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A capital delta ($\Delta$) is commonly used to indicate a difference (especially an incremental difference). For example, $\Delta x = x_1 - x_0$

My question is: is there an analogue of this notation for ratios?

In other words, what's the best symbol to use for $[?]$ in $[?]x = \dfrac{x_1}{x_0}$?

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What you have there is $1+\dfrac{\Delta x}{x_0}$, i.e. one more than the proportionate change, which tends to $1$ as the change gets smaller. I suspect that in most cases you may actually want the proportionate change itself. –  Henry Jun 11 '11 at 13:58
    
@Henry if you'd like some context, I'm using this to express changes in a product in terms of changes of its individual terms. The reason I'm using ratios is that the product of ratios is the ratio of the products. –  trutheality Jun 11 '11 at 14:08
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2 Answers

up vote 6 down vote accepted

Not entirely standard, but in Peter Henrici's discussion of the (justly famous) quotient-difference (QD) algorithm in the books Elements of Numerical Analysis (see p. 163) and Essentials of Numerical Analysis (see p. 155), he defines the quotient operator as

$$Q\,x_n=\frac{x_{n+1}}{x_n}$$

in complete analogy with the (forward) difference operator $\Delta$.

Henrici's a pretty sharp mathematician, so I wouldn't mind borrowing notation from him if I were in your shoes...


Here's a screenshot of the relevant page of the first book (sorry, I don't have a digital copy of the other book):

Henrici

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The best symbol to use is $\exp\Delta\log$: $$\exp\Delta\log x = \exp(\log x_1-\log x_0) = \frac{x_1}{x_0}.$$ The point is that this operation isn't "qualitatively different" from $\Delta$, so it may be reduced to $\Delta$. So far, I haven't used any new symbols but if you want some multiplicative new creative symbols, see e-percentages and units of evidence:

http://motls.blogspot.com/2010/01/exponential-percentages-useful-proposed.html
http://motls.blogspot.com/2010/08/units-of-evidence.html

There is no compact symbol for $\exp\Delta\log$. If you want an ally who would endorse the idea to introduce such a symbol, you may count on me. What about $\Delta^\times$?

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What if $x_1$ and $x_0$ are negative? Also, $\Delta^{\times}$ doesn't seem particularly good, as it mixes the D in difference with multiplication. –  t.b. Jun 11 '11 at 15:06
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Dear Theo, $\log(-1)=\pi i$ and $\exp(\pi i)=-1$: is there any problem with that? Something's changing multiplicatively but changing sign would be a discontinuous process in the real numbers, anyway, so it only makes good sense in the complex realm. Concerning the notation, not sure whether I understand which $D$ you mean. –  Luboš Motl Jun 11 '11 at 18:01
    
Dear Luboš, 1) no, no problem with that, but it might be worth pointing out. The process is not really discontinuous, but simply undefined at $0$, as it should be. 2) The $\Delta$ is a Greek D(elta) for the D in difference. –  t.b. Jun 11 '11 at 18:08
    
Dear Theo, what I mean by "discontinuous" is that there is no continuous $f(x)$ such that $f(0)$ is positive and $f(1)$ is negative so that $f(k)/f(0)$ which is $\Delta^\times x$ at some moment would be well-defined for all $k$ between $0$ and $1$. That's a warning sign - if one uses $\Delta^\times$ for things that change sign, it could be an unnatural thing that can go awry at moment... I know that $\Delta$ is the Greek counterpart of $D$ but I think it's a good idea to distinguish them. Did your $D$ mean $\Delta$? –  Luboš Motl Jun 12 '11 at 4:02
    
Thank you for the input, I was hoping there was something more compact than $\exp \Delta \log$ and while $\Delta^{\times}$ has some appeal it may confuse some readers because of the association of $\Delta$ with differences. I think I might just end up switching all the expressions to log-space or coming up with some other symbol (perhaps an unambiguous placement of $\div$?) –  trutheality Jun 12 '11 at 23:44
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