A QR-method for computing the singular values via semiseparable matrices

A QR–method for computing the singular values via semiseparable matrices. Abstract The standard procedure to compute the singular value decomposition of a dense matrix, first reduces it into a bidiagonal one by means of orthogonal transformations. Once the bidiagonal matrix has been computed, the QR–method is applied to reduce the latter matrix into a diagonal one. In this paper we propose a new method for computing the singular value decomposition of a real matrix. In a first phase, an algorithm for reducing the matrix A into an upper triangular semisep-arable matrix by means of orthogonal transformations is described. A remarkable feature of this phase is that, dependding on the distribution of the singular values, after few steps of the reduction, the large singular values are already computed with a precision depending on the gaps between the singular values. An efficient implementation of the implicit QR–method for upper triangular semiseparable matrices is derived and applied to the latter matrix for computing its singular values. The numerical tests show that the proposed method can compete with the methods available in the literature for computing the singular values of a matrix. A QR–method for computing the singular values via semiseparable matrices Abstract The standard procedure to compute the singular value decomposition of a dense matrix, £rst reduces it into a bidiagonal one by means of orthogonal transformations. Once the bidiagonal matrix has been computed, the QR–method is applied to reduce the latter matrix into a diagonal one. In this paper we propose a new method for computing the singular value decomposition of a real matrix. In a £rst phase, an algorithm for reducing the matrix A into an upper triangular semiseparable matrix by means of orthogonal transformations is described. A remarkable feature of this phase is that, depending on the distribution of the singular values, after few steps of the reduction, the large singular values are already computed with a precision depending on the gaps between the singular values. An ef£cient implementation of the implicit QR–method for upper triangular semiseparable matrices is derived and applied to the latter matrix for computing its singular values. The numerical tests show that the proposed method can compete with the methods available in the literature for computing the singular values of a matrix.