Re: Re: Simplify with NestedLessLess?
- To: mathgroup at smc.vnet.net
- Subject: [mg106606] Re: [mg106531] Re: [mg106487] Simplify with NestedLessLess?
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Mon, 18 Jan 2010 02:35:06 -0500 (EST)
- References: <201001141049.FAA19892@smc.vnet.net>
- Reply-to: drmajorbob at yahoo.com
I think it will work whenever Mathematica can Collect terms and find the
Limit... for some value of the word "work". (I don't think you've defined,
precisely, what you're looking for.)
If M can't Collect linear terms (some terms involve radicals, for
instance), you might need to use Normal@Series (to SOME order, in the
variables except C and Cf) before Limit.
Bobby
On Sun, 17 Jan 2010 18:00:54 -0600, Dave Bird <dbird at ieee.org> wrote:
> This works for the test expression, obviously. But, can it be made to
> work for a more complex expression containing say a radical?
>
> I don't understand it well enough to critique at this point; I'll have
> to study it more tomorrow.
>
> Thanks,
>
> Dave
>
> DrMajorBob wrote:
>> Oops... Normal was extraneous there. I had another solution using
>> Series, but I switched to Limit.
>>
>> Clear[h]
>> h[expr_] /; FreeQ[expr, C] || FreeQ[expr, Cf] := expr
>> h[expr_] := Limit[expr /. C -> delta Cf, delta -> 0]
>>
>> expr = -Cf^2 L2^2 Rg^2 Vg^4 +
>> 3 (4 C Rg^2 Vd^2 + 4 Cf Rg^2 Vd^2 + 2 C Rg^2 Vd Vg);
>> vars = Complement[Variables@expr, {C, Cf}];
>> Collect[expr, vars, h]
>>
>> -Cf^2 L2^2 Rg^2 Vg^4 + Rg^2 (12 Cf Vd^2 + 6 C Vd Vg)
>>
>> Bobby
>>
>> On Sat, 16 Jan 2010 23:00:46 -0600, DrMajorBob <btreat1 at austin.rr.com>
>> wrote:
>>
>>>> Vg is potentially much larger than Vd (last two terms). In other
>>>> words, the magnitude of Vd, Vg and L2 could conspire together to make
>>>> the last term significant even though C<<Cf.
>>>
>>> The same can be said for ANY term you want to ignore, can it not?
>>>
>>> But never mind; I think you want to apply the technique I just showed
>>> you, but only to individual coefficients in
>>>
>>> Collect[-Cf^2 L2^2 Rg^2 Vg^4 +
>>> 3 (4 C Rg^2 Vd^2 + 4 Cf Rg^2 Vd^2 + 2 C Rg^2 Vd Vg), {L2, Rg, Vg}]
>>>
>>> -Cf^2 L2^2 Rg^2 Vg^4 + Rg^2 (12 C Vd^2 + 12 Cf Vd^2 + 6 C Vd Vg)
>>>
>>> ...but ONLY if C and Cf both appear in the same coefficient.
>>>
>>> That's a strange set of conditions, in my opinion, but here's a way to
>>> do it:
>>>
>>> Clear[h]
>>> h[expr_] /; FreeQ[expr, C] || FreeQ[expr, Cf] := expr
>>> h[expr_] := Normal[Limit[expr /. C -> delta Cf, delta -> 0]]
>>>
>>> expr = -Cf^2 L2^2 Rg^2 Vg^4 +
>>> 3 (4 C Rg^2 Vd^2 + 4 Cf Rg^2 Vd^2 + 2 C Rg^2 Vd Vg);
>>> vars = Complement[Variables@expr, {C, Cf}];
>>> Collect[expr, vars, h]
>>>
>>> -Cf^2 L2^2 Rg^2 Vg^4 + Rg^2 (12 Cf Vd^2 + 6 C Vd Vg)
>>>
>>> % == -Cf^2 L2^2 Rg^2 Vg^4 +
>>> 3 (4 Cf Rg^2 Vd^2 + 2 C Rg^2 Vd Vg) // Simplify
>>>
>>> True
>>>
>>> Bobby
>>>
>>> On Sat, 16 Jan 2010 21:09:38 -0600, Dave Bird <dbird at ieee.org> wrote:
>>>
>>>> Yes, but when we make delta->0 we eliminate _all_ C terms, thus
>>>> missing the correct answer of
>>>>
>>>> -Cf^2 L2^2 Rg^2 Vg^4+3 (4 Cf Rg^2 Vd^2+2 C Rg^2 Vd Vg)
>>>>
>>>> Vg is potentially much larger than Vd (last two terms). In other
>>>> words, the magnitude of Vd, Vg and L2 could conspire together to make
>>>> the last term significant even though C<<Cf.
>>>>
>>>> Dave
>>>>
>>>> DrMajorBob wrote:
>>>>> You said C<<Cf, so let's say C = delta CF, where delta is small. We
>>>>> compute a series around delta=0 and, after giving it a look, we set
>>>>> delta = 0.
>>>>>
>>>>> expr =
>>>>> Normal@Series[-Cf^2 L2^2 Rg^2 Vg^4 +
>>>>> 3 (4 C Rg^2 Vd^2 + 4 Cf Rg^2 Vd^2 + 2 C Rg^2 Vd Vg) /.
>>>>> C -> delta Cf, {delta, 0, 6}]
>>>>> firstTry = expr /. delta -> 0
>>>>>
>>>>> 12 Cf Rg^2 Vd^2 - Cf^2 L2^2 Rg^2 Vg^4 +
>>>>> delta (12 Cf Rg^2 Vd^2 + 6 Cf Rg^2 Vd Vg)
>>>>>
>>>>> 12 Cf Rg^2 Vd^2 - Cf^2 L2^2 Rg^2 Vg^4
>>>>>
>>>>> In a more complicated situation (if a few Series terms were
>>>>> unconvincing), we might use Limit, instead:
>>>>>
>>>>> expr = -Cf^2 L2^2 Rg^2 Vg^4 +
>>>>> 3 (4 C Rg^2 Vd^2 + 4 Cf Rg^2 Vd^2 + 2 C Rg^2 Vd Vg) /. C -> delta
>>>>> Cf
>>>>> secondTry = Limit[expr, delta -> 0]
>>>>>
>>>>> -Cf^2 L2^2 Rg^2 Vg^4 +
>>>>> 3 (4 Cf Rg^2 Vd^2 + 4 Cf delta Rg^2 Vd^2 + 2 Cf delta Rg^2 Vd Vg)
>>>>>
>>>>> Cf Rg^2 (12 Vd^2 - Cf L2^2 Vg^4)
>>>>>
>>>>> In this case, both answers are the same:
>>>>>
>>>>> firstTry == secondTry // Simplify
>>>>>
>>>>> True
>>>>>
>>>>> Bobby
>>>>>
>>>>> On Sat, 16 Jan 2010 13:38:22 -0600, Dave Bird <dbird at ieee.org> wrote:
>>>>>
>>>>>> Interesting! But, I don't think I am correctly communicating what
>>>>>> I'm after yet. (Although, I admit that I am struggling some to keep
>>>>>> up with you guys in your Mathematica replies due to my
>>>>>> inexperience.)
>>>>>>
>>>>>> The original expression that I put up for illustration is:
>>>>>>
>>>>>> -Cf^2 L2^2 Rg^2 Vg^4+3 (4 C Rg^2 Vd^2+4 Cf Rg^2 Vd^2+2 C Rg^2 Vd Vg)
>>>>>>
>>>>>> We compare 4 C Rg^2 Vd^2 to 4 Cf Rg^2 Vd^2 because the two terms
>>>>>> share common coefficients so that they "reduce" to (4 Rg^2 Vd^2+4
>>>>>> Rg^2 Vd^2) (C+Cf) . Thus it becomes obvious that C may be discarded
>>>>>> w.r.t. Cf.
>>>>>>
>>>>>> Please forgive if I have missed the correct application of your
>>>>>> suggestion, and thanks for the interest.
>>>>>>
>>>>>> Dave
>>>>>>
>>>>>> DrMajorBob wrote:
>>>>>>> Series[-Cf^2 L2^2 Rg^2 Vg^4 +
>>>>>>> 3 (4 C Rg^2 Vd^2 + 4 Cf Rg^2 Vd^2 + 2 C Rg^2 Vd Vg), {C, 0,
>>>>>>> 5}] // Simplify
>>>>>>>
>>>>>>> SeriesData[C, 0, {
>>>>>>> Cf Rg^2 (12 Vd^2 - Cf L2^2 Vg^4), 6 Rg^2 Vd (2 Vd + Vg)}, 0, 6, 1]
>>>>>>>
>>>>>>> Bobby
>>>>>>>
>>>>>>> On Fri, 15 Jan 2010 02:21:09 -0600, Dave Bird <dbird at ieee.org>
>>>>>>> wrote:
>>>>>>>
>>>>>>>> Not infinitesimals. I'm working in analog circuit
>>>>>>>> design/analysis. I
>>>>>>>> have a 3 pole symbolic circuit response (third order) which is not
>>>>>>>> easily separable. I can use Mathematica to find the three roots
>>>>>>>> of the
>>>>>>>> response. But, the roots are, of course, very messy. I know that
>>>>>>>> certain
>>>>>>>> elements in the circuit are orders of magnitude larger than other
>>>>>>>> like
>>>>>>>> elements - capacitors in this case. For example, one small
>>>>>>>> section of
>>>>>>>> one root is
>>>>>>>>
>>>>>>>> -Cf^2 L2^2 Rg^2 Vg^4+3 (4 C Rg^2 Vd^2+4 Cf Rg^2 Vd^2+2 C Rg^2 Vd
>>>>>>>> Vg)
>>>>>>>>
>>>>>>>> I know that C<<Cf. By careful inspection, I can see that the
>>>>>>>> first term
>>>>>>>> in the parens will drop out compared to the second term in the
>>>>>>>> parens. I
>>>>>>>> would like Mathematica to do this without my having to examine it
>>>>>>>> so
>>>>>>>> closely since there are many other like situations.
>>>>>>>>
>>>>>>>> This kind of situation occurs in many other engineering
>>>>>>>> situations.
>>>>>>>>
>>>>>>>> Hope this helps clarify.
>>>>>>>>
>>>>>>>> Thanks for the interest.
>>>>>>>>
>>>>>>>> Dave
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> Daniel Lichtblau wrote:
>>>>>>>>> Dave Bird wrote:
>>>>>>>>>> Thanks Daniel for the observation. I forgot to add that both a,
>>>>>>>>>> and b
>>>>>>>>>> are real positive. That, of course would have to be added to the
>>>>>>>>>> assumptions.
>>>>>>>>>>
>>>>>>>>>> Dave
>>>>>>>>>
>>>>>>>>> It's still not obvious what you are wanting to do. I have the
>>>>>>>>> idea you
>>>>>>>>> are working in some sense with infinitesmals. If so, I doubt
>>>>>>>>> Simplify
>>>>>>>>> would be the best tool for removing them; it really can only do
>>>>>>>>> that
>>>>>>>>> if it is told, in some way, to replace them with zero. How might
>>>>>>>>> one
>>>>>>>>> instruct Simplify to figure that out?
>>>>>>>>>
>>>>>>>>> Daniel
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>> Daniel Lichtblau wrote:
>>>>>>>>>>> dbird wrote:
>>>>>>>>>>>> Please excuse if this has been answered before, but I can't
>>>>>>>>>>>> find it.
>>>>>>>>>>>>
>>>>>>>>>>>> Is there some way to do a Simplify with assumptions using a
>>>>>>>>>>>> NestedLessLess or something similar? For example:
>>>>>>>>>>>>
>>>>>>>>>>>> d=a+b
>>>>>>>>>>>> Simplify[d,NestedLessLess[a,b]]
>>>>>>>>>>>>
>>>>>>>>>>>> Answer is:
>>>>>>>>>>>> a+b
>>>>>>>>>>>>
>>>>>>>>>>>> Answer should be:
>>>>>>>>>>>> b
>>>>>>>>>>>>
>>>>>>>>>>>> Thanks,
>>>>>>>>>>>>
>>>>>>>>>>>> Dave
>>>>>>>>>>>
>>>>>>>>>>> I fail to see why the result should be b.
>>>>>>>>>>>
>>>>>>>>>>> Daniel Lichtblau
>>>>>>>>>>> Wolfram Research
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>>
>>>>>
>>>>>
>>>
>>>
>>
>>
--
DrMajorBob at yahoo.com
- References:
- Simplify with NestedLessLess?
- From: dbird <dbird@ieee.org>
- Simplify with NestedLessLess?