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Re: Re: Simplify with NestedLessLess?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg106584] Re: [mg106531] Re: [mg106487] Simplify with NestedLessLess?
*From*: DrMajorBob <btreat1 at austin.rr.com>
*Date*: Sun, 17 Jan 2010 07:11:29 -0500 (EST)
*References*: <201001141049.FAA19892@smc.vnet.net>
*Reply-to*: drmajorbob at yahoo.com
> 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
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