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Numeric methods Solve the following equation for one root using Newton-Raphson Method, with an initial guess of 1.0 f(x) = x – cos(x) = 0 Use an accuracy of 0.001 Give the root. How many iterations will it be needed to achieve that accuracy? 1. Solve the following equation for one root using Newton-Raphson Method, with an initial guess of 1.0 f(x) = x – cos(x) = 0 Use an accuracy of 0.001 Give the root. How many iterations will it be needed to achieve that accuracy? 2. Using the following data, construct Newton’s difference table: x y (=f(x)) F’s -1 2 1 -4 3 6 5 10 Then determine the Newton’s interpolation polynomial of degree 3. Using the polynomial determine the value of y at x = 4 3. Numerically integrate the following function: with no of intervals, n = 4 using both (a) Trapezoid rule and (b) Simpson’s 1/3rd rule. What would be the %age error of the numerical solution as compared to the analytical solution for both cases? Given the analytical solution of 4. Solve the following two equations using Gauss-Seidel iterative method. 3x1 – x2 = 8 x1 + 6x2 = 9 Use the initial guess to be (0, 0). The maximum error should be less than 0.01. If the equation sequence is reversed, that is if the equations are given as below: x1 + 6x2 = 9 3x1 – x2 = 8 Will that lead to convergence faster? Why or Why not? 5. Illustrate Gaussian Elimination method to solve the following system of equations: 2X1 – X2 + X3 = 1 -X1 + 4X2 = 2 X1 – 2X2 +2 X3 = 4 Show each step.