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Linearization of a function

  1. Nov 14, 2009 #1
    Hi,

    I am trying to understand an example from a FEM software manual. The manual mentions a nonlinear equation http://aamir-pc:2080/v6.9/books/exa/graphics/exa_eqn00137.gif [Broken] and this equation is linearized to obtain http://aamir-pc:2080/v6.9/books/exa/graphics/exa_eqn00152.gif [Broken].[/URL] Can any one please explain how this has been done?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Nov 14, 2009 #2

    mathman

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    You need to fix your messages. The two equations don't show up.
     
  4. Nov 14, 2009 #3
    The nonlinear equation is Y= G^(-1) * X + a * X^3 where G and a are constants. The linearized equation is Y(i+1) = (G^(-1) + a * X(i)^2) * X(i+1) where i and i+1 are superscripts.
     
  5. Nov 15, 2009 #4

    HallsofIvy

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    The first thing done is factor out an "X": Y= (G-1+ aX2)X. The next thing done was convert to a recursive form by treating the separate "X"s as if they were different terms in a sequence: Yi+1= (G-1+ aXi2)Xi+1. Given a starting value, X1, you could then calculate a sequence of "Y"s. If that sequence convertes, then [itex]Y= \lim_{i\to\infty}Y^i[/itex] will satisfy that equation: [itex]\lim_{i\to \infty} Y^i= (G^{-1}+ a(\lim_{i\to\infty}X^i)^2)(\lim_{i\to\infty}X^{i+1})[/itex] and, since "Xi" and "Xi+1" refer to the same sequence they both converge to the same limit, X.
     
  6. Nov 15, 2009 #5
    Thank you for your reply. Is the linearization carried out using the first two terms of the taylor series in incremental form?
     
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