School of Sciences Indira Gandhi National Open University Maidan Garhi, New Delhi (For January 2012 cycle)
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1 MTE-0 ASSIGNMENT BOOKLET Bachelor's Degree Programme Numerical Analysis (MTE-0) (Valid from st January, 0 to st December, 0) School of Sciences Indira Gandhi National Open University Maidan Garhi, New Delhi-0068 (For January 0 cycle)
2 Dear Student, Please read the section on assignments in the Programme Guide for Elective courses that we sent you after your enrolment. A weightage of 0 per cent, as you are aware, has been earmarked for continuous evaluation, which would consist of one tutor-marked assignment for this course. The assignment is in this booklet. Instructions for Formating Your Assignments Before attempting the assignment please read the following instructions carefully. ) On top of the first page of your answer sheet, please write the details exactly in the following format: ROLL NO : NAME : ADDRESS : COURSE CODE:. COURSE TITLE :. ASSIGNMENT NO.. STUDY CENTRE:.... DATE:.... PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY. ) Use only foolscap size writing paper (but not of very thin variety) for writing your answers. ) Leave 4 cm margin on the left, top and bottom of your answer sheet. 4) Your answers should be precise. ) While solving problems, clearly indicate which part of which question is being solved. 6) This assignment is to be submitted to the Study Centre as per the schedule made by the study centre. Answer sheets received after the due date shall not be accepted. We strongly suggest that you retain a copy of your answer sheets. 7) This assignment is valid only upto December, 0. If you have failed in this assignment or fail to submit it by December, 0, then you need to get the assignment for the year 0 and submit it as per the instructions given in the programme guide. 8) You cannot fill the exam form for this course till you have submitted this assignment. So solve it and submit it to your study centre at the earliest. We wish you good luck.
3 Assignment (MTE 0) (January 0 December 0) Course Code: MTE-0 Assignment Code: MTE-0/0 Maximum Marks: 00. a) A negative root of smallest magnitude of the equation x + x + 0 = 0 is to be determined i) Find an interval of unit length which contains this root ii) iii) Perform two iterations of the bisection method Taking the end points of the last interval as initial approximations perform one iteration of the secant method. () b) Find a root of the equation x + 0x + 0x + 7 = 0 which is close to. 0 using the Birge-Vieta method. Perform two iterations of the method. () c) Obtain the cube root of using Newton-Raphson formula. (). a) Derive a suitable iteration function φ (x), such that the sequence of iterates obtained from the formula x k = φ(x k ), k = + 0,,, converge to the root of f (x) = 0 for f (x) = x log0 x 7 = 0. Using this formula and initial approximation x 0 =. 8, find the root correct to four decimal places. (4) b) Set up the Gaussi-Jacobi iteration scheme in matrix form for the linear system of equations x + 4x x 4x x x + 4x = = = Show that the iteration scheme is convergent. Hence find the rate of convergence of this method. (6). a) Find all the roots of the polynomial x 6x + x 6 = 0 by the Graeffe s root squaring method using three squarings. (7) b) How many maximum positive and negative roots does the equation 8x 4 + x 0x + 7x 8x + = 0 has? () 4. a) The Gauss elimination method is used to solve the system of equations x + 4x + αx = x x + αx = α x + x + x = 6 Find the value of α for which the system has (i) a unique solution (ii) no solution (iii) infinitely many solutions. (4) b) Find the eigenvalue of the matrix A, nearest to and also the corresponding eigenvector using four iterations of the inverse power method where
4 4 0 A = 4 (6) 0 4. a) i) Set up the Gauss-Seidel iteration scheme in matrix form for solving the system of equations x x 7 ii) iii) = x + x x = x x = Show that this iteration scheme converges and find the rate of convergence. Perform two iterations of this method taking the zero vector as the initial approximation. (6) b) Find the inverse of the matrix A = 4 using LU decomposition method. (4) 6. a) Find the interpolating polynomial that fits the following data: x f(x) () b) Using the Lagrange s form of an interpolating polynomial find the value of x when y = from the following table of values: x y 4 () c) Using Lagrange s interpolation formula, prove that y = y 0.(y y ) + 0.(y y ) approximately. (4) 7. a) Given log0 64 =.86, log0 68 =.88, log0 69 =.889, log0 66 =. 80, find log () b) Prove that the third divided differences with arguments a, b, c, d of the function is equal x to. () abcd c) Determine the spacing h in a table of equally spaced values of the function f (x) = x between 0 and, so that quadratic interpolation in this table yields accuracy of 0. (4) 8. a) Find the value of f () from the following table 6 the constants x f(x) a, b, c in the numerical differentiation formula () b) Find the value of 4
5 y (x i ) = ay(x i h) + by(x i ) + cy(x i + h) such that the method is of highest possible order. Derive the corresponding Richardson extrapolation scheme. () c) Using Stirling s formula find the number of persons at age years, given where, y =, y 0 = 49, y 40 = 46, y 0 0 = 4 y x represents the number of persons at age x years in a life table. (4) 9. a) The following table of values of f (x) is given Find f (0.) using an 0(h ) method (using all the three values.) () b) Evaluate by Simpson s one-third rule an approximate value of sin x + cos x dx using 7 ordinates. (4) c) Determine the value of the integral I x f(x) = x( + 0 x / ) dx by composite trapezoidal rule with and ordinates. Improve the result by using extrapolation technique. (4) 0. a) Find the solution of the difference equation y k + 4y k+ + 4y k = 0; k = 0,,. Also find the particular solution when y 0 = and y = 6. () b) Solve the IVP, y = ; y(4) = 4 using Euler s method. Find y (4.) with h = 0. x 4y and 0. and extrapolate the value y (4.). () c) Solve the IVP y = + y, y(0) = 0 using classical R-K method of 0(h 4 ). Find y (0.4) taking h = 0.. Compare the solution obtained with the exact solution and find the error. (6) 0
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