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Assignment_3

# Assignment_3 - 3 1 2 4 1 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = x...

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ME 303 ADVANCED ENGINEERING MATHEMATICS Assignment 3 Question 1. As part of solving some PDE problems, it is necessary to find the roots of an equation of the type, x x 1 ) tan( = . a) Use a sketch to identify the number of positive roots and their approximate values. b) Apply Newton’s method to find the values of the first two positive roots to 4 decimal places. c) Try to find the first positive root by a direct iteration with ) tan( 1 old new x x = with an initial guess 1 = old x . d) Investigate the effect of relaxation added to part (c). Question 2. It is of interest to find the smallest positive root of 3 2 4 4 x x = a) From a sketch, show that the root is near x=0.9. b) Try a direct iteration with the rearrangement ( ) 2 / 1 3 4 4 x x = and explain what happens. c) Try a direct iteration with the rearrangement
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Unformatted text preview: 3 / 1 2 4 1 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = x x and explain what happens. Question 3. Consider the following system of equations: ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = + + + − = + + + = + + + − = + + + 1 2 1 2 1 2 1 2 w z y x w z y x w z y x w z y x a) Write the system in matrix form and solve using Gaussian elimination. b) Solve by Gauss-Seidel iteration, without rearranging the equations. Use initial guesses 1 = = = = w z y x . c) Investigate whether relaxation would help convergence for part (b). d) Repeat part (b), but rearrange the equations first to get as strong diagonal elements as possible. e) Investigate whether relaxation can speed up convergence for part (d)....
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