MHB Polynomials and Numerical Analysis

AI Thread Summary
Polynomials are favored in numerical analysis due to their straightforward structure, ease of manipulation, and well-understood behavior in various mathematical operations. Their ability to be differentiated, integrated, and approximated numerically makes them highly practical for solving problems. Additionally, polynomials are dense in large function spaces, allowing for the approximation of many functions, which is particularly useful in differential equations. This versatility is exemplified in applications like modeling the motion of a drum head using Bessel functions expressed as infinite polynomial sums. Overall, polynomials serve as a fundamental tool in numerical analysis for their flexibility and effectiveness.
Suvadip
Messages
68
Reaction score
0
Why polynomials are used in numerical analysis?
 
Mathematics news on Phys.org
Re: Numerical analysis

suvadip said:
Why polynomials are used in numerical analysis?

This is a very broad question.. I suppose because their structure is relatively straightforward and well-understood, and they are flexible yet easy to manipulate (it's trivial to differentiate/integrate/add/multiply polynomials, they are well-behaved with respect to numerical approximation methods, we know exactly when they cross the x-axis, we can easily find their minima and maxima, they work the same in the complex plane, and so on..) Can you be more specific?​
 
Re: Numerical analysis

suvadip said:
Why polynomials are used in numerical analysis?

Polynomial are based on the elentary operators of sum and multiplication, the most feasible for humans and computers... that's why N.A., the scope of which is to solve numerically problems, is pratically based on polynomials...

Kind regards

$\chi$ $\sigma$
 
Re: Numerical analysis

I think another reason why numerical analysis uses polynomials is that they are dense in some very large function spaces. That means (in case you weren't already aware of what it means) that you can approximate a very large number of functions with polynomials. This is quite useful in differential equations, with series methods like Frobenius. Indeed, the motion of, say, a round drum head when you hit it can be modeled using Bessel functions, which are written as an infinite sum of polynomials.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Back
Top