Undergrad Some questions from a numerical analysis book

[awholenumber]

https://www.amazon.com/dp/1305253663/?tag=pfamazon01-20

Are all these called root finding algorithms? All of it ?

<Moderator note: Link changed because of undetermined copyright situation. fresh_42>

 

[BvU]

Can't find what this is about ...

 

[DrClaude]

Are all these called root finding algorithms? All of it ?

This is a 300 page book. You will have to specify what you are talking about.

 

[awholenumber]

for example , for a polynomial ,

a solution of a polynomial equation is also called a root of the polynomial .

a value for the variable that makes the polynomial zero

if you can't find an exact expression, then you can use numerical methods to get approximations .

with numerical methods you can choose how close to zero you want, and it will give you a value that's at least that close

Are we trying to find the roots with all these types of numerical methods ? or is it called finding something else ?

 

[DrClaude]

Generally speaking, a numerical algorithm that allows you to solve a problem of the type $\mathbf{F}(\mathbf{x}) = 0$ is indeed called a root-finding algorithm.

 

[awholenumber]

So what are we trying to do when we use numerical differentiation ? I don't think its called root finding ? is it ?

 

[DrClaude]

So what are we trying to do when we use numerical differentiation ? I don't think its called root finding ? is it ?

No, it is called numerical differentiation.

 

[awholenumber]

OK , thanks

 

[Mark44]

A book on Numerical Analysis typically discusses techniques for solving equations, finding derivatives numerically, and calculating integrals numerically. It might also discuss techniques for finding matrix inverses, finding eigenvalues and eigenvectors of matrices, and other topics.

 

[awholenumber]

Thanks

 

[Mark44]

Really, though, @rosiekidcute, if you're struggling with basic concepts like fractions, factoring, and trig (as in recent threads), it's not very likely that you will understand topics in numerical analysis or partial differential equation.

 

[awholenumber]

I keep thinking of naturally occurring things whenever i think of derivatives , differential equations and partial differential equations .Like electricity , magnetic field .
I am always like if there is a derivative in a differential equation , how could have that happened ? what could have made it like that . Maybe i should stop thinking about it that way .Its just some rules , Maybe its my lack of experience .

 

[jbriggs444]

I keep thinking of naturally occurring things whenever i think of derivatives , differential equations and partial differential equations .Like electricity , magnetic field .
I am always like if there is a derivative in a differential equation , how could have that happened ? what could have made it like that .

No need to invoke electricity or magnetism. Consider Newton's second law for an object with a fixed mass: F=ma. Force equals mass times acceleration.

That acceleration is a the first derivative of velocity. $F=m \frac{dv}{dt}$.
That velocity is the first derivative of position. $F = m \frac{d^2x}{dt^2}$.
If we are dealing with a mass on a spring, that force is also given by Hooke's Law: F = -kx.
So we have $-kx = m \frac{d^2x}{dt^2}$

That's a differential equation. (A second order homogeneous linear differential equation. There is a straightforward crank-and-grind approach to solving those).

 

[awholenumber]

Thanks ,

Same things right ?

 

[Mark44]

View attachment 205477

Same things right ?

No, these are different quantities: displacement, velocity, acceleration, and jerk.

 

[awholenumber]

Right , sorry about that .