More accurate bidiagonal reduction for computing the singular value decomposition (Q2784379)

As a preliminary stage for computing the singular value decomposition of a matrix \(A \in R^{m \times n}\) (\(m \geq n\)) the reduction of \(A\) to a bidiagonal form is discussed, i.e. one has to find orthogonal matrices \(U \in R^{n \times n}\) and \(V \in R^{m \times m}\) such that \(U^T A^T V = (B 0)\) with a bidiagonal \((n \times n)\) matrix \(B\). Then, several algorithms for computing the singular value decomposition of bidiagonal matrices are known. NEWLINENEWLINENEWLINEThe reduction algorithm consists of two steps. In the first step the matrix \(A\) is reduced to a lower triangular matrix \(C \in R^{n \times n}\) by using a Householder factorization method. In the second step a new bidiagonal reduction algorithm is applied to the matrix \(C\). In the new algorithm Givens rotations instead of Householder transformations are used in the construction of the right orthogonal transformation matrix. A detailed error analysis of the new algorithm is presented. The proposed algorithm allows to compute the singular value decomposition with more guaranteed accuracy than by standard bidiagonal reduction procedures.