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I am currently researching a QR-based root finding algorithm encountered two operations that I don't understand. I'd love to look them up, but I can't find the names of these operations/notations.

The first was an unfamiliar usage of a colons and commas in a possibly set related way - (1 : 2, 1). The term seemed to be expressed as the coefficient to another term (I.E. F(1 : 2, 1)). I later encountered a similar term (1 : 2, 2 : 3) that also seemed represented as a coefficient to a variable.

NOTE: After running a few more searches and rereading the article I encountered this on, I realized this notation may be matrix related rather than set related. F is actually a unitary matrix used to generate another matrix, it seems. I thought these might be matrix indices, but the typical notation for this used in the article is two comma separated subscript numbers.

Specifically, I encountered this in equation (21) under Theorem 2.8 of the paper A fast implicit QR eigenvalue algorithm for companion matrices. The equation describes a vector hk as the result of a formula Fk (1 : 2, 1) and a matrix Bk + 1 as the result of an equation Fk (1 : 2, 2 : 3).

The second is the one I had trouble running a search on. It seemed to be two terms, one over top of the other, in parenthesis as if functioning as a coefficient. I came up with the term "binomial coefficient" at some point, but I believe this might not be correct.

I'd settle for just the names of these two operations/notations, but if anyone is willing to explain what exactly they mean, it'd be appreciated. Thank you for taking the time to read this question. I apologize if this is not a suitable question for this forum or the details are inadequate.

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    I have updated the question to include the information requested. I'm sorry that I had not previously included this.2012-03-13

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Thanks for linking the source. It looks to me like this notation is used to refer to submatrices. So if $A$ is a matrix, $A(i:j,k:l)$ is the $(j-i+1) \times (l-k+1)$ matrix consisting of rows $i$ through $j$ and columns $k$ through $l$ of $A$. Variations are $A(i, k:l)$ to refer to the $i$th row, columns $k$ through $l$, and $A(i, :)$ to refer to the $i$th row, all columns.

Also, it appears that notation like $\begin{pmatrix} A \\ B \end{pmatrix}$ refers to the block matrix whose first rows come from $A$ and whose last rows come from $B$.

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    The "colon" notation here is inherited from MATLAB; in fact, books on numerical linear algebra like Golub and Van Loan's sometimes borrows idioms from MATLAB to express algorithms compactly. This paper seems to be no exception.2018-03-28
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Regarding your second one, I'm not sure what kind of search you tried: did you try Wikipedia for "binomial coefficient"?

The definition is $ {n\choose{k} }=\frac{n!}{k!(n-k)!}, $ usually refered in English as "$n$ choose $k$". They appear very naturally in combinatorics, but they get their name from the binomial formula $ (a+b)^n=\sum_{k=0}^n{n\choose k}a^kb^{n-k} $

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    Thank you for your answer. It seems I made a misconception about what exactly a binomial coefficient was.2012-03-12