Re: Re: When is a List not a List?
- To: mathgroup at smc.vnet.net
- Subject: [mg90992] Re: [mg90956] Re: [mg90947] When is a List not a List?
- From: DrMajorBob <drmajorbob at att.net>
- Date: Sat, 2 Aug 2008 03:26:28 -0400 (EDT)
- References: <200807310656.CAA07700@smc.vnet.net>
- Reply-to: drmajorbob at longhorns.com
By the by, should all three branches in the following have the same color
(and style, if styles were specified)?
Plot[x /. Solve[x^2 + x - 1/Sqrt[x + a] == 1, x], {a, 2, 4}]
In this case Solve gives FIVE solutions:
Solve[x^2 + x - 1/Sqrt[x + a] == 1, x]
{{x -> Root[-1 +
a + (1 - 2 a) #1 + (-2 - a) #1^2 + (-1 + 2 a) #1^3 + (2 +
a) #1^4 + #1^5 &, 1]}, {x ->
Root[-1 +
a + (1 - 2 a) #1 + (-2 - a) #1^2 + (-1 + 2 a) #1^3 + (2 +
a) #1^4 + #1^5 &, 2]}, {x ->
Root[-1 +
a + (1 - 2 a) #1 + (-2 - a) #1^2 + (-1 + 2 a) #1^3 + (2 +
a) #1^4 + #1^5 &, 3]}, {x ->
Root[-1 +
a + (1 - 2 a) #1 + (-2 - a) #1^2 + (-1 + 2 a) #1^3 + (2 +
a) #1^4 + #1^5 &, 4]}, {x ->
Root[-1 +
a + (1 - 2 a) #1 + (-2 - a) #1^2 + (-1 + 2 a) #1^3 + (2 +
a) #1^4 + #1^5 &, 5]}}
However:
Cases[Plot[
x /. Solve[x^2 + x - 1/Sqrt[x + a] == 1, x], {a, 2.5, 2.7}], _Line,
Infinity] // Length
3
There are three lines; there's no reason Plot could not assign colors and
styles to them. (Although, perhaps, we couldn't know which branches would
match which colors, a priori.)
Bobby
On Fri, 01 Aug 2008 11:00:34 -0500, Andrzej Kozlowski <akoz at mimuw.edu.pl>
wrote:
> Just in case I have not made my point clearly enough, I suggest
> pondering over the difference between:
>
>
> Plot[x /. Solve[x^3 + x - 1/Sqrt[x + a] == 1, x], {a, -1, 3}]
>
> and
>
> Plot[Evaluate[x /. Solve[x^3 + x - 1/Sqrt[x + a] == 1, x]], {a, -1, 3}]
>
>
> The second graph uses three colors, just as you and AES would have
> wished, but unfortunately there is a slight problem with it :it is quite
> wrong.
>
> Understanding a bit of mathematics often helps to understand Mathematica.
>
>
> Andrzej Kozlowski
>
>
>
> On 1 Aug 2008, at 17:30, Andrzej Kozlowski wrote:
>
>> Because there are lots of plots of the kind which I just sent (Plot[x
>> /. Solve[x^3 - 3 x^2 + a == 0, x], {a, -3, 5}]) but where you have
>> FindRoot or NMaximize etc., instead of Solve and the first argument
>> can't be evaluated until the value of the parameter a has been
>> supplied. There have been lots of post of this kind on this forum so
>> its kind if weird you managed to miss them all.
>>
>> Andrzej Kozlowski
>>
>>
>>
>>
>> On 1 Aug 2008, at 16:29, DrMajorBob wrote:
>>
>>> Agreed.
>>>
>>> And I'm wondering what advantage HoldAll has, in the case of Plot,
>>> since a plot can't result without evaluating the arguments?
>>>
>>> Why NOT evaluate the first argument, at least, before making style
>>> decisions/assignments?
>>>
>>> The defense that "everybody already knows this" is irrelevant to the
>>> question of how we'd LIKE Plot to behave.
>>>
>>> Naturally, we won't necessarily GET what we want... but we're entitled
>>> to say what that is.
>>>
>>> I'm trying to think how David Park's Presentations package handles
>>> such things.
>>>
>>> Bobby
>>>
>>> On Thu, 31 Jul 2008 22:33:50 -0500, AES <siegman at stanford.edu> wrote:
>>>
>>>> At 7:44 PM -0500 7/31/08, DrMajorBob wrote:
>>>>> Plot COULD assign colors after evaluation, OTOH... the fact that it
>>>>> doesn't is a design choice/artifact, not a necessity preordained by
>>>>> fate.
>>>>>
>>>>> That being so, users are entitled to find it odd at first glance.
>>>>> (Or even second... maybe third.)
>>>>
>>>> If the above is true -- and I'd suppose it is -- then I'd say it's
>>>> also very much an unfortunate, not to say flat-out *bad* design
>>>> choice or artifact.
>>>>
>>>> An innocent novice-level user creates a Plot[ ] with three curves
>>>> using the {f1, f2, f3} List form of the first argument, and discovers
>>>> that M by default assigns a different color to each curve -- a good
>>>> and helpful default design choice on M's part, I'd say.
>>>>
>>>> Maybe this novice user wants to go a bit further: Thicken certain
>>>> curves, change the Dashing, and so on. He or she discovers the
>>>> PlotStyle option (or equivalent); learns how to do this; is happy.
>>>>
>>>> And then this user also realizes: Hey, I could plot 8 or 10 curves
>>>> this way, without having to type in f1 thru f8 by just using a Table[
>>>> ] command for the first argument and iterating over some appropriate
>>>> parameter. A Table[ ] creates a List, right?
>>>>
>>>> So he/she does this; the 8 or 10 curves appear exactly as desired;
>>>> except the styling behavior is suddenly all screwed up. Once again,
>>>> a classic M-style Gotcha!!! -- and a particularly nasty Gotcha: Am I
>>>> getting this unwanted result because of the way I structured the
>>>> PlotStyle commands I used? -- or because of something mysterious with
>>>> using Table[ ]? The coloring and so on in the default {f1, f2, f3}
>>>> case has the nice default cycling behavior for the styling -- Why am
>>>> I not getting it now?
>>>>
>>>> Andrzej says this unfortunate result _has_ to be the case because
>>>> Plot[ ] doesn't "pre-Evaluate" the first argument.
>>>>
>>>> Well, somehow, if the first argument is {f1, f2, f3, f4, f5, f6, f7,
>>>> f8}, Plot[ ] somehow "pre-evaluates" (lower-case pre-evaluate) this
>>>> argument at least enough to know that it's not only a List, but how
>>>> many elements that List has. Is it somehow impossible for Plot[ ] to
>>>> know that Table will also produce a List, and to similarly
>>>> pre-evaluate how many elements that List will have? I suspect it's
>>>> not impossible.
>>>>
>>>> And if that is indeed impossible with the PlotStyle option in Plot[ ]
>>>> then can Andrzej explain how it _is _ possible for Plot[ ] to
>>>> somehow handle the PlotRange->All option correctly (i.e.,
>>>> identically) with either form of the first argument -- even though
>>>> that option needs to determine not only the number of curves in the
>>>> first argument, but the maximum and minimum values over all those
>>>> curves, in order to set the axes and axis Tick locations and values
>>>> for the plot. Is just getting the number of curves and picking the
>>>> colors for them really harder than that?
>>>>
>>>> I very much like DrMajorBob's wording here: I'll bet the coloring
>>>> problem with List vs Table is precisely "an [accidental] design
>>>> choice/artifact, not a necessity preordained by fate" -- and an
>>>> unfortunately unfortunate "design choice/artifact".
>>>>
>>>> The only things more unfortunate are (a) that M has a fair (and
>>>> increasing?) number of these Gotchas; (b) M's documentation is
>>>> substantially less helpful than it could or should be either in
>>>> diagnosing or in warning about them; and (c) it's far from clear that
>>>> anyone at WRI really recognizes these points.
>>>>
>>>>
>>>>
>>>>
>>>
>>>
>>>
>>> --DrMajorBob at longhorns.com
>>
>
>
--
DrMajorBob at longhorns.com