Numerical Methods — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

1) Define order of convergence. Add the following numbers from left to right and right to left, rounding to 3 digits at each step: Find the relative error in each case and the number of significant digits to which the two values approximate the exact solution. b) Determine the number of significant digits in the following numbers: c) If the true value of a number x = 1.6180339 and its approximate value is given by 1.618, find the absolute, relative and percentage errors. 2) Find a root of the equation f(x) = 3x3 - x' - 18x + 6 in the interval [0,1], using Bisection Method. Perform 5 iterations. Write the order of Convergence of Bisection b) Find a root of the equation f(x) = 2* - x3 taking Xo = 1 and x1 = 1.5, using Secant Method. Perform 5 iterations. Write the order of Convergence of Secant Method. of Regula-Falsi Method. 3) a) Solve the system of equations using the Gauss elimination with partial pivoting. b) Solve the following system of linear equations using Gauss-Jacobi method, with XO= (0,0,0). Perform three iterations. c) Solve the following system of equations using the Gauss-Seidel iteration method. Take X(®)= (0,0,0) and perform three iterations. 4. (a). Define the interpolating polynomial, and using the Lagrange formula, find the polynomial that passes through the points (1,2), (2,3), (4,1). (b). Prove the following relations: (ii) where each symbol has its usual meaning. (c). Show that the third divided difference with arguments ab, cd of the function f(x) = 4 is 5. (a). Use the three-point backward difference formula to approximate the derivative of f(x) = sin x + cos x at x, = 5, taking h = (b). For the following data, find f" (7.50) by using the formula f"(x) = (xi) - 2f(xith) +/ (x1+2h) (c). Evaluate Jo I+* dx by using Trapezoidal rule. Hence obtain the approximate value of r. 6. (a). Given dy = -x with the initial condition x = 0, y = 1. Find y (0.1) using Euler's method with step size h = 0.025. (b). Using mid-point (Runge Kutta second order) method, solve the initial value problem dy= x'+ Jy from x = 0 to x = 0.6 where y (0) = 1 by using h = 0.2. (c). Apply Runge Kutta method of fourth order to find the approximate value of y for x = 0.2 , in steps of 0.1, if"=x+y, given that y = 1 where x = 0.