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Re: Re: Simplify with NestedLessLess?

• To: mathgroup at smc.vnet.net
• Subject: [mg106557] Re: [mg106531] Re: [mg106487] Simplify with NestedLessLess?
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Sat, 16 Jan 2010 06:11:51 -0500 (EST)
• References: <201001141049.FAA19892@smc.vnet.net> <4B4F39E7.1070002@wolfram.com> <4B4FAC81.7000108@ieee.org> <4B4FB26F.7050702@wolfram.com> <201001150821.DAA29881@smc.vnet.net>

Dave Bird 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
> [...]

Yes, that clarifies. This is along the lines of what I had meant by 
infinitesmals; you want to drop terms of certain order in certain 
settings. One respondant suggested using Series and then Normal to 
remove terms in that way. I think that is probably a good way to go 
about it, and I have seen that approach in the past.

In the setting of your example above, what you might do is replace each 
small variable such as C by C*t (and if, say, L2 was much smaller than 
C, maybe use  L2*t^2). For a series expansion in powers of t, dropping 
all terms beyond a certain point. If the "large" terms have varying 
orders of magnitude, that could be emulated by multiplying them by 
powers of 1/t.

Another approach would use polynomial reduction. Short examples of each 
method can be found at the location below.

http://forums.wolfram.com/mathgroup/archive/2007/Dec/msg00476.html

Daniel Lichtblau
Wolfram Research

• References:
  • Simplify with NestedLessLess?
    • From: dbird <dbird@ieee.org>
  • Re: Simplify with NestedLessLess?
    • From: Dave Bird <dbird@ieee.org>