Major Quiz1 Soln.pdf - Major Quiz 1 Solution Q1 The following set of equations is to be solved to get the value of x for a given[8 =2(1 − = 1 a For

Major Quiz1 Soln.pdf - Major Quiz 1 Solution Q1 The...

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Major Quiz 1 Solution Q1. The following set of equations is to be solved to get the value of x for a given 𝛿 : [8] ? + ? = 2 ? + (1 − 𝛿 ) ? = 1 a) For what values of 𝛿 will this problem be well-conditioned? (5) Ans a. ? + ? = 2 ? + (1 − 𝛿 ) ? = 1 Solving for x , we get x= f(δ)= 2𝛿−1 𝛿 The condition number of the problem is C p = | 𝛿𝑓 ′(𝛿) 𝑓(𝛿) | = | 1 2𝛿−1 | For the problem to be well-conditioned, ? 𝑝 < 1 (3 marks) −1 < 1 2δ − 1 < 1 or −1 > 2δ − 1 > 1 δ < 0 or δ > 1 (2 marks) b) Calculate the % relative change in x for a 1% change in 𝛿 , for 𝜹 = 2. Perform computations using infinite precision but report intermediate steps up to 4 significant digits with rounding. (3) Ans b. Calculate the relative change Approach 1: ∆𝛿 = 0.02 𝛿 = 2 ?? = | (∆?/?)/(∆𝛿/𝛿)| = ∆𝑥/𝑥 0.02/2 ? = ?(𝛿) = 2𝛿−1 𝛿 = 3 2 = 1.50 ? = ?(2.02) = 2∗2.02−1 2.02 = 4.04−1 2.02 = 1.50495 | ∆𝑥 𝑥 | = (1.50495 − 1.50) ∗ 100 1.5 = 0.33% Approach 2: 𝐹?𝑟 𝛿 = 2, ?? = | (∆?/?)/(∆𝛿/𝛿)| |∆?/?| = ?? ∗ | ∆𝛿 𝛿 | = | 1 2𝛿−1 | ∗ 1% = 1 3 ∗ 1% = 0.33%
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