Summary: Most methods for calculating the SVD (singular value decomposition) require to first bidiagonalize the matrix. The blocked reduction of a general, dense matrix to bidiagonal form, as implemented in ScaLAPACK, does about one half of the operations with BLAS3. By subdividing the reduction into two stages dense \(\rightarrow\) banded and banded \(\rightarrow\) bidiagonal with cubic and quadratic arithmetic costs, respectively, we are able to carry out a much higher portion of the calculations in matrix–matrix multiplications. Thus, higher performance can be expected. This paper presents and compares three parallel techniques for reducing a full matrix to banded form. (The second reduction stage is described in another paper [B. Lang, Parallel Comput. 22, 1–18 (1996; Zbl 0873.65044)].) Numerical experiments on the Intel Paragon and IBM SP/1 distributed memory parallel computers demonstrate that the two-stage reduction approach can be significantly superior if only the singular values are required.