Numerical integration - polynomial order confusion

Mondo

Junior Member
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Apr 23, 2021
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107
Hi,

I am a bit confused by the the meaning of order in approximating polynomials. I though a constant is a zero order, a line is order one, a quadratic function is of second order and so on. In a simple rectangular approximation we do [imath]\int{f(x)} = \sum_{n=1}^{\infty} f_n dx[/imath]. In the trapezoidal method we do [math]\int{f(x)} = \sum_{n=1}^{\infty} \frac{h}{2}(f_n + f_{n+1}) dx[/math]. Now, it is said that in the rectengular method we integrate 0 order polynomial (a constant) to get an integral which is of order one. Here is my first question, how can the integral be an integral of order one? Likewise in the trapezoid method - it is said that we integrate a first order polynomial (a line) to get an integral of second order. So why in the average of two consecutive points in the trapezoid method considered a line? And more importantly why is the resulting integral second order? After all it is the same plain multiplication as it was in the rectengular case, just one of the arguments in an average of two points instead of a single one.

Thanks!
 
Seems that more context is needed, i.e., "it is said" where ?
My best guess: the order of error relative to [imath]h[/imath] in the rectangular method is 1, i.e. the error is the same order of magnitude as [imath]h[/imath], but for the linear method the error is the same order as [imath]h^2[/imath]. Would this make sense in your context ?
 
it is said that in the rectengular method we integrate 0 order polynomial (a constant) to get an integral polynomial which is of order one. Here is my first question, how can the integral be an integral polynomial of order one?
In your statement, the second "integral" should be replaced of polynomial. Does that answer your question?
 
I presume you know the basics of integration:
\(\displaystyle \int a dx= ax+ c\)
\(\displaystyle \int ax dx= \frac{a}{2}x^2+ c\)
\(\displaystyle \int ax^2 dx= \frac{a}{3}x^3+ c\)
and in general
\(\displaystyle \int ax^n dx= \frac{a}{n+1}x^{n+1}+ c\).

That is, the integral of a polynomial of degree n is a polynomial of degree n+ 1.
(Notice that I am using the word "degree" not "order". A polynomial has a "degree", not an "order". If you mean, as blamocur suggests, the "order of error" in a numerical integration, please clarify.)
 
@HallsofIvy, ahh right, I got blinded for a moment. Yes it makes sense now, expect one thing "So why is the average of two consecutive points in the trapezoid method considered a line?" I am still not sure why the trapezoid method is said to be an equation of a line?
 
"So why is the average of two consecutive points in the trapezoid method considered a line?"
The statement above is within quotes - can you provide a reference to that "quotation"?

Do you know the equation of area of a trapezoid?
 
"So why is the average of two consecutive points in the trapezoid method considered a line?
That makes no sense. Numbers have averages, not points! Of course, it is true that there exist a unique line through two given points. Is that what you mean? That is a basic property, in fact, a postulate, of Euclidean Geometry.
 
"So why is the average of two consecutive points in the trapezoid method considered a line?" The average of two consecutive points is multiplied by another dimension (horizontal distance "h" between two consecutive points) to give the area of the trapezoid. But the trapezoid method is synonymous with line approximation because the top of the trapezoid is a line connecting two neighboring discrete points to approximate the values of function f, i.e. first order approximation.
 

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@blamocur, ok this way it makes sense. I got confused because the way you described it really suggest it is a line, however if you think of trapezoid method as the average distance between two adjacent points then it has nothing to do with a line.
 
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