@Article{Deadman:2015:TMF,
  author =       "Edvin Deadman and Nicholas J. Higham",
  title =        "Testing Matrix Function Algorithms Using Identities",
  journal =      "{ACM} Transactions on Mathematical Software",
  volume =       "42",
  number =       "1",
  accepted =     "14 January 2015",
  upcoming =     "true", 
  abstract =     "
                  Algorithms for computing matrix functions are
                  typically tested by comparing the forward error
                  with the product of the condition number and the
                  unit roundoff. The forward error is computed with
                  the aid of a reference solution, typically computed
                  at high precision. An alternative approach is to
                  use functional identities such as the ``round
                  trip tests'' $e^{\log A} = A$ and $(A^{1/p})^p = A$, as
                  are currently employed in a SciPy test module. We
                  show how a linearized perturbation analysis for a
                  functional identity allows the determination of a
                  maximum residual consistent with backward stability
                  of the constituent matrix function evaluations.
                  Comparison of this maximum residual with a
                  computed residual provides a necessary test for
                  backward stability. We also show how the actual
                  linearized backward error for these relations can
                  be computed. Our approach makes use of Fr\'echet
                  derivatives and estimates of their norms. Numerical
                  experiments show that the proposed approaches are
                  able both to detect instability and to confirm
                  stability.",
}