There are many equivalent ways of defining a function $f(A)$ of a matrix $A$. We focus on Jordan canonical form, Hermite interpolation and Cauchy integral.
What is the difference between methods for defining a matrix function (in applications)? Which method can be evaluated more efficiently? Can you give references?
None: they are all equivalent and all three return the same result for any function that's defined on the spectrum of $A$. (Theorem 1.12 in Higham's book Matrix Functions.) The only minor drawback of the Cauchy integral one is that you need the function to be analytic on a suitable region including the eigenvalues for it to make sense.
I'm not sure why you care about definitions "in application"; they are just definitions, and typically that's not how you compute them. If you are interested in their effectiveness as a method to actually compute the matrix function, then that's another question. :)
A rather complete overview of methods (20 in total!) to compute the exponential function of a matrix is given in Nineteen Dubious Ways to Compute the Exponential of a Matrix (2003). Much of what is said there applies to other functions. The first question to ask is whether you need the full matrix $f(A)$ or only its application to a vector. If $A$ is sparse then the Krylov method is recommended (number 20 on the list). The paper also contains an extensive bibliography for each method.
