1.what is linear approximation used for?

2.why would you use the linear approximation of a function than the actual function

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- Oct 16th 2006, 10:01 AMbobby77What are Linear approximations used for
1.what is linear approximation used for?

2.why would you use the linear approximation of a function than the actual function - Oct 16th 2006, 03:49 PMCaptainBlack
A nice example of the use of a linear approximation is in the Newton Raphson

method for finding numerical roots of equations. You start with an estimate

of the root, you then replace the function by its tangent (a linear

approximation) find the zero of the equation of the tangent, to give a

new approximation to the root. Repeat this process as often as needed.

Quote:

2.why would you use the linear approximation of a function than the actual function

usually simpler than with the function itself. Or to use known results from

a linear theory to give an approximate result in a non-linear case.

A nice example which is probably beyond your current experience is the

Kalman filter, which is an optimal estimator for certain types of linear

systems. With mildly non-linear systems one linearises about an appropriate

estimate, and then uses the KF equations to give a sub-optimal but good

estimator for the non-linear case (known as the Extended Kalman Filter (EKF)).

RonL - Oct 16th 2006, 05:41 PMThePerfectHacker
- Oct 17th 2006, 12:18 AMCaptainBlack
- Oct 17th 2006, 06:20 AMtopsquark
- Oct 17th 2006, 07:17 AMThePerfectHacker
I cannot possibly see where in math these approximations can be used. You cannot create equality when two quantitites are close. In science you can (this is where that entire derivative thing comes int).

- Oct 17th 2006, 08:37 AMCaptainBlack
- Oct 17th 2006, 09:03 AMThePerfectHacker
- Oct 17th 2006, 11:02 AMAfterShock
With linear approximations, there are also error bounds. For problems that you're unable to get an equality, an approximation is the next best thing. They can be especially useful (more so when you know HOW useful it is); I completely disagree with:

"If you are asking for mathematics, then the answer is no purpose at all."

Consider other approximations, such as Riemann sums or taylor polynomials. Do these serve no purpose at all, for an integral (for instance) that cannot be found exactly? - Oct 17th 2006, 11:07 AMThePerfectHacker
- Oct 17th 2006, 11:58 AMCaptainBlack
- Oct 17th 2006, 12:12 PMJakeD
Serge Lang in

*Real Analysis*proves a Mean Value Theorem of which one form is

|f(z) - f(x)| <= |z - x| sup |f'(v)|

for v on the line segment connecting x and z. That is a linear approximation.

He then uses this theorem to prove the Inverse Mapping Theorem, Implicit Mapping Theorem and Existence Theorem for Differential Equations. Theses theorems all require functions to be differentiable so that a linear approximation may be used.

In fact, whenever any theorem or definition requires differentiability, it is probably to use a linear approximation. That's almost true by definition: a function is differentiable iff it can approximated locally by linear map. Consider a C1 manifold in R3 in Differential Topology. It is a surface that can be approximated locally by its tangent plane, which is a linear approximation. - Oct 17th 2006, 12:19 PMJakeD
- Oct 17th 2006, 12:49 PMCaptainBlack
- Oct 17th 2006, 02:00 PMThePerfectHacker
What JakeD said about the MVT is exactly what I meant when inequalities are used non-stop in number theory. However, when in physics we drop a term because we assume it does not exist (since it is almost zero) we need to consider in math, which is what the error terms accomplish.