Reference no: EM133157812

CS 417 Computational Methods and Software - Old Dominion University

Question 1. Let f(x) = √(1 - x)ⁿ - 1 where n ≥ 2.
a) Explain the difficulty of computing f(x) for a small value of |x| (i.e., x → 0).
b) Show how the difficulty in computing f (x) can be circumvented.
c) Compute the condition number of f (x) for x → 0.

Question 2. Assume you have computed φˆ = c₀x + c₁x³ where x ≥ 1. Compute the relative error in evaluating this function (i.e., evaluate the impact of finite precision on the result). Do not bound. Assume:
• all error originates from representing c₀, c₁, and x.
• arithmetic operations introduce no error. Start with φˆ* = C*₀x^∗ + c*₁(x*)⁴.

Question 3. Continue the previous problem. Derive an upper bound for the relative error of

φˆ = c₀x + c₁x³ where x ≥ 1.

Consider defining ∈_max = max(|∈₀|, |∈₁|, |∈₂|, . . . , |∈_n|).

Question 4. Assume you have computed φˆ = c₀x+c₁x³. Compute the relative error in evaluating this function (i.e., evaluate the impact of finite precision on the result). Do not bound. Assume:

(a) error originates from representing c₀, c₁, and x.
(b) arithmetic operations do introduce error.
(c) xⁿ can be computed without error. Start with φˆ∗ = ((c^∗0x^∗)^∗ + (c^∗₁(x^∗)⁴)^∗)^∗.

Question 5. Continue the previous problem. Derive an upper bound for the relative error of φˆ = c₀ + c₁x³.
Consider defining ∈_max = max(|∈₀|, |∈₁|, |∈₂|, . . . , |∈_n|).

Question 6. Can one claim that

(cond z)(x) ≤ (cond f )(x) + (cond g)(x) + (cond h)(x) + (cond j)(x)

where z(x) = f (x) ∗ g(x) ∗ h(x) ∗ j(x)? You may genaralize the problem to n functions.

(a) Start by computing z′(x) = d/dx (f (x) ∗ g(x) ∗ h(x) ∗ j(x)).
(b) Compute z′(x)/z(x) and simplify the result.
(c) Use the general form of the condition number.
(d) State your final answer.

Attachment:- Computational Methods and Software.rar