Considering the problem, $$\text{Evaluate } \log_{10} (7)$$
Assuming that a numerical algorithm was used to obtain an approximate value of $0.8$,
Find the perturbation of the argument to the original function, so that the perturbed problem has $0.8$ as its exact value? That, is, how to find a $δ$ such that $\log_{10} (7 + δ) = 0.8$
So, what I worked out so far was:
$\log_{10} (7) = 0.84509804...$
$\log_{10} (x) = 0.8$
$x= 10^{0.8} = 6.309573...$
$\log_{10} (10^{0.8})= \log_{10}\ 6.309573...$
$7 + δ = 6.3095345\ldots$
$δ = 6.3095345\ldots- 7$
I was just wondering whether the method I used above would be the correct way of approaching the solution and if there was any way to present the solution better.
I would be grateful is anyone could also confirm whether:
- the backward error for the problem above would be $δ$;
- and if, the relative backward error would then be $\frac{δ}{7}$
Thank you.
Yes, you have the right idea and the right result. However, this is not necessarily enough. You have written a sequence of mathematical statements which are not tied together using logical symbols or written sentences. I know what you mean, but you are entirely dependent on your instructor being willing to fill in the blanks. It is your instructors job to certify that you know why your reasoning is correct and this is not possible here.
Consider a the following alternative. Let $f : (0,\infty) \rightarrow \mathbb{R}$ be given by $$f(x) = \log_{10}(x).$$ Suppose that the computed value $\hat{y}$ of $y=f(x)$ is given by $\hat{y}$ for $x=0.8$. Find the backward error and the relative backward error in this case.
By definition, the absolute backward error is given by $$ \omega(\hat{y}) = \min \{ \delta > 0 : \exists \Delta x \: : \: \hat{y} = f(x+\Delta x) \wedge |\Delta x| \leq \delta \}.$$ Now since the $f$ is a bijection with inverse $g : \mathbb{R} \rightarrow \mathbb{R}$ given by $g(x) = 10^x$ we have $$\hat{y} = f(x + \Delta x)$$ if and only if $$x + \Delta x = g(\hat{y}).$$ It follows that $$\Delta x = g(\hat{y}) - x = 10^{0.8} - 0.7 \approx -0.69042655519.$$ This shows that the absolute backward error is given by $$\omega(y) = |g(\hat{y}) - x| \approx 0.69042655519.$$
By definition, the relative backward error is given by $$\eta(\hat{y}) = \min \{ \delta > 0 : \exists \Delta x \: : \: \hat{y} = f(x+\Delta x) \wedge |\Delta x| \leq \delta |x|\}.$$ This definition requires $|x| \not = 0$. By the previous analysis there is only one choice for $\Delta x$ and we find that the relative backward error is $$\eta(\hat{y}) = \frac{|g(\hat{y}) - x|}{|x|} \approx 0.09863236502.$$
The main difference is the reference to original definitions as wells as the key property of the function $f$ which allows us to isolate a single value of $\Delta x$ such that $\hat{y} = f(x+\Delta x)$. It is entirely possible that your instructor is not as fanatical as mine were, but this level of details will make it clear to the reader that you know what you are doing and why it is correct.