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Root Finding and Equation Solving Questions

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Root Finding and Equation Solving Questions NUMERICAL COMPUTATION QUESTION ONE (15 MARKS) a) The Newton-Raphson method is generally given as: f ( xi ) xi 1 = xi − f ( xi ) Give the Algorithm of finding the root of a function using this method. (2 Marks) b) Use Newton-Rahpson method to find the root of the function f ( x) = x 3 − 467 to the third iteration to 5 decimal places, where f ( x) = 3 x 2 . Take initial guess to be xo = 6 . (3 Marks) c) Find a root of x4-x-10 = 0 using the fixed point iteration technique. Perform 5 iterations for the start points x0=1.0, 2.0 and 4.0, all to 5 decimal places. (5 Marks) d) Consider the set of equations: x1 x2 = 4 x1 − 2×2 = 1 Use eight rounds of the Gauss-Seidel Iteration to solve for x1 and x2 QUESTION TWO (15 MARKS) a) Use Gaussian elimination technique to solve the following system of equations i. x − 3y z = 4 ii. 2x − 8y 8z = −2 iii. −6x 3y − 15z = 9 b) i) Given (5 Marks) x = (4 Marks) b−a b , show the formula for approximating the integral  f (x )dx for n partitions n a using the trapezoidal rule. (2Marks)  ii) Approximate the integral  sin (x)dx using trapezoidal rule for n = 8 . 0 a) i) Use the equation below to define what is meant by Linearization of Differential Equations (3 Marks) a) ii) Linearize the following differential equation with an input value of u=16. (6 Marks) Page 1 of 2 END Page 2 of 2

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