Algorithms for the case of a symmetrical full matrix
Calculation of eigenvalues

In this section, we adapt the orthogonal transformation introduced in to the case where the starting matrix is symmetrical. Any orthogonal transformation Q ^T AQ of the matrix A then remains symmetrical. The final matrix obtained by the procedure is therefore both upper Hessenberg and symmetrical: it is a symmetrical tridiagonal matrix.

When applying Householder transformations, symmetry can be taken into account to reduce the number of operations required. In particular, this can be achieved by using the BLAS library procedure, which updates a matrix with a rank-2 correction. This is programmed in the DSYTRD procedure of the LAPACK library