Certainly better than the Taylor series I suggested above It's (not) funny, but the only time I actually used a Padé approximant was at the exam to the course in algorithmics by a Professor who looked a bit like Stallman, if you recollect the (in-)famous toe jam eating. Then I turned a physicist and my world was truncated at the first order term of the power series expansion. I once ventured an order farther and I still keep receiving inquiries if I've seen any dragons and sea monsters.

Would differentiating a Pade approximation of exp(H) over H_{kl} give a good approximation of d exp(H) / dH_{kl}? I suppose not...

No.

Would differentiating a Pade approximation of exp(H) over H_{kl} give a good approximation of d exp(H) / dH_{kl}? I suppose not...

No.

I thought so...

Take the Pade (0,1) (or was it (1,0) ?) [$]e^x[$] approximate by [$]1+x[$] on [-1,1]

1: Compute maximum error using 101 calculus
2. Compute derivative
3. GOTO 1

Gets worser and worser.

Thinking out loud (don't know if it will work, but it might). Differentiate 

du./dt = Au in time to get 

d^2u/dt^2 = d(Au)/dt (A = A(t) in general

Then write as an ODE system

dv/dt = Bv  where B is a nested matrix.

Only(?) issue is to compute du/dt at t = 0 (usually by heuristic handwaving) (*)

Boost odeint could test it..

(*) this is the trick used in Keller's box scheme
https://wwwf.imperial.ac.uk/~ajacquie/I ... uffyCN.pdf

You could try it in the 1d case and know soon enough if it works.
du/dx = u, u(0) = 1.
aka GO/NOGO

I'd rather not go down the ODE solving route

The reason being?
What about a home-grown solver?

No.

I thought so...

Take the Pade (0,1) (or was it (1,0) ?) [$]e^x[$] approximate by [$]1+x[$] on [-1,1]

1: Compute maximum error using 101 calculus
2. Compute derivative
3. GOTO 1

Gets worser and worser.

It's not a Padé approximant.

exp x = exp(x/2) / exp(-x/2) for Padé.

I thought so...

Take the Pade (0,1) (or was it (1,0) ?) [$]e^x[$] approximate by [$]1+x[$] on [-1,1]

1: Compute maximum error using 101 calculus
2. Compute derivative
3. GOTO 1

Gets worser and worser.

It's not a Padé approximant.

exp x = exp(x/2) / exp(-x/2) for Padé.

It is Pade table (1,0)
https://en.wikipedia.org/wiki/Pad%C3%A9_table
Not sure what your above identity is telling me.

There's no such a thing as Padé(.,0). It would kill the whole idea of the approximant. My identity was a hint how to calculate the Padé approximant of exp. Nevermind, I need to go and do something mathematically sound now.

Read the Wiki link.
The identity is cute but far away from rational approximation (IMO). But I might be wrong.Or you.

Feller was an ebullient man, who would rather be wrong than undecided. 

We essentially don't agree whether the Taylor series is a Padé approximant. I think the motivation for the latter is to achieve(/extend the radius of) or accelerate the convergence of the first, hence it has the form of a quotient of two Taylor series expansions:
[$]\sum_{i=0}^M a_i x^i / ( 1 + \sum_{j=1}^N a_j x^j )[$]

Thus the Pade approximant of exp x should be derived from exp(x/2) / exp(-x/2) = T(x/2) / T(-x/2).

I think ISayMoo's could use Padé approximant to calculate exp(A), but needs to assure that the norm of A is low (e.g. by applying the Gershgorin - a.k.a. some Belarusian guy who's name I forgot - circle theorem, which we have once intensely discussed in the forum with Lovenatalya).

BTW, why do you use the word "cute" so often when we talk?

This is the first time I ever used that word Cute foto.

Thus the Pade approximant of exp x should be derived from exp(x/2) / exp(-x/2) = T(x/2) / T(-x/2).
What's 'T'?

It's *relatively* often then.
T is for Taylor expansion.

Simple question which stumps me: I  have a complex square matrix H. There are some nice methods for calculating exp(H). What about calculating the derivative of exp(H) over elements of H? To be precise: let M = exp(H). I want to calculate dM_{jk} / dH_{mn} numerically, accurately and (relatively) quickly.

What about somefing on the lines of BCH?

https://math.stackexchange.com/question ... -of-matrix

A successful implementation would improve the code (e.g. maintainability and readability) for backpropagation etc.

Is what you call Higham's method simply a complex step derivative, which has been used by statisticians for sensitivity analysis since complex was added in the C99 standard? Who's Higham?