Hello,
I was curious about the pivoting strategy of Eigen's Full Pivoting Householder QR algorithm. I wrote a little test program and debugged throug the decomposition computation. What I found out is, that the pivoting strategy is exactly the same as in a full pivoting gaussian elimination:

1. Search for the element "P" with the biggest absolute value in the bottom right sub-matrix
2. Interchange rows and columns so that "P" is in the upper left corner

I write this post because I wonder if that simple strategy is really adequate for householder QR decompositions. I would rather expect to select the columns by the biggest squared norm and then to select the rows so that the off-diagonal elements in the tail of the row are as small as possible in comparison to the diagonal element.

Could you give me a reference to a paper or to the literature on which your algorithm is based?

Best Regards
Piotr Trebisz

Indeed, looking at the code it seems that it behaves as you described. I agree with you that it should start by looking at the column with biggest norm (as in the column pivoting strategy) and then search within this column for row interchange. Patch welcome