Numerical Analysis — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

Name of thei Paper : DSC-Numerical Analysis Choice is given within the question. 1. (ci) Define -the order `of convergence of a sequence p„- which converges to a number p. Show that the sequence generated by the formula (a) Verify that the equation a33 + #2 - 3# - 3 I 0 has a root on the interval (1, 2). Perform the bisectiop method to determine p4, the fourth approximation to the location of the Toot. (c) Perform four iterations of the method of false position to approximate the \~ root of the equation #4 -|8ff2 + 45=0 `in the interval (1, 2). 15 2. ~(cb) Let g be a coitinuous function on the interval [a, b] such` t`hat g([ci, b].) c [o., b]. Suppose g is differentiable function on (a, a) and satisfies A sequchce p„ is generated by the fixed point iteration pn = g(p-n_1). re 21. Show that the sequence {p7a} converges to a fixed point p of g for any initial. approximation po e [cE, b]. (a) Use Netwton's method to approximate the root of the equation #5 + 2# - 1 = 0 in the interval (`0, 1). Use error tolerance 104, and initial approximation po = 0.5. (c') Perform the Secant method to deterlhine p42 the fourth `apprQximation to the location of the ro.ot of the equation tan (") - 7t - 6 ± 0 with starting approximations po = 0, pi = 0.48. 15 (a) Find an LU decomposition of the matrix and use it to solve the system AX = [14 36 71F. (b) Perform three iterations of Jacobi method to solve the system of equations, for the given coefficient matrix and right hand side vector, starting with the initial vector xlo) = 0, : (c) State and prove existence and uniqueness theorem for Interpolating. 1. (a) Find the Lagrange form of interpolating polynomial for the given data set and hence interpolate at x= 1.5. 0) If a =then evaluate the eth divided difference fIx, 2, 22, an. (c) Obtain the piecewise Linear Interpolating Polynomial for the function f(x) defined by the given data set and hence estimate the value of f(1:5) • approximation • 'formula (a) Prove the forward difference f'(X) = -3f(x)+4f(x, +h)+ f(x6+2h) and determine the approximate value of the derivative of f(x) = 2x2 - 1 at xo = 1, with the step length. h = 0.1, 0.01 and 0.001, and absolute error. (b) Verify that the second-order central difference approximation for the second derivative provides the exact value of the second derivative, regardless the value of h, for the functions f(x) = 1, f(x) = x, f(x) = x, and f(x) =x", but not for the function f(x) =*.. (c) Derive the expression of the Simpson's rule to approximate definite integral and hence approximate the value of the integral f. e* dx. 15 (a) Define, the degree of precision of the quadrature rule. Further, calculate the values of the coefficients A, Ay, and Az so that the quadrature precision at least 2. (b) Use the Modified Euler Method to determine the approximate solution of the initial value problem = (1) = 1,1 ≤ t ≤ 2, taking the step size as h = 0.5.- (c) Use the Runge-Kutta Method fourth order to determine the approximate