Improve documentation of numerical_approx()

[saraedum]

In a class on computer algebra, students were surprised by the behavior of the N() function. Here are some minimal examples that should illustrate the issues we ran into:

sage: x=N(pi,digits=3); x
3.14  
sage: y=N(3.14,digits=3); y
3.14  
sage: x==y
False

sage: N(pi,digits=1)
3.2

sage: N(pi-3,prec=2)
0.00

Imo, none of these is a bug, but I believe that the documentation should be improved.

---

Apply trac_12120_numerical_approx.patch​ and trac_12120-review.patch​ to the Sage library.

[saraedum]

The second example is also discussed in #11647.

[eviatarbach]

There's some other problems with the digits argument; it accepts non-integer values and zero. Can you add a sanity check?

[eviatarbach]

The last example is odd; I made a thread about it at ​https://groups.google.com/forum/#!topic/sage-devel/nOdFzn6U06o.

[saraedum]

I'll try to incorporate what I've written on sage-devel into the patch.

[saraedum]

Replying to eviatarbach:

There's some other problems with the digits argument; it accepts non-integer values and zero. Can you add a sanity check?

This ticket tries only to improve on the docstring of numerical_approx(). Adding sanity checks would make sense, though. I suggest you open a new ticket if you want to add sanity checks.

[eviatarbach]

Yes, I'll create another patch.

However, could you change the wording from "numerical instability"? This suggests there is something wrong. I like how Julian Rüth explained it: "This approximates both pi and 3 with two bits of precision and then computes their difference".

[saraedum]

Ok. I have no clue about numerical analysis, but isn't that what numerical instability is about, i.e., addition not being numerically stable? Maybe I got the term wrong here. In any case, feel free to add this explanatory sentence.

[eviatarbach]

I think "numerical instability" refers to propagation of errors in numerical algorithms, which is not what is occurring here. According to the sage-devel thread it is the intended behaviour, although perhaps unintuitive.

[saraedum]

Replying to eviatarbach:

Yes, I'll create another patch.

However, could you change the wording from "numerical instability"? This suggests there is something wrong. I like how Julian Rüth explained it: "This approximates both pi and 3 with two bits of precision and then computes their difference".

Are you happy with the wording now?

[eviatarbach]

Yes, looks good now. Sorry for delaying this.

[kcrisman]

Hi, looks nice! This will be quite valuable. Note that Trac #11647 should probably use the new style of :trac:`11647`.

[saraedum]

Ok. I changed Trac reference.

[kcrisman]

I like the following reviewer patch; to me, it clarifies one or two things. If you think I've misstated them, let me know. If it's fine with you, then click positive review!

Also, I think that #11647 probably should count as a dup of this. My guess is that there is no way to ensure at least one correct digit without occasionally bringing along a second for the ride.

Eviatar, feel free to open that sanity check ticket as well.

[kcrisman]

Patchbot: apply trac_12120_numerical_approx.patch and trac_12120-review.patch

[saraedum]

I would like to keep the examples without the *100. At least that's what my students ran into. The other changes you made are certainly an improvement.

[kcrisman]

I would like to keep the examples without the *100. At least that's what my students ran into. The other changes you made are certainly an improvement.

Okay, I was just trying to make it clearer about the significant digits. Otherwise one could have imagined that digits really means digits, because it so happens that with pi, the number of digits equals the number of significant digits!

[kcrisman]

Okay, I just kept both! That can't hurt, and makes it clearer to me, at least, what's going on.

Nice work.

[saraedum]

Sure. Thanks :)

[jdemeyer]

improve docstring of numerical_approx()

[jdemeyer]

Re-created the patch using hg export (which you should always use to create a patch).