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**Additional resources for An introduction to differentiable manifolds and riemannian geometry**

**Example text**

We will not dive deeper into this. Specific exercises for this section: Exercises 20–21. ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 32 Chapter 2. 5 Exercises 0. Review questions (a) What is a normalized floating point number and what is the purpose of normalization? (b) A general floating point system is characterized by four values (β, t, L,U ). Explain in a few brief sentences the meaning and importance of each of these parameters. (c) Write down the floating point representation of a given real number x in a decimal system with t = 4, using (i) chopping and (ii) rounding.

D1 d2 d3 · · · dt −1dt dt +1 · · · ) × β e using only t digits, it is possible to use one of a number of strategies. The two basic ones are • chopping: ignore digits dt , dt +1, dt +2 , dt +3 . d1 d2 d3 · · · dt −1 + β 1−t × β e , fl(x) = dt < β/2, dt > β/2. In case of a tie (dt = β/2), round to the nearest even number. 3. 4. Since humans are used to decimal arithmetic, let us set β = 10. 6666 . . = 3 6 6 6 6 2 + 1 + 2 + 3 + 4 + · · · × 100. 0 10 10 10 10 10 This number gives an infinite series in base 10, although the digit series has finite length in base 3.

Why is the former considered more damaging than the latter? (f) What is a cancellation error? Give an example of an application where it arises in a natural way. (g) What is the rounding unit for base β = 2 and t = 52 digits? (h) Under what circumstances could nonnormalized floating point numbers be desirable? (i) Explain the storage scheme for single precision and double precision numbers in the IEEE standard. 1. The fraction in a single precision word has 23 bits (alas, less than half the length of the double precision word).