Polynomials and Numerical Analysis

[Suvadip]

Why polynomials are used in numerical analysis?

 

[Nono713]

Re: Numerical analysis

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?​

 

[chisigma]

Re: Numerical analysis

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

$\chi$ $\sigma$

 

[Ackbach]

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.

 

[CellCoree]

Polynomials are a fundamental mathematical concept that is used extensively in numerical analysis for several reasons. First, polynomials are simple and easy to work with, making them a convenient choice for representing and approximating more complex functions. Additionally, polynomials have a well-defined algebraic structure, which allows for efficient mathematical operations such as addition, subtraction, multiplication, and division. This makes it easier to perform numerical computations and obtain accurate results.

Furthermore, polynomials are flexible and versatile, as they can accurately approximate a wide range of functions, including non-polynomial ones. This is especially useful in numerical analysis, where real-world problems often involve complex functions that cannot be solved analytically. By approximating these functions with polynomials, numerical methods can be used to find solutions that are close enough to the exact values.

Moreover, polynomials have a smooth and continuous nature, which makes them suitable for interpolation and curve fitting. In numerical analysis, interpolation refers to the process of estimating the value of a function at a point within a given range, based on known values at surrounding points. The use of polynomials in interpolation allows for the creation of smooth and continuous curves, which can provide more accurate approximations of the original function.

In conclusion, polynomials are an essential tool in numerical analysis due to their simplicity, versatility, and ability to accurately approximate complex functions. Their use allows for efficient and accurate numerical computations, making them a valuable asset in the field of scientific research and problem-solving.