Each of the following problems describes an algorithm implemented on a computer satisfying the axioms (13.5) and $(13.7) .$ For each one, state whether the algorithm is backward stable, stable but not backward stable, or unstable, and prove it or at least give a reasonably convincing argument. Be
sure to follow the definitions as given in the text.
(a) Data: $x \in \mathbb{C} .$ Solution: $2 x,$ computed as $x \oplus x$
(b) Data: $x \in \mathbb{C}$. Solution: $x^{2},$ computed as $x \otimes x$
(c) Data: $x \in \mathbb{C} \backslash\{0\} .$ Solution: $1,$ computed as $x \oplus x .$ (A machine satisfying (13.6) will give exactly the right answer, but our definitions are based on the weaker condition $(13.7) .)$
(d) Data: $x \in \mathbb{C}$. Solution: $0,$ computed as $x \ominus x .$ (Again, a real machine may do better than our definitions based on $(13.7) .)$
(e) Data: none. Solution: $e,$ computed by summing $\sum_{k=0}^{\infty} 1 / k !$ from left to right using $\otimes$ and $\oplus,$ stopping when a summand is reached of magnitude
$<\epsilon_{\text {machine }}$
(f) Data: none. Solution: $e,$ computed by the same algorithm as above except with the series summed from right to left. (g) Data: none. Solution: $\pi,$ computed by doing an exhaustive search to find the smallest floating point number $x$ in the interval [3,4] such that $s(x) \otimes$ $s\left(x^{\prime}\right) \leq 0 .$ Here $s(x)$ is an algorithm that calculates $\sin (x)$ stably in the given interval, and $x^{\prime}$ denotes the next floating point number after $x$ in the floating point system.