ASSIGNMENT BOOKLET. Numerical Analysis (MTE-10) (Valid from 1 st July, 2011 to 31 st March, 2012)

Transcription

1 ASSIGNMENT BOOKLET MTE-0 Numerical Analysis (MTE-0) (Valid from st July, 0 to st March, 0) It is compulsory to submit the assignment before filling in the exam form. School of Sciences Indira Gandhi National Open University Maidan Garhi New Delhi-0068 (For July, 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 Formatting 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. 5) While solving problems, clearly indicate which part of which question is being solved. 6) This assignment is valid only upto March, 0. If you have failed in this assignment or fail to submit it by March, 0, then you need to get the assignment for the January, 0 cycle and submit it as per the instructions given in that assignment. 7) It is compulsory to submit the assignment before filling in the exam form. We strongly suggest that you retain a copy of your answer sheets. We wish you good luck.

3 Assignment (MTE-0) (July 0 March 0) Course Code: MTE-0 Assignment Code: MTE-0/TMA/0- Maximum Marks: 00. a) Find an interval of unit length which contains the negative real root of f (x) = 8x x + = 0. Construct a fixed-point iteration x = g(x), which converges. Verify the condition of convergence. Take the mid-point of this interval as a starting approximation and iterate thrice. (6) b) Errors in two successive approximations of a general linear iteration method x k + = φ(x k ), k = 0,,, are given by ε k+ = A εk, εk+ = Aε k+ where εk = x k ξ and ξ is the exact root. Using this information, obtain a better estimate of the solution whenever three successive approximations are available. (4). a) A negative root of smallest magnitude of the equation x + 5x + 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. (5) b) The equation x + x 5 = 0 has a positive root in the interval ], [. Write a fixed point iteration method and show that it converges. Starting with initial approximation x 0 =. 5 find the root of the equation. Perform two iterations. (5). a) Perform four iterations of the Newton-Raphson method to find an approximate value of / ( 0) starting with the initial approximation x 0 =. 5. () b) Find all the roots of the polynomial x 6x + x 6 = 0 by the Graeffe s root squaring method using three squarings. (7) 4. a) How should the constant α be chosen to ensure the fastest possible convergence with the iteration formula αx n + x n + x n+ =. α + () b) Find the interval of unit length which contains the smallest positive root of the equation x 5x + x 5 = 0. Perform one iteration of the bisection method. Using the left endpoint of this interval as an initial approximation perform two iterations of the Birge-Vieta method. () c) Find the inverse of the matrix A = 5 4 using LU decomposition method. (4)

4 5. a) Find the positive root of x cos = 0 by bisection method. Perform six iterations of the method. () b) Using Newton Raphson method find the smallest positive real root of x log0 x =. correct to five decimal places. (4) c) Do six iterations of the secant method to solve the equation x + x 6 = 0, starting with x 0 = and x =. () 6. a) Consider the system of equations a x b = a x b where a is a real constant. For what values of a, the Jacobi and Gauss-Seidal methods converge. (5) b) Estimate the eigenvalues of the matrix using the Gershgorin bounds. Draw a sketch of the region where the eigenvalues lie. (5) 7. a) A numerical differentiation formula for finding f (x ) is given by h where f (x ) = f (x nh) [ f 8f + 8f f ] f k = k k k+ k+ k n k. Using Taylor series expansion, find the truncation error of the formula. (4) b) The second divided difference f[x 0, x, x ] can be written as f[x 0, x, x ] = af (x 0 ) + bf (x) + cf (x ) Find the expressions for a, b, c. () c) Linear interpolation is to be used in a table of values. Derive the error bound for linear interpolation. () 8. a) Find the largest step length than can be used for constructing a table of values for the function 4 f (x) = x + 5ln x, 0 x 0, so that a quadratic interpolate can be used with an accuracy 6 of 5 0. (5) b) Obtain the approximate value of the integral + 0 dx k x, using composite Simpson s rule with h = 0.5 and h = Also obtain the improved value using Romberg integration. (5) 9. a) Consider the table of values of x f (x) = xe given below x f (x) Find f (.0 ) using the central difference formula of 0(h ) for h = 0. and h = 0.. Calculate TE in both the cases. (4) 4

5 b) Determine the constants α, β, γ in the differentiation formula y (x 0 ) = α y(x 0 h) + β y(x 0 ) + γ y(x 0 + h) so that the method is of the highest possible order. Find the order and the error term of the method. (6) 0. a) Solve the IVP, y = log0 (x + y), y(0) = using modified Euler s method at x = with h = 0.. (5) b) Using R-K method of fourth order, solve the IVP y x y = with y (0) = y + x at x = 0., 0. 4 with h = 0.. (5) 5