# Is there any convenient notation for a vector entry j at iteration step i?

I have a stupid question: I'm writing a master thesis and I have to describe an iterative method of an algorithm. The method uses vectors which are manipulated at each iteration step.

Now my question: is there any convenient method of writing for example entry $j$ of vector $\mathbf{a}$ on iteration step $k$?

I thought of $\mathbf{a}^k_j$ or $(\mathbf{a}_k)_j$ or $\mathbf{a^{\langle k \rangle}_j}$ but these are all quite ambiguous.

Any ideas?

With kind regards,

Filip M

• That's not a stupid question. – GFauxPas Mar 13 '15 at 21:55
• With the possible exception of the latter of the three, they all seem like reasonable notations. As long as you make it clear in the text and the notation is unambiguous you're good, I'd say. – Dan Mar 13 '15 at 21:57
• I would prefer $a_j^{(k)}$. Another variant would be $a^{(k)}\cdot e_j$. – mvw Mar 13 '15 at 22:05
• My preference would be for $a_j^{(k)}$ as well. – Alijah Ahmed Mar 13 '15 at 22:09
• All seem to work, so long as you stay consistent. And not just with notation for this scenario; for example, if you're going to use $(a_k)_j$ to denote entry $j$ at iteration step $k$, then I suggest you use $(b)_j$ for entry $j$ for any vector $b$. – Gyu Eun Lee Mar 13 '15 at 22:32

Presumably you already have a notation to indicate the $j$-the element of any vector $\mathbf{v}$, and you should be consistent with this.

Also, for this particular algorithm (or for iterative algorithms in general), you probably have a way to indicate the value of variable $x$ during the $k$-th iteration. You ought to be consistent with this, too.

My personal choices would be:

(1) $\mathbf{v}[j]$ denotes the $j$-the element of the vector $\mathbf{v}$.

(2) $x_k$ denotes the value of the variable $x$ during the $k$-th iteration.

So, putting these two together, $\mathbf{a}_k[j]$ or $\mathbf{a}[j]_k$ is the value of $\mathbf{a}[j]$ during the $k$-th iteration. I prefer the former.

There are many other choices, of course; it's the consistency that's important, in my view.

Thanks everyone for all the answers!

I have chosen for $\mathbf{a}_j^{(k)}$ because after consulting some books on numerical analysis, most books use more or less the same notation. The brackets in the exponent clearly show that it is not a power or something and I didn't want to change the very convenient way to indicate the $j$th-element of a vector.