In this problem, you will prove that the absolute value of the true relative rou

Question

In this problem, you will prove that the absolute value of the true relative round-off error |E_t/x| associated with the approximate representation of any number x in any normalized floating-point system(binary, decimal,.....) that relies on chopping is always less than the machine epsilon Iµ of that system. Consider a normalized (regular normalization as described in the textbook, not IEEE normalization) floating-point system with t significant figures: in that system, a number x_FP can be represented exactly as x_FP = plusminus 0 n_1 n_2...n_1 times b^e, where the n_1 are the t integers of the normalized mantissa, b is the base (which can be anything: 2, 8, 10,....) of the system and e is the exponent of the number. The number x that you are trying to represent can either be among the finite number of x_FP values (in which lucky case E_t = 0), or be located in between 2 consecutive x_FP values with the same exponent e, x_m and x_m + 1, and it will be chopped to x_m, as illustrated below: a. For a given exponent e, express the absolute value of the spacing |Delta r| in terms of t, b and e. b. For the same exponent e as in part a), calculate the largest possible value of |Delta r/x| in this normalized system, and express your result in terms of t and b. c. Use your class notes and part b) to prove that, for any value of d, |E_t/x|

Explanation / Answer

Database normalization is process used to organize a database into tables and columns. The idea is that a table should be about a specific topic and that only those columns which support that topic are included. For example, a spreadsheet containing information about sales people and customers serves several purposes:

Identify sales people in your organization
List all customers your company calls upon to sell product
Identify which sales people call on specific customers.

By limiting a table to one purpose you reduce the number of duplicate data that is contained within your database, which helps eliminate some issues stemming from database modifications. To assist in achieving these objectives, some rules for database table organization have been developed. The stages of organization are called normal forms; there are three normal forms most databases adhere to using. As tables satisfy each successive normalization form, they become less prone to database modification anomalies and more focused toward a sole purpose or topic.

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