Preference for row ordering is more related to the way people view tabular data, I think. For many applications/data, there are many more rows than columns (think of database tables or csv files), and it's slightly unnatural to read those into a transposed data structure for analysis. Putting a time series into a TxN matrix is natural (that's likely how the data is stored), but inefficient if data access is sequentially through time. Without a "TransposeView" or similar, we're forced to make a choice between poor performance or an unintuitive representation of the data.
I can appreciate that matrix operations could be more natural with column ordering, but in my experience practical applications favor row ordering. On Fri, Oct 30, 2015 at 11:44 AM, John Gibson <johnfgib...@gmail.com> wrote: > Agreed w Glenn H here. "math being column major" is because the range of a > matrix being the span of its columns, and consequently most linear algebra > algorithms are naturally expressed as operations on the columns of the > matrix, for example QR decomp via Gramm-Schmidt or Householder, LU without > pivoting, all Krylov subspace methods. An exception would be LU with full > or partial row pivoting. > > I think preference for row-ordering comes from the fact that the textbook > presentation of matrix-vector multiply is given as computing y(i)= sum_j > A(i,j) x(j) for each value of i. Ordering the operations that way would > make row-ordering of A cache-friendly. But if instead you understand > mat-vec mult as forming a linear combination of the columns of A, and you > do the computation via y = sum_j A(:,j) x(j), column-ordering is > cache-friendly. And it's the latter version that generalizes into all the > important linear algebra algorithms. > > John > > > On Friday, October 30, 2015 at 10:46:36 AM UTC-4, Glen H wrote: >> >> >> On Thursday, October 29, 2015 at 1:24:23 PM UTC-4, Stefan Karpinski wrote: >>> >>> Yes, this is an unfortunate consequence of mathematics being >>> column-major – oh how I wish it weren't so. The storage order is actually >>> largely irrelevant, the whole issue stems from the fact that the element in >>> the ith row and the jth column of a matrix is indexes as A[i,j]. If it were >>> A[j,i] then these would agree (and many things would be simpler). I like >>> your explanation of "an index closer to the expression to be evaluated >>> runs faster" – that's a really good way to remember/explain it. >>> >>> >> To help understand, is "math being column major" referring to matrix >> operations in math textbooks are done by columns? For example: >> >> >> http://eli.thegreenplace.net/2015/visualizing-matrix-multiplication-as-a-linear-combination/ >> >> While the order is by convention (eg not that is has to be that way), >> this is how people are taught. >> >> Glen >> >
