# low_rank_smith¶

- Low-rank smith algorithm for Stein equation
A.T X A - X = -Q Q.T

The algorithm can only be used if T is symmetric positive-definite, but this is not checked in this routine for computational performance. The solution X is provided in its factorised form:

X=Z Z.T

As in the most general case, a solution X exists only if the eigenvalues of S are stricktly smaller than one, and the algorithm will not converge otherwise. The algorithm can not exploits parsity, hence, while convergence can be improved for very large matrices, it can not be employed if matrices are too large to be stored in memory.

Parameters: - tol: tolerance for stopping convergence of Smith algorithm - Square: if true the squared-Smith algorithm is used - tolSVD: tolerance for reduce Z matrix based on singular values - kmax: if given, the Z matrix is forced to have size kmax - tolAbs: if True, the tolerance - fullOut: not implemented - Convergence: ‘Zk’,’res’.

If ‘Zk’ the iteration is stopped when the inf norm of the incremental

matrix goes below tol. - If ‘res’ the residual of the Lyapunov equation is computed. This strategy may fail to converge if kmax is too low or tolSVD too large!

Ref. P. Benner, G.E. Khoury and M. Sadkane, “On the squared Smith method for large-scale Stein equations”, 2014.