Implement symbolic Kronecker delta and also make current signum (sgn) symbolic

[golam-m-hossain]

We should have a symbolic Kronecker delta in Sage.

Also, current implementation of signum (sgn) function
returns wrong answer for symbolic input.

sage: sgn(x)
1
sage: sgn(-x)
1

So we should make sgn() symbolic aware. Given, implementation of
these functions in new symbolics should be very similar to the existing generalized function Dirac delta and Heaviside, I am putting them together here.

Also, ticket #777 should be closed down as Sign is already in Sage
and this ticket will further enhance it.

Patch:

Apart from implementing these two symbolic functions, attached patch slightly speeds up three other generalized functions by avoiding default __call__ method of PrimitiveFunction. These functions take explicit value either 0,-1,1 so those checks are not needed.

Timing before the patches:

sage: timeit('dirac_delta(1.0)')
625 loops, best of 3: 179 µs per loop
sage: timeit('unit_step(1.0)')
625 loops, best of 3: 345 µs per loop
sage: timeit('heaviside(1.0)')
625 loops, best of 3: 344 µs per loop

Timing after the patches:

sage: timeit('dirac_delta(1.0)')
625 loops, best of 3: 159 µs per loop
sage: timeit('heaviside(1.0)')
625 loops, best of 3: 324 µs per loop
sage: timeit('unit_step(1.0)')
625 loops, best of 3: 323 µs per loop

Also, it does slight re-arrangements of references.

Component: symbolics

Author: Golam Mortuza Hossain

Reviewer: Karl-Dieter Crisman

Merged: sage-4.3.alpha0

Issue created by migration from https://trac.sagemath.org/ticket/6803

[golam-m-hossain]

comment:1

Attachment: trac_6803-symbolic-kronecker-n-signum.patch.gz

[golam-m-hossain]

Author: Golam Mortuza Hossain

[kcrisman]

Reviewer: Karl-Dieter Crisman

[kcrisman]

comment:2

Overall this is good, and should get positive review. However, there is a doctest failure (very typical when one adds a new symbolic function in line 204 of symbolic/random_tests.py, with random_expr(). This should be easy to fix.

[kcrisman]

comment:3

I have attached the patch, but with the random test fixed as a reviewer patch. Apply only this patch.

[kcrisman]

Based on 4.2.1.alpha0, apply only this patch.

[kcrisman]

comment:4

Attachment: trac_6803-symbolic-kronecker-n-signum-review.patch.gz

[mwhansen]

comment:5

I had to add a small patch to the doctest for sage.quadratic_forms.extras.sgn to make sure that the doctest was actually doctesting that function.

[mwhansen]

Merged: sage-4.3.alpha0

[kcrisman]

comment:6

Replying to @mwhansen:

I had to add a small patch to the doctest for sage.quadratic_forms.extras.sgn to make sure that the doctest was actually doctesting that function.

Can you post that here, or at least the code in braces - just for posterity?

[sagetrac-mvngu]

apply after previous

[sagetrac-mvngu]

comment:7

Attachment: trac_6803-sgn.patch.gz

Replying to @kcrisman:

Can you post that here, or at least the code in braces - just for posterity?

mhansen's patch is contained in trac_6803-sgn.patch