Re: [eigen] Specialized QR

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On Wed, 20 May 2009, Benoit Jacob wrote:

Unless someone who knows better disagrees, I think that in plain QR
decompositions (not RRQR) it's pretty safe to assume that the Q matrix
is wanted.

I'm not so sure. Suppose you solve Ax = b via QR. The steps are: factor
A = QR, compute y = Q^{-1} b = Q^T b, solve the triangular system Rx = y. So you don't need Q, but only Q^T b. To form Q, you apply O(n^2) Givens rotations to the identity matrix, total cost O(n^3). But you can compute Q^T b by applying the Givens rotations directly to the vector b, for a cost of only O(n^2).

I'm not sure about the details. And I don't know a use case for solving Ax = b by QR instead of LU. But I note that LAPACK has routines for the factorization (with and without pivoting) without computing Q, for computing Q, and for computing Q (or Q^T) times a vector [1]. And Golub & van Loan say something similar in the context of Householder QR for least squares problems.


Givens rotations, that is the loop

for (int i = k1; i < cols; ++i){
  tmp = m_R.coeff(k1, i);
  m_R.coeffRef(k1, i) = ei_conj(o1)*m_R.coeff(k1, i)
                        + ei_conj(o2)*m_R.coeff(k2, i);
  m_R.coeffRef(k2, i) = o1*m_R.coeff(k2, i) - o2*tmp;

are used in many algorithms, for instance SVD (in the algorithm we're using at the moment) and GMRES (iterative method for solving Ax = b). So if possible we should have an optimized version in a separate function.


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