AFAIK this method is quite slow for larger matrices, and best algorithms are those published by N. Higham and coauthors, as posted above by Farid. NB, TensorFlow also uses a similar algo: https://www.tensorflow.org/api_docs/python/tf/linalg/expm

- Cuchulainn
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Yes, but I prefer to ask the experts directly.

- katastrofa
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Is the question in physics? Infinitesimal symmetry transformations? exp(a*H) = 1+ a*H + O(H^2)?

No, another thing we talked about

- katastrofa
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Oh, OK. That's just 99 topics to parse (from this month)...

- Cuchulainn
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katastrofa wrote:Is the question in physics? Infinitesimal symmetry transformations? exp(a*H) = 1+ a*H + O(H^2)?

https://www.tu-braunschweig.de/Medien-D ... cture6.pdf

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

- katastrofa
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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.

- Cuchulainn
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ISayMoo wrote: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.

katastrofa wrote: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.

One respected (?) Polish professor of physics said in my presence "if the 2nd order expansion is not enough then the 4th order one won't suffice either, and you should just give up"

Cuchulainn wrote:ISayMoo wrote: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...

- Cuchulainn
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ISayMoo wrote:Cuchulainn wrote:ISayMoo wrote:

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.

I know. I was hoping for some $MAGIC cancellation.

- Cuchulainn
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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

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