A \(QR\) decomposition for matrix pencils (Q1569900)

An efficient and numerically stable modification of the \(QR\) decomposition for solving a linear least squares problem with a matrix of the form \(A+\lambda B\) is given. The idea is to proceed by columns and in step \(i\) the algorithm is driven by data from column \(i\) of the transformed matrices \(B\) and \(A\) in turn. The resultant matrix is staircase triangular with \(2i\) nonzeros at most in column \(i\). For a given value of \(\lambda\) it is transformed via Givens rotation into an upper triangular matrix.