# Linearization of a function

1. Nov 14, 2009

### aamirmub

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. Nov 14, 2009

### mathman

You need to fix your messages. The two equations don't show up.

3. Nov 14, 2009

### aamirmub

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.

4. Nov 15, 2009

### HallsofIvy

Staff Emeritus
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 $Y= \lim_{i\to\infty}Y^i$ will satisfy that equation: $\lim_{i\to \infty} Y^i= (G^{-1}+ a(\lim_{i\to\infty}X^i)^2)(\lim_{i\to\infty}X^{i+1})$ and, since "Xi" and "Xi+1" refer to the same sequence they both converge to the same limit, X.

5. Nov 15, 2009

### aamirmub

Thank you for your reply. Is the linearization carried out using the first two terms of the taylor series in incremental form?