{\sl SRRIT} is a FORTRAN program to calculate an approximate orthonormal basis for a dominant invariant subspace of a real matrix $A$ by the method of simultaneous iteration \cite{stewart76a}. Specifically, given an integer $m$, {\sl SRRIT} attempts to compute a matrix $Q$ with $m$ orthonormal columns and real quasi-triangular matrix $T$ of order $m$ such that the equation [ AQ = QT ] is satisfied up to a tolerance specified by the user. The eigenvalues of $T$ are approximations to the $m$ largest eigenvalues of $A$, and the columns of $Q$ span the invariant subspace corresponding to those eigenvalues. {\sl SRRIT} references $A$ only through a user provided subroutine to form the product $AQ$; hence it is suitable for large sparse problems. (Also cross-referenced as UMIACS-TR-92-61)