  For solving the continuous Sylvester equation, a class of the multiplicative
splitting iteration method is presented. We consider two symmetric positive
definite splittings for each coefficient matrix of the continuous Sylvester
equations and it can be equivalently written as two multiplicative splitting
matrix equations. When both coefficient matrices in the continuous Sylvester
equation are (non-symmetric) positive semi-definite, and at least one of them
is positive definite; we can choose Hermitian and skew-Hermitian (HS)
splittings of matrices $A$ and $B$, in the first equation, and the splitting of
the Jacobi iterations for matrices $A$ and $B$, in the second equation in the
multiplicative splitting iteration method. Convergence conditions of this
method are studied and numerical experiments show the efficiency of this
method.
