch). describe the weights W (k), abscissas x (k), f (x (k)) and I as double precision variables. (7)

Q.11 a. Consider the subsequent Runge-Kutta method for the solution of the initial value issue .


calculate y (0.4) when and h=0.2 (Keep five decimal places). (6)

b. The initial value issue is provided. Taylor series method of order four is to be used for computing y in the interval [a, b] with step length h. Write a C program for finding the solutions. describe double precision functions fD1, fD2, fD3, fD4 for finding . presume that . describe the arguments x,y also as double precision variables. Input a, b, h (in double precision) and calculate the number of steps as . Output y (x) at all step points. (You may use arrays for x and y). presume that maximum value of M is 500. (8)

Earning:   Approval pending.