Distribution of matrix nonzeros
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A few months ago, frmky and I were looking for a new angle on matrix multiplies, and he noticed something peculiar when plotting a histogram of the column counts of nonzeros in the matrix for 2,1036+. You would expect a smooth distribution of light columns that peaks at the matrix average, and then trails off into a smaller number of heavier columns.
What we actually see is attached. Why do you think it is that we get 'beats' at the lowest column weights? The matrix average is 81 nonzeros per column; if I had to make up a reason, I would think that the relations that survive the filtering have a bias towards an odd or even number of ideals, and combining a small number of relations into a matrix column magnifies that bias. But it still mystifies me; really I'm worried that the NFS filtering in msieve is actually doing something dumb that I don't know about :) I'm also reminded for some reason of the 'minimax' phenomenon in function approximation, where the approximation with the lowest worstcase error will alternately overshoot and undershoot a true value. 
[QUOTE=jasonp;235808]A few months ago, frmky and I were looking for a new angle on matrix multiplies, and he noticed something peculiar when plotting a histogram of the column counts of nonzeros in the matrix for 2,1036+. You would expect a smooth distribution of light columns that peaks at the matrix average, and then trails off into a smaller number of heavier columns.
What we actually see is attached. Why do you think it is that we get 'beats' at the lowest column weights? The matrix average is 81 nonzeros per column; if I had to make up a reason, I would think that the relations that survive the filtering have a bias towards an odd or even number of ideals, and combining a small number of relations into a matrix column magnifies that bias. But it still mystifies me; really I'm worried that the NFS filtering in msieve is actually doing something dumb that I don't know about :) I'm also reminded for some reason of the 'minimax' phenomenon in function approximation, where the approximation with the lowest worstcase error will alternately overshoot and undershoot a true value.[/QUOTE] Keep in mind that for the algebraic norms, not every prime is in the factor base. The algebraic polynomial must have a root mod p. Some primes are excluded ....... 
A more pedestrian explanation would be that most relations have about the same number of factors, but only whole relations are merged together to form matrix columns, so it makes sense that there are large numbers of matrix columns with about the same weight, separated by troughs that are 1015 factors wide :)

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