Add Vector == ZZ method

[d-torrance]

Allows calling "zero" to test whether a vector is the zero vector.

Also add a unit test.

Before

i1 : zero vector {0, 0, 0}
stdio:1:4:(3): error: no method for binary operator == applied to objects:
                          3
            0 (of class ZZ )
     ==     0 (of class ZZ)

After

i1 : zero vector {0, 0, 0}

o1 = true

[mahrud]

1. Did you mean to also add ZZ == Vector? This might be a simple way to do both:

Suggested change

	Vector == ZZ := (v,i) -> matrix v == i
	Vector == ZZ := ZZ == Vector := (u, v) -> matrix u == matrix v

2. What you wrote also makes vector {1} == 1 to be true, which is probably fine, but just making sure this is intended.
3. Is this automatically caught in a generic documentation node?

[d-torrance]

1. I didn't think to, but that's a good idea! I just pushed a new version adding this, and also supporting comparison using == with Number and RingElement, matching the corresponding Matrix methods.
2. Yes, that's intended.
3. No, but I think that's a wider issue with all the == methods. None of them seem to be documented!

[mahrud]

While you're at it, maybe also define Vector == Matrix?

I don't particularly care about all of these nodes being documented, as long as it's obvious what they do. I do care about them not "missing documentation" though, so either marking as undocumented or globbing all missing nodes involving == seem like fine solutions to me. This would still allow documenting the ones that are noteworthy, like for modules/ideals/sheaves.

[d-torrance]

Yeah, == is really bad for missing documentation nodes! https://macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/Macaulay2Doc/html/__eq_eq.html

Some sort of automatic way for adding otherwise undocumented methods for common operators like +, -, and == to the main node for that operator sounds like great idea. But I think it's outside the scope of this PR.

[mahrud]

Okay, then maybe don't touch the documentation at all.

[d-torrance]

I figured I'd at least document the + and - methods that I added. The spacing between all the various keys for these nodes were super messy and inconsistent, so I decided I might as well clean it up a bit.

[mahrud]

If you want to clean up the messy documentation nodes then use globbing, otherwise why touch a hundred lines just to fix the indentation?

[d-torrance]

Fair enough. I removed that commit and just added the new keys for now.

[mahrud]

Does this need to be written this way? This is slow due to #3035. For instance try v == 1 with v the matrix defined in that issue.

[mahrud]

Also, Vector == Number covers Vector == ZZ, no?

[d-torrance]

Do you have a suggestion for a better way to define Matrix == Number and friends? That's probably outside the scope of this PR.

Regarding Vector == ZZ, we currently need it to avoid a recursion error. Otherwise, something like v == 0 would call v - 0 == 0.

[mahrud]

I guess I'm wondering what's wrong with m == matrix f? The slight edge case is when the target of m has relations and you want to reduce f first, but maybe the user should explicitly write m - f * id_(target m) in that case.

[d-torrance]

That would break the current behavior since matrix f is 1x1:

i1 : matrix {{3, 0}, {0, 3}} == 3

o1 = true

i2 : matrix {{3, 0}, {0, 3}} == matrix 3

o2 = false

[mahrud]

Sure, but this doesn't apply to vectors.

[d-torrance]

Calling matrix on the Number/RingElement won't work for the zero vector if the rank of the module is greater than one, since then we'll be comparing matrices with different sizes:

i1 : matrix vector {0, 0} == matrix 0

o1 = false

I did change it to just matrix v == f, which works just fine:

i2 : matrix vector {0, 0} == 0

o2 = true

Then we don't have to worry about defining Vector == ZZ.

[mahrud]

Instead of a long list of methods like this, which requires constant maintenance, I'd rather you glob them all like in 7bfcb62.

[mahrud]

i1 : Number + Vector := RingElement + Vector := (r,v) -> vector(r + matrix v);

i2 : 1 + (coker matrix {{ 10 }})_0
stdio:1:58:(3):[1]: error: expected source to be free with rank 1

[d-torrance]

This is coming from vector(Matrix):

i1 : code (vector, Matrix)

o1 = -- code for method: vector(Matrix)
     M2/Macaulay2/m2/modules.m2:66:17-69:5: --source code:
     vector Matrix := f -> (
          if not isFreeModule source f or numgens source f =!= 1 then error "expected source to be free with rank 1";
          new target f from {f}
         )

Should Vector objects be elements of free modules? Or arbitrary modules?

[mahrud]

That's not the problem. Single generators of any module whatsoever are vectors. The implementation of Number + Vector isn't correct because it uses the identity matrix on the target.

[d-torrance]

Hmm, this seems to be a problem with matrix addition:

i1 : M = coker matrix 10;

i2 : A = map(M, ZZ^1, 1)

o2 = | 1 |

                    1
o2 : Matrix M <-- ZZ

i3 : 1 + A

o3 = | 2 |

o3 : Matrix M <-- M

i4 : A + 1

o4 = | 2 |

                    1
o4 : Matrix M <-- ZZ

Shouldn't they both have ZZ^1 as their source?

[d-torrance]

I just pushed a new commit that should fix this