  In this paper, we perform a roundoff error analysis of an integration-based
method for computing the matrix sign function recently proposed by Nakaya and
Tanaka. The method expresses the matrix sign function using an integral
representation and computes the integral numerically by the double-exponential
formula. While the method has large-grain parallelism and works well for
well-conditioned matrices, its accuracy deteriorates when the input matrix is
ill-conditioned or highly nonnormal. We investigate the reason for this
phenomenon by a detailed roundoff error analysis.
