  We study a stability preserved Arnoldi algorithm for matrix exponential in
the time domain simulation of large-scale power delivery networks (PDN), which
are formulated as semi-explicit differential algebraic equations (DAEs). The
solution can be decomposed to a sum of two projections, one in the range of the
system operator and the other in its null space. The range projection can be
computed with one shift-and -invert Krylov subspace method. The other
projection can be computed with the algebraic equations. Differing from the
ordinary Arnoldi method, the orthogonality in the Krylov subspace is replaced
with the semi-inner product induced by the positive semi-definite system
operator. With proper adjustment, numerical ranges of the Krylov operator lie
in the right half plane, and we obtain theoretical convergence analysis for the
modified Arnoldi algorithm in computing phi-functions. Lastly, simulations on
RLC networks are demonstrated to validate the effectiveness of the Arnoldi
algorithm with structured-orthogonalization.
