The orthogonal qd-algorithm is presented to compute the singular value decomposition of a bidiagonal matrix. This algorithm represents a modification of Rutishauser's qd-algorithm, and it is capable of determining all the singular values to high relative precision. A generalization of the Givens transformation is also introduced, which has applications besides the orthogonal qd-algorithm. The shift strategy of the orthogonal qd-algorithm is based on Laguerre's method, which is used to compute a lower bound for the smallest singular value of the bidiagonal matrix. Special attention is devoted to the numerically stable evaluation of this shift. (Also cross-referenced as UMIACS-TR-94-9.1)