Code: A-07 Subject: NUMERICAL ANALYSIS & COMPUTER PROGRAMMING

Time: 3 Hours Max. Marks: 100

JUNE 2003

NOTE: There are 11 Questions in all.

· Question 1 is compulsory and carries 16 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

· Answer any THREE Questions each from Part I and Part II. Each of these questions carries 14 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1 Choose the correct or best alternative in the following: (2x8)

a. Using one-point Gauss-Chebyshev formula an approximate value of the integral is

(A) 0. (B) 2.40.

(C) 3.14. (D) 1.57.

b. The postfix notation for the following infix notation is

(A) . (B) .

(C) . (D) .

c. The n-point Gauss Quadrature formula is exact for all polynomials of degree upto

(A) n. (B) 2n.

(C) 2 n – 1. (D) 2n +1.

d. If int s[5] is a one dimensional array of integers, which of the following refers to the third element in the array?

(A) . (B) .

(C) s + 3. (D) s + 2.

e. The interpolating polynomial that fits the data

x		0	2	4	5
f(x)	1		1	11	19

(A) . (B) .

(C) . (D) .

f. If is an eigenvalue of A, then the eigenvalue of is

(A) . (B) .

(C) . (D) .

g. Given that k = 4 and m = 1, the value of k after execution of the statement k + = k > m is

(A) equal to 4. (B) equal to 8.

(C) equal to 9. (D) cannot be found because the

statement is invalid.

h. The rate of convergence of the Newton-Raphson method for finding a simple root of the equation f (x) = 0 is

(A) 1 (B) 3.

(C) 5. (D) 2.

PART I

Answer any THREE Questions. Each question carries 14 marks.

Q.2 Show that the following two sequences have convergence of second order with the same limit

(i) (ii)

If is a close approximation to , show that the error in the first formula is about one-third of that in the second formula and deduce that the formula gives a sequence with third-order convergence. (14)

Q.3 a. Apply Cholesky’s method to the matrix to obtain upper triangular matrix U such that A = UU^T. (7)

b. Write a program in C to solve a system of linear equations using Jacobi’s iteration method. (7)

Q.4 a. Derive Euler’s method for solving the initial value problem . Use this method to obtain an approximate value of y(0.4) for the equation with h = 0.1. (5)

b. Find the inverse of the matrix

using Gauss-Jordan method. (5)

c. Find the linear least squares polynomial approximation for the data (4)

x_i	-2	-1	0	1
f_i	6	3	2	2

Q.5 Describe power method for finding the largest eigen value in magnitude of a square matrix. Perform four iterations of this method to find the largest eigen value in magnitude and the corresponding eigen vector of the matrix

Take the initial approximation to the eigenvector as . (14)

Q.6 a. Find a bound on the truncation error in quadratic interpolation based on equispaced points with spacing h. Determine the largest step size h that can be used in tabulation of in so that error in quadratic interpolation is less than . (8)

b. Write a program in C that reads a variable x and computes the sum of series containing 25 terms. (6)

PART II

Answer any THREE Questions. Each question carries 14 marks.

Q.7 a. Write a program in C using a recursive function for calculating values of Chebyshev polynomial by

(7)

b. For the given data, use Newton divided difference interpolation to find f (2) and f (8) : (7)

x	4	5	7	10
f(x)	48	100	294	900

Q.8 a. Find a quadratic factor of the polynomial

starting with and using one iteration of the Bairstow’s method. (7)

b. Use the classical Runge-Kutta fourth order method to obtain an approximate value of y(1.2) for the initial value problem with . (7)

Q.9 a. Use the two point Gauss-Legendre quadrature formula to evaluate . (7)

b. Write a program in C that reads a text file and then computes the total number of words and sentences in it. (7)

Q.10 a. Calculate the n^th divided difference of based on the points . (7)

b. Write the differences between a structure and union in C. Give an example of each. (7)

Q.11 a. Determine p and q such that the order of the iteration method for calculating becomes as high as possible. Find the order and the error constant of this method. (8)

b. Gauss – Seidel iteration method is used to solve the system of equations

Find the iteration matrix. Show that the method converges. (6)

Code: A-07 Subject: NUMERICAL ANALYSIS & COMPUTER PROGRAMMING

Time: 3 Hours December 2003 Max. Marks: 100

NOTE: There are 11 Questions in all.

· Question 1 is compulsory and carries 16 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

· Answer any THREE Questions each from Part I and Part II. Each of these questions carries 14 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1 Choose the correct or best alternative in the following: (2x8)

a. . Then is

(A) . (B) .

(C) . (D) .

b. The value of where are Lagrange fundamental polynomials is

(A) 0. (B) 1.

(C) x. (D) .

c. The rate of convergence of the Regula – Falsi method for finding a simple root of the equation is

(A) 1. (B) 1.5.

(C) 2.0. (D) 2.25.

d. If is

(A) . (B) .

(C) . (D) .

e. The result of mode bit noreplace is

(A) error when opening if file does not already exist.

(B) error when opening for output if file already exists unless ate or app is already set.

(C) error when opening for input if file already exists, unless ate or app is set.

(D) error when closing if file does not already exist.

f. The meaning of bit is

(A) No bit set, no errors. (B) at EOF.

(C) Hard error. (D) Read or write failed.

g. Consider 3-point Gauss-Legendre formula: Then for i=0,1,2 are given by

(A) . (B) .

(C) . (D) .

h. The order of the numerical differentiation method

where , for all i, is

(A) 1. (B) 2.

(C) 3. (D) 4.

PART I

Answer any THREE Questions. Each question carries 14 marks.

Q.2 a. Using Cholesky factorisations method, solve the system of equations

. (7)

b. For the Newton’s method , to find a simple root of , find C and s such that . (7)

Q.3 a. What conditions must be placed on a, b and c so that the following system of equations has a solution?

. (7)

b. The system of equations

has a solution near Set up the Newton’s iteration scheme for finding the solution and iterate once. (7)

Q.4 a. A program in C contains the following statements

float a = 0.001;

float b = 0.003;

float c, *pa, *pb;

pa = &a;

*pa = 2 * a;

pb = & b;

c = 3* (*pb - *pa);

Suppose each floating-point number occupies 4 bytes of memory. If the value assigned to a begins at (hexadecimal) address 1130 the value assigned to b begins at address 1134 & c at 1138, then

(i) what value is assigned to &a, &b and &c?

(ii) what value is represented by *pa, & (*pa)? (7)

b. Find the interpolating polynomial for the following data :

x	1		0	2
f(x)	3		3

(7)

Q.5 a. Describe the output generated by the following program:

* include <stdio.h>

int a = 100, b = 200;

main()

{

int count, c, d; int funct1(int a, int b);

for (count = 1; count <=5; ++count){

c = 20 *(count – 1);

d = 4 * count * count;

printf(“%d %d”, funct1(a, c), funct1(b, d));

}}

funct1 (int x, int y)

{ return(x – y);

} (7)

b. For the generalized Simpson’s rule of the form

where a, b, p are constants and find the constants a, b, p and the error term. (7)

Q.6 a. Determine the least squares approximation of the type to the function at the points . (7)

b. Write a program with a macro to find the roots of a quadratic equation = 0 where a, b, c are any real numbers. (7)

PART II

Answer any THREE Questions. Each question carries 14 marks.

Q.7 a. Determine the inverse of the matrix using partition method. (7)

b. Find the largest eigenvalue in magnitude and the corresponding eignevector of the matrix using power method. Start the iteration using the unit vector as an approximation to the eigenvector and obtain two successive iterations. (7)

Q.8 a. Find all the eigenvalues and eigenvectors of the matrix using Jacobi method. Perform only two iterations. (7)

b. Write a program in C which sorts a list of strings into alphabetical order using an array of pointers. (7)

Q.9 a. Obtain a bound on the truncation error in linear interpolation based on the nodal points . Determine the maximum step-size that can be used in the tabulation of in [0, 1] so that the error in linear interpolation will be less than . (6)

b. Determine the constants a, b, c in the following integration rules :

Which of these two rules will give more accurate results? Justify. (8)

Q.10 a. Write a loop that will generate every third integer, beginning with i = 2 and continuing for all integers that are less than 100. Calculate the sum of those integers that are evenly divisible by 5. Use the following two different methods:

(i) use the conditional operator (? :).

(ii) use an if-else statement. (7)

b. The Gauss-Seidal iterations method is used to solve the linear system of equations . Find the iteration matrix and show that the iterations converge. (7)

Q.11 a. Find an approximate value of for the initial value problem using the fourth order classical Runge Kutta method with h = 0.1. (9)

b. State the theorem concerning the existence and uniqueness of the solution of

the initial value problem. Prove that the initial value problem

has a solution on . (5)

Code: A-07 Subject: NUMERICAL ANALYSIS & COMPUTER PROGRAMMING

Time: 3 Hours JUNE 2004 Max. Marks: 100

NOTE: There are 11 Questions in all.

· Question 1 is compulsory and carries 16 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

· Answer any THREE Questions each from Part I and Part II. Each of these questions carries 14 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1 Choose the correct or best alternative in the following: (2x8)

a. The integration rule is given. The error is of the form . Then, the value of c is given by

(A) . (B) .

(C) . (D) .

b. The iteration method , q is a positive constant, is being used to evaluate a certain quantity. If the iteration converges, then the method is finding the value of

(A) . (B) .

(C) (D) .

c. An integral I is being evaluated by the composite trapezoidal rule. The values of I for two different step lengths and are obtained as I=0.6123, I =0.6011. A better approximation using Romberg integration is

(A) 0.5974. (B) 0.6004.

(C) 0.1893. (D) 0.5867.

d. Using the data f (0.1) = 0.9950 and f (0.2) = 0.9801, an approximate value of f (0.15) is determined by Lagrange interpolation. If over , then the bound on the error in determining f (0.15) is given by where the value of C is

(A) 0.0025. (B) 0.00125.

(C) 0.0004. (D) 0.00375.

e. The least squares straight line approximation to the data

X x	1	2	3	4
f (x)	6	9	14	21

is given by

(A) 5 x +3. (B) 6 x +1.

(C) 5x. (D) 3 x +1.

f. Gauss-Jacobi iteration method is used to solve the system of equations

with arbitrary initial approximation. Then, the system

(A) diverges. (B) has rate of convergence 0.6253,

(C) has rate of convergence 0.2342. (D) has rate of convergence 0.4532.

g. What will be the output of the following program.

main()

{

static int a [4] = {1, 2, 3, 4};

int *p, i;

p = a;

printf(“%d”, *p);

p = p*a;

printf (“%d”, *p);

}

(A) 1 2 3 4 (B) 4 3 2 1

(C) 4 2 3 1 (D) Error.

h. What will be the output of the following program.

main()

{

struct name{

char aname [10];

char *bname;

} p1 = { “ Numerical”, “Analysis”};

printf (“%c..% c\n”, p1.aname [0], *p1.bname);

printf (“%s..%s\n”, p1.aname, p1.bname);

}

(A) N..n (B) N..A

Numerical..Analysis Numerical..Analysis

(C) N..s (D) N..A

Numerical.. .. Analysis

PART I

Answer any THREE Questions. Each question carries 14 marks.

Q.2 a. Locate the negative root of smallest magnitude, in an interval of length 1, for the equation Take the middle point of this interval as the starting approximation and find the root correct to three decimals using the Newton-Raphson method. (7)

b. Write a C program to find a simple root of f (x)=0, using the Newton-Raphson method. Read the initial approximation as xinitial, maximum number of iterations as n and error tolerance as epsilon. Define f(x) and df/dx using functions. Assume f(x) is defined as in part (a). (7)

Q.3 a. Using BairStow’s method, extract a quadratic factor of the form for the equation Assume the initial approximation as p = 0.4, q = 1.1 and perform one iteration. (7)

b. Find the inverse of the matrix using the Choleski factorization method, in which the factorization is taken as , where U is an upper triangular matrix. (7)

Q.4 a. Solve the following system of equations using the Gauss-seidel iteration method

Take the initial approximation as and . Perform two iterations. Find the iteration matrix and hence find the rate of convergence of the method. (8)

b. Find the largest eigen value in magnitude and the corresponding eigen- vector of the matix using the power method. Assume the initial approximation to the eigen vector as . Perform five iterations. (6)

Q.5 a. Show that the order of the secant method for finding a simple root of the equation f (x) = 0 is 1.62. (7)

b. The following formulas for finding the yearly depreciation for some

depreciable items like a car or machine are used.

(i) Straight line method: Depreciation each year = x/n.

(ii) Sum of the year’s digits method: Depreciation =

Where x = original value of the item, i= ith year and n = number of years.

Write a C program using switches to choose one of the depreciation methods. Output the year, depreciation and current value. (7)

Q.6 a. Find the smallest eigen value in magnitude, correct to two decimals, of the matrix using the inverse power method. Assume the initial eigen-vector as . (7)

b. Reduce the matrix to tridiagonal form by the Givens method. Form the sturm sequence and locate the largest eigen value in magnitude in an interval of length one unit. (7)

PART II

Answer any THREE Questions. Each question carries 14 marks.

Q.7 a. Determine the step size that can be used in the tabulation of in the interval [0, 1], at equally spaced nodal points using quadratic interpolation such that . (7)

b. Construct an interpolation polynomial that fits the data

x	-1	0	1	2	3	4
f(x)	1.2	1.0	3.8	9.6	18.4	30.2

Interpolate the value at 0.5. (7)

Q.8 a. Evaluate using the Simpson’s rule with three and five points. Improve the value of the integral using the Romberg integration. (7)

b. Write a C program to evaluate the integral using trapezoidal rule. Input a, b and the number of subintervals N. Define the integrand as a function. Assume . Output a, b, n and the value of the integral. (7)

Q.9 a. Construct the least squares approximation of the form to the data. (7)

x	0.1	0.2	0.4	0.5	1	2
y	21	11	6	5	3	5

b. Construct the Hermite interpolation polynomial that fits the following data.

X	f(x)	(x)
1	9	23
2	87	163

Interpolate f(x) at x = 1.5. (7)

Q.10 a. Find the uniform (Chebyshev) linear approximation P (x) = a + bx to the function on [0, 1]. (7)

b. The Fibonacci numbers are generated using the recurrence formula.

Write a C program that generates the first n Fibonacci numbers, where n is specified by the user. Write a function called Fibonacci to generate the numbers. Use this function in the main program. The Fibonacci numbers are to be represented as long integers. (7)

Q.11 a. The initial value problem is given. Use the second order Taylor series method to find an approximation to y (1.2) with h = 0.1. (7)

b. Find an approximate value of y (0.4) for the initial value problem

using the Runge-Kutta method of order four with h=0.2. (7)

Code: A-07 Subject: NUMERICAL ANALYSIS & COMPUTER PROGRAMMING

Time: 3 Hours DECEMBER 2004 Max. Marks: 100

NOTE: There are 11 Questions in all.

· Question 1 is compulsory and carries 16 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied.

· Answer any THREE Questions each from Part I and Part II.

Q.1 Choose the correct or best alternative in the following: (2x8)

a. The number 0.015625 is rounded as 0.0156. Then, the relative error in this approximation is

(A) -0.0016. (B) 0.0016.

(C) 0.016. (D) 0.000025.

b. A real root of f(x) = 0 lies in the interval [0, 1]. Bisection method is applied to find this root. If the permissible error in the approximation is , then the number of iterations required is greater than or equal to

(A) . (B) .

(C) . (D) .

c. Gauss-Seidel method is applied to solve the system of equations p real constant. The method converges for

(A) . (B) all p.

(C) . (D) .

d. The divided difference for the function is given by

(A) 6. (B) .

(C) . (D) .

e. The least squares approximation to the data

X x	1	2	3	4
f (x)	6	9	14	21

is given as f(x) = 5x. Then, the least squares error is given as

(A) 0.04. (B) 4.

(C) 6. (D) 0.004.

f. The error in the numerical differentiation formula is given by , where the value of M is

(A) . (B) .

(C) . (D) .

g. The value of the integral

using Simpson’s rule is

(A) (B)

(C) (D)

h. The following C program is given

# include <stdio.h>

main( )

{

switch (choice = toupper(getchar( )))

{

case ‘B’:

printf(“BLUE”);

break;

case ‘P’:

printf(“PINK”);

break;

case ‘G’:

printf(“GREEN”);

break;

default:

printf(“ERROR”);

}

If the character ‘g’ is entered, the output is

(A) ERROR (B) GREEN

(C) PINK (D) green

PART I

Answer any THREE Questions. Each question carries 14 marks.

Q.2 a. Solve the system of equations using the Gauss elimination method. (8)

b. Using the Choleski method, find the solution of the following system of equations . (6)

Q.3 a. The error in the Newton-Raphson method for finding a simple root of f(x)=0, can be written as . Determine the values of c and p. What is the order of the method? (6)

b. The equation f(x)=0 has a simple root in the interval (1, 2). The function f(x) is such that and for all x in (1, 2). Assuming that the Newton-Raphson’s method converges for all initial approximations in (1, 2), find the number of iterations required to obtain the root correct to . (8)

Q.4 a. Locate the negative root of smallest magnitude of the equation in an interval of length 1. Taking the end points of this interval as the initial approximations to the root, perform five iterations using secant method (use five decimal places). (7)

b. Write a C program to find a simple root of f(x) = 0 by the secant method. Input (i) two initial approximations to the root as a and b, (ii) maximum number of iterations m, that the user wants to be done. (iii) error tolerance epsilon. Evaluate f(x) as a function. Output (i) number of iterations taken to obtain the root, (ii) the value of the root, (iii) value of f(root). If the iterations m, are not sufficient, output that “Number of iterations given are not sufficient”. (7)

Q.5 a. Solve the system of equations using the Gauss-Jacobi method, with the starting approximations taken as , , , . Perform three iterations. (5)

b. For the system in 5 (a) above, write the Gauss-Jacobi method in matrix

form. Hence, find the rate of convergence of the method. (9)

Q.6 a. The system of equations xy + yz + zx = 0.59, has a solution near (x, y, z) = (1, 2, -0.5). Derive the Newton’s method for solving this system. Iterate once using the given initial approximations. (7)

b. Write a C program to rearrange a given set of integer numbers into ascending order. Use the following:

(ii) Define initially an array f as an 100 element array.

(iii) Read n, the number of given integer numbers followed by the numerical values.

(iv) Write a function prototype called “reordering”, whose arguments are n and f.

(v) The program for reordering in ascending order is to be given in “reordering”. (7)

PART II

Answer any THREE Questions. Each question carries 14 marks.

Q.7 a. Construct an interpolating polynomial that fits the data

x	0	1	2	5	7	10
f(x)	-2.5	-0.5	10.5	187.5	515.5	1502.5

Hence, or otherwise interpolate the value of f (8). (7)

b. A table of values for f (x)= in [0, 1] is to be constructed with step size h = 0.1. Find the maximum total error if quadratic interpolation is to be used to interpolate in this interval. (7)

Q.8 a. A mathematical model of a periodic process in an experiment is taken as f (t) = a + b cos (t) and a data of N points , i = 1, 2, ……, N is given. If the parameters a and b are to be determined by the method of least squares, find the normal equations. Use these equations to find a, b for the following data (keep four decimal accuracy). (3+5)

t(radians)	0.5	1.0	1.5	2.0	2.5
f(t)	0.9082	0.6552	0.3031	-0.0621	-0.3509

b. Write a C program for interpolation using Lagrange interpolation. Input the following (i) Limit to number of points as 10. (ii) Number of points for any application as n. (iii) (Abscissas, Ordinates) = . (iv) The value of x for which interpolation is required. Output x and the interpolated value. (6)

Q.9 a. A differentiation rule of the form is given. (i) Determine a, b, c so that the rule is exact for polynomials of degree 2. (ii) Find the error term. (iii) If the roundoff errors in computing are where i =1, 2, 3, then obtain the expression for the bound of roundoff error in computing . (8)

b. Use the formula , to compute from the following table of values with h=0.4 and h=0.2. Perform Richardson extrapolation to compute a better estimate for . (6)

X	0.2	0.4	0.6	0.8	1.0
f(x)	1.3016	2.5256	3.8296	5.3096	7.1

Q.10 a. Find the values of a,b,c such that the numerical integration formula is of as high order as possible. Find the error term. (7)

b. Write a C program for solving the initial value problem by Euler’s method. (i) Input the initial values the final value x = xf and step length h. (ii) Use a subprogram for evaluating . (iii) Create a file named “result” and put the computed values, for each value of x, in it. (7)

Q.11 a. Evaluate the integral by (i) two point Gauss-Legendre formula, (ii) two point Gauss-Chebyshev formula. (6)

b. An approximate value of u(0.2) for the initial value problem

with h = 0.2 is to be obtained. Find this value using

(i) Euler’s method, (ii) Taylor series method of order four. (8)