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I can't post image with below-10 reputation. The image of the cited definition/problem description in the book is uploaded to Imgur and labeled as [Image:$\dots$]. Sorry for the inconvenience.

Context

I've been studying on my self on the part III (Conditioning and Stability) of the book Numerical Linear Algebra by Trefethen and Bau. The model of computation satisfies Image 13.5 and Image 13.7

After reading the definition in the book Image: 13.3, 13.4, other's post, and Nick Higham, my understanding of stability is that it's desirable if the value $\tilde{f}(x)$ ($\tilde{f}$ is the implementation of $f$ on computers.) suffered from rounding error in the operations in the algorithm (Forward error I think.) can be modeled as if the original problem $f(x)$ deviate it's input to neighboring $\tilde{x}$ satisfying (13.5) (Backward error) with small relative error. Additionally, backward stable means the relative error is actually zero.

Problem

Image: Exercise 15.1(a)~(f)

Image: Exercise 15.1(g)

  1. I don't know how to relate the method in (a), (b), (d) for (c) (If it's the right way).
  2. As you can see my understanding of stability comes with reformulate error into $\tilde{x}$ terms. How do I even discuss backward stability if there is no input at the first place like those in (e), (f), (g). What do I miss out on stability?

Attempts

  • (a): $\tilde{f}(x) = x \bigoplus x = (x+x)*(1+\epsilon) = 2[x(1+\epsilon)] = 2\tilde{x} = f(\tilde{x})$. Hence it's backward stable, which implies stable. (d) is similar, we can distribute the ratio into $x$.
  • (b): $\tilde{f}(x) = x \bigotimes x = (x^2)(1+\epsilon)$, which has the same order as $(x^2)(1+ 2\epsilon + \epsilon^2) = [x(1+\epsilon)]^2 = \tilde{x}^2 = f(\tilde{x})$. Hence it's backward stable, which implies stable.
  • (c) If it is $x/y$, I will use $(1+\epsilon) \approx (1+\epsilon)^2$ to argue that $\tilde{f}(x,y) = [x(1+\epsilon)]/[y(1-\epsilon)] = f(\tilde{x},\tilde{y})$, or simply $\tilde{f}(x,y) = [x(1+\epsilon)]/y = f(\tilde{x},y)$. However, x = y in (c), so I was wondering if I have to make the deviation the same, i.e. reformulate $x/x(1+\epsilon)$ into something like $[x(1+\epsilon_2)]/[x(1+\epsilon_2)] $

Thanks in advance for your consideration. p.s. This is my first post in Math Stackexchange. If there's any formatting/notation/layout suggestion on the writing please let me know.

user45277
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  • Please do not over use bold face text. – amWhy Feb 01 '22 at 17:33
  • This particular problem from Trefethen and Bau generates questions from time to time. It would be good to have the solution written out. Take the time to replace the images with text and equations so that it is readable and searchable. This will significantly increase the chance of getting a good answer. – Carl Christian Feb 02 '22 at 13:11
  • I understand. Thank you Christian I will edit the post as soon as I have enough reputation – user45277 Feb 03 '22 at 08:31

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