15 Marks Deadline November 10, 2024 11:59 PM

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📄 Submission form Q1 📄 ⇒ https://forms.gle/7EXGU7YL43HSJDa1A

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Three number representations are -

  1. (Conversion 01) Standard Format: $\pm (0.d_1d_2d_3\dots)_\beta\times \beta^e$, where $d_1\ne 0$
  2. (Conversion 02) Normalised Format: $\pm (1.d_1d_2d_3\dots)_\beta\times \beta^e$
  3. (Conversion 03) Denormalised Format: $\pm (0.1d_1d_2d_3\dots)_\beta\times \beta^e$

Question 01

Consider a number system to have the following properties: $\beta = 2, m = 3 \text \ {and} −3≤ e ≤ 3$. Now answer the following three questions for all three convensions.

  1. Find the maximum non negative number in the number system (write your answer in base 10 as well). [2 Marks]
  2. Find the minimum non negative number in the number system (write your answer in base 10 as well). [2 Marks]
  3. Find total how many numbers that can be represented through the number system. [1 Marks]
**ANSWERING FORMAT**

Answer to question 1(a):

Convension 01: maximum non-negative = (?)₂ x 2^? = (?)₁₀
Convension 02: maximum non-negative = (?)₂ x 2^? = (?)₁₀
Convension 03: maximum non-negative = (?)₂ x 2^? = (?)₁₀

Question 02

Consider $\beta = 2, m = 6, -3<e<3$ to be a number representation system.

  1. Compute the value of machine epsilon for the above mentioned system in Denomalised form and Standard form. [3 Marks]
  2. If we change the value of $e_{min} = -1\ \text{and} \ e_{max} = 10$ will that affect the value of machine epsilon. [2 Marks]

Question 03

Consider the quadratic equation in the form, $5x^2 − 70x + 4 = 0$. While calculating, at each step consider upto 5 Significant digit.

  1. Find the actual roots. [1 Marks]
  2. Find out where the loss of significance occurs when you calculate the roots. [2 Marks]
  3. Show that the roots evaluated in the previous part do not satisfy the fundamental properties of a polynomial. [1 Marks]
  4. Evaluate the correct roots such that loss of significance does not occur. [1 Marks]