API proposal: derivative by list

[Francesco Bonazzi]

Hi, I think we could support a simple notation for array derivatives.

For example, if would be nice if the Gradient and Hessian could be easily calculated as:

In [5]: f(x, y, z).diff([[x, y, z]])
Out[5]:
⎡∂               ∂               ∂             ⎤
⎢──(f(x, y, z))  ──(f(x, y, z))  ──(f(x, y, z))⎥
⎣∂x              ∂y              ∂z            ⎦

In [6]: f(x, y, z).diff([[x, y, z], 2])
Out[6]:
⎡   2                  2                  2             ⎤
⎢  ∂                  ∂                  ∂              ⎥
⎢ ───(f(x, y, z))   ─────(f(x, y, z))  ─────(f(x, y, z))⎥
⎢   2               ∂y ∂x              ∂z ∂x            ⎥
⎢ ∂x                                                    ⎥
⎢                                                       ⎥
⎢   2                  2                  2             ⎥
⎢  ∂                  ∂                  ∂              ⎥
⎢─────(f(x, y, z))   ───(f(x, y, z))   ─────(f(x, y, z))⎥
⎢∂y ∂x                 2               ∂z ∂y            ⎥
⎢                    ∂y                                 ⎥
⎢                                                       ⎥
⎢   2                  2                  2             ⎥
⎢  ∂                  ∂                  ∂              ⎥
⎢─────(f(x, y, z))  ─────(f(x, y, z))   ───(f(x, y, z)) ⎥
⎢∂z ∂x              ∂z ∂y                 2             ⎥
⎣                                       ∂z              ⎦

That is, if the argument being derived wrt is a nested list, interpret the derivative as an array derivative.

This behaviour would then be similar to Wolfram Mathematica's one: http://reference.wolfram.com/language/ref/D.html

Currently, it's possible to calculate the array derivative provided that the deriving factor is an Array object.

Anyone objecting to introducing this API addition?

[Aaron Meurer]

If this is a generalization (same meaning) as the new matrix
derivative stuff, then I think it's good. I think it would be better
to explicitly require a Matrix object, although lists could
potentially be useful for easier input. Assumedly the output would be
a Matrix (a list output would be less useful for further SymPy
operations).

Also, please consider the proposed syntax at
https://github.com/sympy/sympy/issues/5575 (diff(f(x), (x, n)) to mean
d^n/dx^n f(x), for n symbolic) when looking at any new APIs for
diff().

Aaron Meurer

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[Francesco Bonazzi]

On Monday, 27 November 2017 21:53:08 UTC+1, Aaron Meurer wrote:

I think it would be better to explicitly require a Matrix object, although lists could
potentially be useful for easier input.

Matrix is always a two dimensioanl objects, even if they are row- or column- matrices.
 

Assumedly the output would be a Matrix (a list output would be less useful for further SymPy
operations).

The output is an Array, an the dimension may vary. Matrices always have 2 dimensions.

[Francesco Bonazzi]

For those interested:
https://github.com/sympy/sympy/pull/13655