This report presents several ideas for improving the stability of Gaussian reduction of an arbitrary real matrix to tridiagonal form. First, we analyze conditions under which reduction algorithms break down or become unstable. Second, we discuss how methods of threshold pivoting decrease the probability of these conditions occurring. Finally, we present new methods for recovering from breakdown when it does occur. The class of matrices that can be successfully reduced is significantly broadened by these new recovery algorithms. 16 refs., 4 figs.