 
Summary: RESIDUAL, RESTARTING AND RICHARDSON ITERATION FOR
THE MATRIX EXPONENTIAL
MIKE A. BOTCHEV
To the memory of my father
Abstract. A wellknown problem in computing some matrix functions iteratively is the lack of
a clear, commonly accepted residual notion. An important matrix function for which this is the case
is the matrix exponential. Suppose the matrix exponential of a given matrix times a given vector
has to be computed. We develop the approach of Druskin, Greenbaum and Knizhnerman (1998)
and interpret the soughtafter vector as the value of a vector function satisfying the linear system
of ordinary differential equations (ODE) whose coefficients form the given matrix. The residual is
then defined with respect to the initialvalue problem for this ODE system. The residual introduced
in this way can be seen as a backward error. We show how the residual can be computed efficiently
within several iterative methods for the matrix exponential. This completely resolves the question
of reliable stopping criteria for these methods. Further, we show that the residual concept can be
used to construct new residualbased iterative methods. In particular, a variant of the Richardson
method for the new residual appears to provide an efficient way to restart Krylov subspace methods
for evaluating the matrix exponential.
Key words. matrix exponential, residual, Krylov subspace methods, restarting, Chebyshev
polynomials, stopping criterion, Richardson iteration, backward stability, matrix cosine
AMS subject classifications. 65F60, 65F10, 65F30, 65N22, 65L05
