Relative error machine numbers

Question

Suppose we are working with a machine that does arithmetical calculations with a relative accuracy of $\xi, |\xi| \leq \xi '$. We want to calculate $a^2 - b^2$ in the following two ways; $(A);$ $a^2 - b^2 = (a - b)(a+b)$ and $(B)$; $a^2 - b^2 = a \times a - b \times b$. Suppose $a$ and $b$ are machine numbers. Show that the relative error of the result using method $A$ is at most $3\xi'$ and $\frac{1 + (a^2 + b^2)}{|a^2 - b^2|}\xi'$ using method $B$.

The first part (method $(A)$) is not so difficult but I am making a mistake in the second part. Let's start by calculating $a \times a$. This should be equal to $a^2(1+\xi) = a^2 + a^2\xi$. Similarly, $b \times b = b^2 + b^2\xi$. Substracting these two terms gives $(a^2 + a^2\xi - b^2 + b^2\xi)(1+\xi)$ which is equal to $a^2 + a^2\xi - b^2 - b^2\xi + a^2\xi + a^2\xi^2 - b^2\xi - b^2\xi^2$. If we ignore the terms involving $\xi^2$ then we get $a^2 - b^2 + 2\xi(a^2 - b^2)$ and so the relative error is at most $2\xi'$. Obviously this is not even close to the answer I am supposed to be getting but I am not sure where I went wrong.

For the second part of the question, we assume that $a$ and $b$ are not machine numbers. Show that the relative errors are given by $\frac{3 + 2(a^2 + b^2)}{|a^2 - b^2|}\xi$ and $\frac{1 + 3(a^2 + b^2)}{|a^2 - b^2|}$ respectively. I am not really sure what happens when $a$ and $b$ are not machine numbers, besides the fact that we have some sort of extra approximation error that we need to calculate, however I can't find any info on how to do this.

Answer 1

You have to use different $\xi$ with absolute value bounded by the same $\xi'$. I use the symbols $\oplus$ and $\odot$ for the machine operations. We get $$(a \odot a )\ominus (b \odot b) = \\ (a^2(1+\xi_1) - b^2(1+\xi_2)) (1+\xi_3) = \\ a^2-b^2+a^2(\xi_1+\xi_3+\xi_1\xi_3)-b^2(\xi_1+\xi_2+\xi_1\xi_2) $$ so the relative error is $$\left|\frac{a^2(\xi_1+\xi_3+\xi_1\xi_3)-b^2(\xi_2+\xi_3+\xi_2\xi_3)}{a^2-b^2}\right|\le \\ \frac{a^2(|\xi_1|+|\xi_1||\xi_3|)+b^2(|\xi_2|+|\xi_2||\xi_3|)+|a^2-b^2||\xi_3|}{|a^2-b^2|} \le\\ \frac{a^2(\xi'+\xi'^2)+b^2(\xi'+\xi'^2)+|a^2-b^2|\xi'}{|a^2-b^2|} \approx \\ \left(\frac{(a^2+b^2)}{|a^2-b^2|}+1\right)\xi' $$ if one ignores the quadratic terms.

If $a$ and $b$ are not machine numbers you have rounding errors. You have to calculate $$(\Box a \ominus \Box b)\otimes (\Box a \oplus \Box b)$$ or $$(\Box a \odot \Box a )\ominus (\Box b \odot \Box b) $$ where the function $\Box$ is defined as $\Box a=a(1+\xi_a)$ and $\Box b= b(1+\xi_b)$