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What do the number entries mean in the sympy poly.diff(...) tuple syntax?

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I am looking to take a partial derivative of a sympy polynomial with respect to a symbol in the polynomial.

In the sympy documentation for poly.diff(...) it gives sample code like this:

from sympy import Poly
from sympy.abc import x, y

Poly(x*y**2 + x, x, y).diff((0, 0), (1, 1))

There is no explanation I was able to find as to what the (0, 0), (1, 1) tuples refer to.

I'm really just trying to supplement the sympy documentation here, but here is my attempt at providing an MVCE for a question I already know the answer to:

from __future__ import print_function

from sympy import poly, symbols

from sympy.abc import x, y

base = poly(x*y**2 + x, x, y)
print('base function is', base)

# appears to be the same as doing diff(y)
deriv_mysterious1 = base.diff((0, 0), (1, 1))

print('deriv_mysterious1 is', deriv_mysterious1)

deriv_mysterious2 = base.diff(y)

print('deriv_mysterious2 is', deriv_mysterious2)

# same as base
deriv_mysterious3 = base.diff((0,0))
print('deriv_mysterious3 is', deriv_mysterious3)

# same as diff(y)
deriv_mysterious4 = base.diff((1,1))
print('deriv_mysterious4 is', deriv_mysterious4)

# same as diff(x)
deriv_mysterious5 = base.diff((0,1))
print('deriv_mysterious5 is', deriv_mysterious5)

The output of which is

base function is Poly(x*y**2 + x, x, y, domain='ZZ')
deriv_mysterious1 is Poly(2*x*y, x, y, domain='ZZ')
deriv_mysterious2 is Poly(2*x*y, x, y, domain='ZZ')
deriv_mysterious3 is Poly(x*y**2 + x, x, y, domain='ZZ')
deriv_mysterious4 is Poly(2*x*y, x, y, domain='ZZ')
deriv_mysterious5 is Poly(y**2 + 1, x, y, domain='ZZ')

How do I use this tuple syntax? What do these numbers refer to?

Why should this post be closed?

2 comments

I don't really know how to interpret a downvote on this, if you found official documentation somewhere on this please link it. I had to piece this together from 5 or 6 random websites to figure it out. There's this but it has diff(expr, <symbol>, <order>) not poly.diff((<mysterious symbol index>, <order>)), and poly.diff(...) is not taking the partial derivative with respect to the number 0 or a symbol named 0. ‭jrh‭ 30 days ago

It's kinda funny how the "Run code block in SymPy Live" on the docs gives an error for that example ‭Moshi‭ 29 days ago

2 answers

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It appears that the tuple syntax works like this:

(variable index, order of derivative)

Where something like:

base = poly(x*y**2 + x, x, y)
deriv_mysterious5 = base.diff((0,1))
print('deriv_mysterious5 is', deriv_mysterious5)

Means: "Take first derivative of base with respect to symbol[0]", in this case symbol 0 is x.

  • I was not able to find information on how sympy determines which symbol is symbol 0.

I think I would recommend the alternative syntax instead:

base.diff(<symbol>)

e.g.,

base.diff(x)

Though, interestingly, it seems that for higher order derivatives, poly.diff has some behavior that seems to differ from diff(expr,...) as seen on the Calculus page.

This syntax works:

base.diff(y, y)

I was not able to get this syntax working:

base.diff(y, 2)

But a slightly modified version of this does work,

base.diff((y, 2))

0 comments

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Based on the source code which delegates to this implementation among others, base.diff((x, n)) means to compute the n-th derivative of base with respect to x. Any arguments to diff which aren't tuples get tupled with 1, e.g. base.diff(x) is more or less equivalent to base.diff((x, 1)). Multiple arguments essentially correspond to repeated differentiation, e.g. base.diff((x, m), (y, n)) is more or less the same as base.diff((x, m).diff((y, n)).

Numeric arguments are indexes into the array of generators provided when the polynomial was created. That is, if base is defined by Poly(..., x, y), then base.diff((0, n)) and base.diff((x, n)) are more or less the same. Similarly, base.diff((1, n)) and base.diff((y, n)). This is handled by the method _gen_to_level. That method also allows negative numbers to index backwards into the array of generators, i.e. base.diff((-1, n)) is also equivalent to base.diff((y, n)).

The example from the other answer of base.diff(y, 2) gets interpreted as base.diff((y, 1), (2, 1)), however, since there is only 2 generators (x and y), this should lead to an exception being thrown in _gen_to_level as 2 is not a valid index into the array of generators.

That said, the example in the documentation is rather cryptic, and it is a bit bizarre to have base.diff((0,0)) which would mean "take the 0-th derivative of the first generator" which is well-defined but corresponds to doing nothing.

1 comment

This is a useful answer but readers should note that generally implementation is not a strong of a source as documentation, and could in theory change at any time. Though to be fair I think this implementation is likely to stick and I doubt they would change the meaning of the parameters. I wasn't able to find any official answer to this question at all in the documentation (other than what I linked), and this is probably the best we're going to get. ‭jrh‭ 26 days ago

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