  Two quadrature-based algorithms for computing the matrix fractional power
$A^\alpha$ are presented in this paper. These algorithms are based on the
double exponential (DE) formula, which is well-known for its effectiveness in
computing improper integrals as well as in treating nearly arbitrary endpoint
singularities. The DE formula transforms a given integral into another integral
that is suited for the trapezoidal rule; in this process, the integral interval
is transformed to the infinite interval. Therefore, it is necessary to truncate
the infinite interval into an appropriate finite interval. In this paper, a
truncation method, which is based on a truncation error analysis specialized to
the computation of $A^\alpha$, is proposed. Then, two algorithms are presented
-- one computes $A^\alpha$ with a fixed number of abscissas, and the other
computes $A^\alpha$ adaptively. Subsequently, the convergence rate of the DE
formula for Hermitian positive definite matrices is analyzed. The convergence
rate analysis shows that the DE formula converges faster than the Gaussian
quadrature when $A$ is ill-conditioned and $\alpha$ is a non-unit fraction.
Numerical results show that our algorithms achieved the required accuracy and
were faster than other algorithms in several situations.
