MHB Polynomials and Numerical Analysis

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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
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Why polynomials are used in numerical analysis?
 
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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

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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.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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