Fast algorithms to QR factor circulant matrices

The authors introduce a fast algorithm for computing the QR factors of a complex vertically circulant matrix C. The complexity of the algorithm is 5mn+2n²+O( m) complex-fixed point operations or 3mn+n ²+O(m) complex floating-point operations, where m is the number of rows in C and n is the number of columns (it is supposed that m<n). This algorithm can be used to solve overdetermined systems in the least-squares sense and to find an orthogonal basis spanning the same subspace as the columns of C. The matrices Q and R can be computed independently if desired. The authors also introduce a block QR factorization with two-by-two blocks, which is based on the split Levinson algorithm