QR factorization for row or column symmetric matrix

The problem of fast computing the QR factorization of row or column symmetric matrix is considered. We address two new algorithms based on a correspondence of Q and R matrices between the row or column symmetric matrix and its mother matrix. Theoretical analysis and numerical evidence show that, for a class of row or column symmetric matrices, the QR factorization using the mother matrix rather than the row or column symmetric matrix per se can save dramatically the CPU time and memory without loss of any numerical precision.