Preferring negative exponents

[hanspi42]

There are several fields where negative exponents are preferrable. Fort example,

(s**2*exp(2*s) + 4*exp(s) - 4)*exp(-2*s)/(s*(s**2 + 1))

is the Laplace transform from the Wester tests Y5 and Y6; the expression

(s**2 + 4*exp(-s) - 4*exp(-2*s))/(s*(s**2 + 1))

would be much simpler to read. Under the z transform, the delayed integrator would be easy to read as

z**(-1)/(1-z**(-1))

but will immediately be written as

1/(z*(1 - 1/z))

I suggest that we make it somehow possible that sympy can be told "please prefer negative exponents". Where is the proper place to do this? And how could it be done?

[samithkavishke]

May I try this issue?

[hanspi42]

May I try this issue?

Of course, do you have an idea how to do it?

[smichr]

You can replace your desired inverse symbol, e.g. dealyed_integrator.subs(1/z, Symbol('z^-1')). Or in case of the first example, simplification will help:

>>> expr = (s**2*exp(2*s) + 4*exp(s) - 4)*exp(-2*s)/(s*(s**2 + 1))
>>> i,d=expr.as_independent(exp)
>>> i*expand(d)
(x**2 + 4*exp(-x) - 4*exp(-2*x))/(x*(x**2 + 1))

[hanspi42]

.subs(1/z, Symbol('z^-1'))

Thanks, but it should work for arbitrary integer exponents, e.g., z**(-2)/(1-z**(-2))

What I wondered: somewhere in the codebase something makes exponents positive. Could that something have an option to make exponents negative?

[asmeurer]

SymPy internally represents x**-1 and 1/x in the same way. There is no Division class. This is just a question of how things are printed in the printers. So in the case of z**(-2)/(1-z**(-2)) this is exactly the same expression as 1/(z**2*(1 - 1/z**2)).

Specifically in the Pow printer (for 1/x) and Mul printer (for x/y) for whatever printer you care about (in your example, the str printer). There aren't currently any options for controlling this, but I think one could be added if you can come up with a reasonable heuristic for it. There's already some heuristics for printing exponentials in the numerator with negative denominator rather than as a fraction:

>>> y/exp(x)
y*exp(-x)

If we can improve the general heuristics without a flag, that would be best, but if it's a case where both types of output might be desired, a flag would be best.

On the other hand, (s**2*exp(2*s) + 4*exp(s) - 4)*exp(-2*s)/(s*(s**2 + 1)) and (s**2 + 4*exp(-s) - 4*exp(-2*s))/(s*(s**2 + 1)) are distinct expressions and both are representable.

[smichr]

The z-expression can also be rewritten as

>>> expand(1/(z*(1 - 1/z)))
1/(z - 1)

And that looks pretty clean to me. For purposes of discussion, it would be better to focus on expressions which can't be rewritten in a nice way already.

[hanspi42]

Ok,

The z-expression can also be rewritten as

>>> expand(1/(z*(1 - 1/z)))
1/(z - 1)

And that looks pretty clean to me. For purposes of discussion, it would be better to focus on expressions which can't be rewritten in a nice way already.

Yes, it is clean, but if it were written as $\frac{z^{-1}}{1-z^{-1}}$, it would be in the standard form for digital filter transfer functions (see https://en.wikipedia.org/wiki/Digital_filter#Characterization) and directly implementable (see https://en.wikipedia.org/wiki/Digital_filter#Direct_form_I) because $z^{-1}$ is just a delay.

So the reason why I am asking this is that we have a whole important field out there, digital signal processing, whose practitioners would love it so much if computer algebra tools could print functions in the standard form

$\displaystyle \frac{{b_{0}+b_{1}z^{-1}+b_{2}z^{-2} + \cdots + b_{N}z^{-N}}}{{1+a_{1}z^{-1}+a_{2}z^{-2} + \cdots +a_{M}z^{-M}}}$

with $z$ only having negative exponents.

[hanspi42]

By the way, Mathematica can do this somewhat, there you can define a z1 as Power[z, HoldForm[-1]], and then z1^4 will remain z^(-4).

[asmeurer]

Something that we do need is better Laurent polynomial and rational functions of Laurent polynomials support. See https://github.com/sympy/sympy/issues/5131. If z**-1/(1 - z**-1) were represented as a Laurent rational function in the polys, then such an expression would print and canonicalize to that form (and more importantly, there would be Laurent polynomial-specific routines implemented for it).

To fix the printing I think what we need is a flag in the printers to print monomials with negative powers as powers instead of using division. I'm not sure if we can change the default behavior because I expect most people will still prefer x + 1/x instead of x + x**-1.

[hanspi42]

Something that we do need is better Laurent polynomial and rational functions of Laurent polynomials support. See #5131. If z**-1/(1 - z**-1) were represented as a Laurent rational function in the polys, then such an expression would print and canonicalize to that form (and more importantly, there would be Laurent polynomial-specific routines implemented for it).

This would help tremendously with everything that is discussed in #25040 !

[skieffer]

I'm sorry I missed this issue. I do feel it would be worth considering #23589 again, as a way to give users better control over printing.

In short, make expression elements addressable, and make a way to set style rules, just like HTML + CSS. I'd be interested in continuing to work on this, if there was a will to have it in SymPy.

In #23589 an alternative system, involving rewrite rules, was also discussed.