  We consider the computation of the matrix logarithm by using numerical
quadrature. The efficiency of numerical quadrature depends on the integrand and
the choice of quadrature formula. The Gauss--Legendre quadrature has been
conventionally employed; however, the convergence could be slow for
ill-conditioned matrices. This effect may stem from the rapid change of the
integrand values. To avoid such situations, we focus on the double exponential
formula, which has been developed to address integrands with endpoint
singularity. In order to utilize the double exponential formula, we must
determine a suitable finite integration interval, which provides the required
accuracy and efficiency. In this paper, we present a method for selecting a
suitable finite interval based on an error analysis as well as two algorithms,
and one of these algorithms addresses error control.
