Linearization of non linear model

[asad1111]

i read that taylor series is used to approximate non linear function at optimal point x0 but i don't understand in which case we use first order approximation and in which cases we use higher order approximations?

 

[jsgruszynski]

Here's two examples:

1. Say you want to build a calculator that does a sine function but you only have addition and multiplication functions. How do you do it. Well you build the Taylor expansion of the sine. That's an infinite series however. But do you need all of those terms? No. Just enough to be meet your accuracy specification. And that's the answer to your question of "how many terms": enough for your application. For this expansion the reference point is 0.

2. Another case has to do with how you use it. An example of that is the SPICE circuit simulator. For every nonlinear component like a bipolar transistor, SPICE creates a Taylor expansion to solve the circuit equations. But how many terms? Well because you want to systematically solve any circuit topology, you need to use something mathematically systematic: linear algebra. Which only can solve linear equations. Not quadratics or higher order. Not directly. So the Taylor expansion is terminated at the first order derivative term to get just a linear approximation. And SPICE gets the higher order by using iterative numerical techniques called a Newton-Euler Forward extrapolation. BTW the reference point for the Taylor expansion for SPICE is the DC bias point for the transistor that was calculated at an earlier phase of the simulation.

 

[asad1111]

thanks repky is very good

 

[epenguin]

i read that taylor series is used to approximate non linear function at optimal point x0 but i don't understand in which case we use first order approximation and in which cases we use higher order approximations?

This approximation is used because it is an approximation. Good enough for some practical calcualtion near x₀ with the advantage that it can be more easily calculated than the full case, or that it can be at all calculated. And often gives you the essence of what you need to know.

You go to higher order if you really need more exact results.

But more significantly when first approximation fails to give you information. This can happen when the derivatives at x₀ are 0. Then you find your answers near x₀ are the same as those at x₀. From memory this happens in treatments of the Gibbs-Donnan equilibrium, a problem of osmotic pressure of mixed solutes.

In non-linear differential equations linearisation is routinely used to see the local nature of the stationary point, whether attracting, repelling etc. This local nature might or might not then be the overall nature, but it is always illuminating. Again it can fail when derivatives are 0 and you have to go to higher approximation.

Overall the importance of linear approximation is at least as much qualitative as quantitative I'd guess.