FWL: So the word / composition of R_i is required to have bounded length, of order n_C_2 . Iterative processes such as Jacobi or Givens reduction (which anyway apply to symmetric rather than orthogonal matrices) seem unserviceable in this context. WDS: No no no. First, given an orthogonal matrix one can, in a NON iterative, totally deterministic totally obvious manner, apply Givens rotations to it from the left to zero the lower triangle elements one by one in raster scan order going down columns. Now it is lower triangular orthogonal, which means it is the identity. OK? So in this way any orthogonal matrix is a product of Givens 2x2s. Second, an orthogonal nXn obviously has (n-1)*n/2 degrees of freedom, and hence cannot be factored into fewer than that number of Givens 2x2 matrices (since each contains one degree of freedom) unless we were doing something weird like a spacefilling curve (which it seems plain we aren't), and this procedure meets that bound so bound is tight. Now in my first post I described a way to do it using only ADJACENT Givens 2x2s, i.e. only involving coordinates i and i+1. That involved n*(n-1)/2 having continuously variable angles, but a cubic(n) number of others each with fixed angle=pi. So it fails to meet Lunnon's desire for a quadratic(n) number, but that might merely be because I was being very crude in my approach. OK, here, try this. Start with an orthogonal nXn. Now zero the lower triangle in the order shown: 7 48 259 136... each by means of one adjacent-Givens applied from the left. Doesn't that work, and further do the whole job with only (n-1)*n/2 of them? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)