Thread: What 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

Originally Posted by bobby77

1.what is linear approximation used for?

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.

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

Because the computation with a linear approximation to a function is
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

Originally Posted by bobby77

1.what is linear approximation used for?
2.why would you use the linear approximation of a function than the actual function

If you are asking for mathematics, then the answer is no purpose at all. Only the sciences rely on it.

Originally Posted by ThePerfectHacker

If you are asking for mathematics, then the answer is no purpose at all. Only the sciences rely on it.

And with one bound you write off at least one branch of Mathematics, and
disallow numerous proofs in many others.

RonL

Originally Posted by ThePerfectHacker

If you are asking for mathematics, then the answer is no purpose at all. Only the sciences rely on it.

Isn't a lot of non-Euclidean geometry based on linear approximations?

-Dan

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).

Originally Posted by ThePerfectHacker

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).

And because you cannot see it it is not so?

RonL

Originally Posted by CaptainBlack

And because you cannot see it it is not so?

Unless, you consider inequalities to be approximations.

For example, in number theory there are more inequalities than eqations. None stop use of them.

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?

Originally Posted by AfterShock

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?

In applied math approximations are useful. But you cannot work with an integral approximated by a power series. However, you can use the infinite series (that has a purpose).

Originally Posted by AfterShock

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?

My memory is a bit hazy, but if I recall correctly the proof of the
central limit theorem relies on the use of a truncated power series
with error term.

RonL

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.

Originally Posted by CaptainBlack

My memory is a bit hazy, but if I recall correctly the proof of the
central limit theorem relies on the use of a truncated power series
with error term.

RonL

Statistics uses linear approximations in many, many contexts. But statistics does not count to TPH. It is applied math.

Originally Posted by JakeD

Statistics uses linear approximations in many, many contexts. But statistics does not count to TPH. It is applied math.

Not when taught from a measure theory point of view surly?

Anybye, his point of view is of no importance to the BIG PICTURE

RonL

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.