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Suppose we are working with a machine that does arithmetical calculations with a relative accuracy of $\xi, |\xi| \leq \xi '$, if we are working with machine numbers. We want to calculate the inner product $\langle x , y \rangle = \sum_{i=1}^{n} x_i y_i$. I want to prove the claim:

$$\langle x , y \rangle ' \equiv \langle x,y \rangle + \langle |x|, |y| \rangle n \xi$$

Suppose $n=2$. We have $\langle x , y \rangle ' = (x_1y_1(1 + \xi_1) + x_2y_2(1+\xi_2))(1 + \xi_3) = x_1y_1 + x_1y_1\xi_1 + x_2y_2 + x_2y_2\xi_2 + x_1y_1\xi_3 + x_1y_1\xi_1\xi_3 + x_2y_2\xi_3 + x_2y_2\xi_2\xi_3$.

If we ignore the higher order terms we get $x_1y_1 + x_1y_1\xi_1 + x_2y_2 + x_2y_2\xi_2 + x_1y_1\xi_3 + x_1y_1\xi_1\xi_3 + x_2y_2\xi_3$

which is $\langle x, y \rangle + x_1y_1(\xi_1 + \xi_3) + x_2y_2(\xi_2 + \xi_3)$. And this is the part where I get stuck. The term $x_1y_1(\xi_1 + \xi_3) + x_2y_2(\xi_2 + \xi_3)$ is close to the desired result but that's all it is. Where is my mistake?

user119470
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1 Answers1

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$$ |x_1y_1(\xi_1+\xi_3)+x_2y_2(\xi_2+\xi_3)|\le |x_1||y_1||\xi_1+\xi_3|+|x_2||y_2||\xi_2+\xi_3|\le (|x_1||y_1|+|x_2||y_2|)2\xi=\langle |x|,|y|\rangle 2\xi$$

Hagen von Eitzen
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  • This only proves that $\langle x , y \rangle ' \leq \langle x,y \rangle + \langle |x|, |y| \rangle 2 \xi$ right? – user119470 Jan 27 '14 at 17:08