Computing the matrix fractional power based on the double exponential formula
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Two quadrature-based algorithms for computing the matrix fractional power A^α 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^α, is proposed. Then, two algorithms are presented – one computes A^α with a fixed number of abscissas, and the other computes A^α 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 α is a non-unit fraction. Numerical results show that our algorithms achieved the required accuracy and were faster than other algorithms in several situations.
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