Bug in convolution in old Piecewise

[kcrisman]

The old Piecewise (pre #14801; now deprecated) has this bug:

sage: x = PolynomialRing(QQ,'x').gen()
sage: f = Piecewise([[(-2, 2), 2 * x^0]])
sage: g = Piecewise([[(0, 2), 3/4 * x^0]])
sage: n = f.convolution(g)

sage: n
Piecewise defined function with 3 parts, [[(-2, 0), 3/2*x + 3], [(0, 2), 6], [(2, 4), -3/2*x + 6]]

But the middle piece should be 3, not 6, apparently.

See the original report at ​this ask.sagemath.org question.

Note this is fixed in the new piecewise (lowercase p, from #14801).

sage: x = PolynomialRing(QQ,'x').gen()
sage: f = piecewise([[(-2, 2), 2 * x^0]])
sage: g = piecewise([[(0, 2), 3/4 * x^0]])
sage: n = f.convolution(g)
sage: n
piecewise(x|-->3/2*x + 3 on (-2, 0], x|-->3 on (0, 2], x|-->-3/2*x + 6 on (2, 4]; x)

Can close this bug when the old Piecewise is removed completely.

[kcrisman]

The fix is to fix the following

    cmd1 = "integrate((%s)*(%s),%s,%s,%s)"%(i1,i2, uu, a1,    tt-b1)    ## if a1+b1 < tt < a2+b1
    cmd2 = "integrate((%s)*(%s),%s,%s,%s)"%(i1,i2, uu, tt-b2, tt-b1)    ## if a1+b2 < tt < a2+b1
    cmd3 = "integrate((%s)*(%s),%s,%s,%s)"%(i1,i2, uu, tt-b2, a2)       ## if a1+b2 < tt < a2+b2
    cmd4 = "integrate((%s)*(%s),%s,%s,%s)"%(i1,i2, uu, a1, a2)          ## if a2+b1 < tt < a1+b2
    <snip>
    if a1-b1<a2-b2:
        if a2+b1!=a1+b2:
           h = Piecewise([[(a1+b1,a1+b2),fg1],[(a1+b2,a2+b1),fg4],[(a2+b1,a2+b2),fg3]])

to have f2 instead of f4. There is a parallel part as well in the other branch of the if, where fg2 should be fg4, it appears.

This should be fixed quickly, but should also be checked to make sure it really does do the right thing! The code is not really commented enough to show what is going on with all these different mini-convolutions, one has to really think about it.

[kcrisman]

​Here's another report that is almost certainly the same thing.

x = PolynomialRing(QQ, 'x').gen()
f1 = Piecewise([[(-1, 1), 1*x^0]])
f2 = Piecewise([[(0, 1), x], [(1, 2), -x + 2]])
g = f2.convolution(f1)
Piecewise defined function with 4 parts, [[(-1, 0), 1/2*x^2 + x +1/2], [(0, 1), -1/2*x^2 + 3*x], [(1, 2), -1/2*x^2 - x + 4], [(2, 3), 1/2*x^2 - 3*x + 9/2]].

but probably should be (according to the report)

Piecewise defined function with 3 parts, [[(-1, 0), 0.5*x^2 + x + 0.5], [(0, 2), -0.5*x^2 + x + 0.5], [(2, 3), 0.5*x^2 - 3*x + 4.5]

[kcrisman]

In fact,

    if a1-b1<a2-b2:
        if a2+b1!=a1+b2:

doesn't even make sense; if the former is true, so is the latter? What on earth is going on here? I think an extra branch snuck in, in addition to the typo.

[kcrisman]

See also #14801, though I don't know that will help with this.

[kcrisman]

See #17793 (possible dup)

[rws]

Replying to kcrisman:

​Here's another report that is almost certainly the same thing.

The two curve parts before addition look like this A working implementation should make sure that, when they are added, only three intervals remain.

sage: f = Piecewise([[(0, 1), -1/2*x^2 + x],[(1, 2), 1/2],[(2, 3), 1/2*x^2 - 3*x + 9/2]])
sage: g = Piecewise([[(-1, 0), 1/2*x^2 + x + 1/2],[(0, 1), 1/2],[(1, 2), -1/2*x^2 + x]])
sage: f+g
Piecewise defined function with 4 parts, [[(-1, 0), 1/2*x^2 + x + 1/2], [(0, 1), -1/2*x^2 + x + 1/2], [(1, 2), -1/2*x^2 + x + 1/2], [(2, 3), 1/2*x^2 - 3*x + 9/2]]

[mkoeppe]

Description modified to comment that this seems fixed in the new piecewise (#14801).

[chapoton]

piecewise_old will be removed in #26865