Test Orthogonal Projections

[arnavs]

Not much code in this one. Just verified that the three different projection schemes come out to the same result.

[jlperla]

Do we really need all of these atol = 1e-10 everywhere? Isn't the default tolerance enough? I would prefer to drop them if possible.

One thing, though, is that you can never test that something @test x ≈ 0 without an atol as it can't figure out the relative tolerance between the two sides. We will need to discuss that in the notes at some point.

[jlperla]

I think we should have using LinearAlgebra at the top of every lecture. Julia without that is pretty confusing for users who start playing with the notebook.

[arnavs]

@jlperla The default tolerance for isapprox is full machine precision (1e-16), I believe. I put these in as a precautionary measure (maybe we're not printing the full float, maybe there are differences with different machines, etc.). But they could be redundant, and we could probably remove them at some point.

Nevermind, this isn't true:

julia> 1 ≈ 1.0000001
false

julia> 1 ≈ 1.00000001
true

So yeah, we could definitely delete these. Will do from now on.

[Nosferican]

Having LinearAlgebra at least at this stage will break some lectures given I and a few other namespace conflicts. Might be sensible during the re-write...

[Nosferican]

I know for some with QuantEcon given a naive (in the lectures) and a high precision one (in the package) I think I needed 1e-6 or 1e-8.

[jlperla]

The default tolerance for isapprox is full machine precision (1e-16),

That isn't true. It uses the machine epsilon, but comparing floating points adjusts for the scale. See https://docs.julialang.org/en/v1.0.0/base/math/#Base.isapprox

Try to avoid them if possible and just add in extra numbers if necessary, but not a big deal.

[jlperla]

Having LinearAlgebra at least at this stage will break some lectures given I and a few other namespace conflicts. Might be sensible during the re-write...

Sure, it can wait. But it also is only a problem if you use the I as a uniform scaling. See

julia> using LinearAlgebra

julia> I = 5
5

julia> I
5

[arnavs]

OK, I stripped the tolerances from the tests and there were no issues. @Nosferican is this good to merge?