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Registration No: Total Number of Pages: 02 B.TECH PMT5H001 5th Semester Regular Examination 2017-18 ADVANCED NUMERICAL METHODS BRANCH: METTA, MME Time: 3 Hours Max Marks: 100 Q.CODE:B516 Answer Part-A which is compulsory and any four from Part-B. The figures in the right hand margin indicate marks. Q1 a) b) c) d) e) f) g) h) i) j) Q2 Q3 Part – A (Answer all the questions) Answer the following questions:multiple type or dash fill up type The Newton Raphson Method fails when _________. Order of convergence of Regula-Falsi method is _______________. ------------ number of starting values is required for Adam’s Method. i) 1, ii) 2, iii) 3, iv) 4. Distinguish between interpolation and extrapolation. To find the value of in the corrector method, the number of prior values are required . i) 1, ii) 2, iii) 3, iv) 4. Which of the following method is called step by step method. i) Taylor’s Method, ii) RK method, c) Milne’s Method, d) Newton’s Method. Define Discrete Fourier Transform. Define Elliptic, Parabolic & Hyperbolic type of partial differential equations ? The _________ type of eigen value obtained using Power Method. Milne predictor formula is ________. Answer the following questions: Short answer type a) What do you mean by piecewise cubic interpolation. b) Write the piecewise interpolating polynomial for the following data X 0 0.1 0.2 y 1.000 0.5242 -0.9037 c) Find the quadratic polynomial that fits ( ) = = , , . d) Describe Spline interpolation. e) What is accelerating convergence ? f) What is shifted power method ? g) What is Fast Fourier Transform. h) What are the methods you use to solve one dimensional wave equation i) Express = in terms of difference quotients. j) Explain the advantages of Implicit schemes over explicit schemes ? Part – B (Answer any four questions) a) Find the interpolating polynomial of the following data using Piecewise Cubic Hermite Interpolation. X 0 1 2 (2 x 10) (2 x 10) (8)

Q4 Y 1 3 35 Y’ 1 6 81 b) Fit a natural cubic spline function for the data X -1 0 1 Y -3 -1 -1 (7) a) Find the values of y’(0.4) from the following table : (8) X 0.1 0.2 0.3 Y 1.10517 1.22140 1.34986 b) Using the trapezoidal rule, taking subintervals length as h, h/2, h/4, h/8 etc (7) the integral ∫ / was evaluated. The values of integral are 0.987116, 0.996785, 0.999196 and 0.9997999 respectively. Using Romberg method, improve the result. Q5 a) Find QR factorization of the following matrix 2 2 1 2 2 0 1 0 2 (10) b) Find the 1st derivative and 2nd derivative of the function tabulated below at x=1 & x=2 X 1.0 1.2 1.4 1.6 1.8 2.0 Y 2.7183 3.3201 4.0552 4.9530 6.0496 7.3891 Q6 a) Find the Eigen Values and eigen vector of the matrix by using power (5) (12) method.. −2 −2 2 2 2 −2 −1 0 3 b) Describe inverse power method. Q7 a) Given that = [ , ], , ( )= , = . (3) (8) Find out y(2) by using Adams- Bashforth predictor-corrector method. b) Given that = , ( )= , (7) ( . )= . , ( )= . ( . )= . Find out y(2) by using Milne-Simpsons predictor-corrector method. Q8 a) Solve the heat equation = satisfying the conditions ( , ) = , ( , )= > 0 and ( , ) = , < <1 . Using Crank-Nicolson Method compute u for two time steps by taking = . , = . . b) Using FFT, find the interpolation function for the data z=(0,1,2,3) Q9 a) Explain wave equation. Derive the iterative schemes for the solution of wave equation using (i) Explicit method , (ii) Implicit Method. b) Write short notes on DFT and FFT (10) (5) (10) (5)

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