Replaced all instances of <RingElement> with <Element>

[vluo13]

Fixes #40611

Performance Measurement

Before

sage: sage: v = vector([1,2/3,pi]); w = vector([-2/3,pi^2,1])
sage: %timeit v._add_(w)
953 ns ± 24.5 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)
sage: sage: V = QQ^5
sage: sage: W = V.span([V.1, V.2])
sage: sage: %timeit W.0 - V.0
11.2 μs ± 148 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
sage: R.<x> = QQ[]
sage: M = span([[x, x^2+1], [1/x, x^3]], R)
sage: %timeit x * M.basis()[0]
18.2 μs ± 228 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
sage: %timeit M.basis()[0] * x
18.1 μs ± 280 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
sage: v = vector([x,x^2,3*x]); w = vector([2*x,x,3+x])
sage: %timeit v.pairwise_product(w)
481 ns ± 4.86 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)
sage: sage: %timeit w.pairwise_product(v)
476 ns ± 2.45 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)

After

sage: v = vector([1,2/3,pi]); w = vector([-2/3,pi^2,1])
sage: %timeit v._add_(w)
942 ns ± 6.55 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)
sage: V = QQ^5
sage: W = V.span([V.1, V.2])
sage: %timeit W.0 - V.0
11.2 μs ± 124 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
sage: R.<x> = QQ[]
sage: M = span([[x, x^2+1], [1/x, x^3]], R)
sage: %timeit x * M.basis()[0]
18.4 μs ± 167 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
sage: %timeit M.basis()[0] * x
18.3 μs ± 50.2 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
sage: v = vector([x,x^2,3*x]); w = vector([2*x,x,3+x])
sage: %timeit v.pairwise_product(w)
500 ns ± 3.55 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)
sage: %timeit w.pairwise_product(v)
502 ns ± 8.06 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

⌛ Dependencies

[whoami730]

pairwise_product seems to incur a significant slowdown.

Before

sage: v = vector([x,x^2,3*x]); w = vector([2*x,x,3+x])
sage: %timeit v.pairwise_product(w)
481 ns ± 4.86 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)
sage: sage: %timeit w.pairwise_product(v)
476 ns ± 2.45 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)

After

sage: v = vector([x,x^2,3*x]); w = vector([2*x,x,3+x])
sage: %timeit v.pairwise_product(w)
500 ns ± 3.55 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)
sage: %timeit w.pairwise_product(v)
502 ns ± 8.06 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)

[user202729]

pairwise_product seems to incur a significant slowdown.

given that in one of the run the standard deviation of 10⁶ runs is 8ns, and the difference is on the order of 20ns, I'd be tempted to dismiss it as random fluctuation.

This difference is… around 2.5σ, so roughly p=0.012?

[user202729]

do add a doctest.

[whoami730]

pairwise_product seems to incur a significant slowdown.

given that in one of the run the standard deviation of 10⁶ runs is 8ns, and the difference is on the order of 20ns, I'd be tempted to dismiss it as random fluctuation.

This difference is… around 2.5σ, so roughly p=0.012?

It is probably happening in each loop of the run and in each run right? One sure shot method to verify is to take a larger dimension say 100 or 1000 and do performance testing then.

[vluo13]

I added a doctest that uses the original code that generated the error. Is the location where I put it ok?

[whoami730]

Consider using lmul or rmul explicitly. We can have both tests - matrix * vector and vector * matrix.

[user202729]

by lmul/rmul you likely mean testing multiply the matrix on left/right side. Calling the internal method explicitly is fine, but not possible if the method is cdef only.

[whoami730]

The method being tested is cpdef _lmul_(self, Element right):, thus I believe both matrix * vector and matrix._lmul_(vector) should be tested, similar to other tests above.

[whoami730]

A test for v + v should be added as well, probably similarly other operations can also be tested for.

[user202729]

while adding the tests for other operations (+, etc.) is fine, I think the risk of regression is very low. Now wait for someone to approve workflows.

[whoami730]

@tscrim requesting workflow approval on this PR

[github-actions]

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