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

Question

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

Answer 1

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.

Answer 2

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.