1.

In error analysis it is sometimes convenient to bound instead of . Obtain inequalities between and .

2.

(Skeel and Keiper [1044, 1993, 1.2]) The number y = e ^??163 was evaluated at t-digit precision for several values of t, yielding the values shown in the following table, which are in error by at most one unit in the least significant digit (the first two values are padded with trailing zeros):

Does it follow that the last digit before the decimal point is 4?

3.

Show how to rewrite the following expressions to avoid cancellation for the indicated arguments.

1. , x ? 0.

2. sin x ? sin y, x ? y.

3. x ² ? y ², x ? y.

4. (1 ? cos x)/sin x, x ? 0.

5. c = ( a ² + b ² ? 2 ab cos ?) ^1/2, a ? b, ? ? 1.

4.

Give stable formulae for computing the square root x + iy of a complex number a + ib.

5.

[570, 1982] Show how to compute log(1 + x) accurately for all x > ?1, including for small x. Assume that the log function is computed with a relative error not exceeding u. (Hint: adapt the...