Nonlinear optimisation

Question 1

Let f(x) = (x-c) e-x+c - 0.3 where c = 3.558.

Using the simple iteration method, find the root of f(x) correct to TWO decimal places using x0= 2.0 + c. Hint: Check that the root satisfies x = ln(x-c) - ln 0.3  AND x = 0.3ex-c + c. You may need to iterate more than 5 times.

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Question 2 

Let f(x) = (x-c)2 e-x+c - 0.3 where c = 3.058.

Using the Newton-Raphson iteration scheme, find the root (correct to 3 decimal places) of f(x) with x0 = 1.0 + c.

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Question 3

Let f(x) = (x-c) sin(x-c) + (x-c) + 5 where c = -0.022. If we set a0 = -6.7 + c, and b0 = -6.3+c and set (a0,b0) as the starting interval for the r=0 iteration of the bisection method. Find a4.

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Question 4

 Let f(x) = (x-c) e-x+c - 0.3 where c = 0.769. Using the simple iteration method, find the root of f(x) correct to TWO decimal places using x0= 0.4 + c.

Hint: Check that the root satisfies x = ln(x-c) - ln 0.3  AND x = 0.3ex-c + c. You may need to iterate more than 5 times.

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Question 5

Let f(x) = a/x where a = 6.207. If R={x: 0.5 <= x  <= 1.0 }

Find the maximum value (correct to 3 decimal places) of | df/dx |  in the interval of R.

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Question 6

Let f(x) = eax + bx where a = 4.463 and b = 1.746. The Newton-Raphson method is used to solve f(x) = 0 with x0 = 0.67. Find x2 correct to 3 decimal places.

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Question 7

We shall  let f(x) = ax2 where a = 0.98. If f(x) fulfills the condition 1 (Refer to SU1-8) of the contraction mapping in the interval

R = {x : 0 <= x <= c}

Find the maximum value of c. Express your answer correct to 3 decimal places.

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Question 8

 If f(x) = ax2 where a = 1.477. If | df/dx |  <=  1 in the interval R = { x:  0 <=  x <=  c }

find maximum c (correct to 3 decimal places).

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Question 9

A bisection method is to be used to find the solution of f(x) = 0 In the 0th iteration (the zeroth iteration, i.e., r = 0), an interval of a <=  x <= b  is used,

where a = -0.401 and b = -0.337.

Find the length of the interval in the third iteration (i.e., r = 3). Give the answer correct to 3 decimal places.

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Question 10

Let f(x) = (x-c)2 e-x+c - 0.3 where c = 3.549.

Using the Newton-Raphson iteration scheme, find the root (correct to 3 decimal places) of f(x) with x0 = 4.0 + c.

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