Linearization of a function

**aamirmub** · Nov14-09, 12:21 PM

Hi,

I am trying to understand an example from a FEM software manual. The manual mentions a nonlinear equation

and this equation is linearized to obtain

. Can any one please explain how this has been done?

**mathman** · Nov14-09, 05:44 PM

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

**aamirmub** · Nov14-09, 06:07 PM

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.

**HallsofIvy** · Nov15-09, 04:55 AM

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

**aamirmub** · Nov15-09, 10:38 AM

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