Relation between Simpson's Rule, Trapezoid Rule and Midpoint Rule. How

telegrafyx

telegrafyx

Answered question

2022-06-21

Relation between Simpson's Rule, Trapezoid Rule and Midpoint Rule.
How 2 n numbers come out at the left side while only n numbers at the right side.

Answer & Explanation

aletantas1x

aletantas1x

Beginner2022-06-22Added 22 answers

For the Trapezoidal Rule, you actually use n + 1 points. For example, in the simple case where you are integrating f ( x ) from 0 to 1, and you want T 4 , you evaluate f at the points 0 / 4, 1 / 4, 2 / 4, 3 / 4, and 4 / 4. It is n + 1 points because we use the endpoints.
For the Midpoint Rule, you use n points, but these are not the same points as for the Trapezoidal Rule. They are the midpoints of our intervals. So in the example discussed above, for M 4 you would be evaluating f at 1 / 8, 3 / 8, 5 / 8, and 7 / 8.
The Simpson Rule S 2 n uses evaluation of f at 2 n + 1 points. If for example n = 4, then you are dividing the interval into 8 subintervals. With n = 4 and the interval 8, you would be using the points 0 / 8, 1 / 8, 2 / 8, 3 / 8, 4 / 8, 5 / 8, 6 / 8, 7 / 8, and 8 / 8.
Note that 1 / 8, 3 / 8, 5 / 8 and 7 / 8 are the points that were used for M 4 . The points 0 / 8, 2 / 8, 4 / 8, 6 / 8, and 8 / 8 are just 0 / 4, 1 / 4, 2 / 4, 3 / 4, and 4 / 4, exactly the points that were used for T 4 .
A more abstract summary: T n uses n + 1 points, and M n uses n points. But the n points used by Mn are completely different from the points used for M n . So altogether, T n and M n carry information about function evaluation at 2 n + 1 points, which is exactly what S 2 n does.
I have not written out a proof of the formula, only tried to deal with your discomfort with the 2 n on one side and n's on the other. The formula is not hard to verify. Let's do it explicitly for n = 4. Write down, say for the interval [ 0 , 1 ], what T 4 is. We have
T 4 = 1 8 ( f ( 0 ) + 2 f ( 1 / 4 ) + 2 f ( 1 / 2 ) + 2 f ( 3 / 4 ) + f ( 1 ) ) .
Now write down M 4 : M 4 = 1 4 ( f ( 1 / 8 ) + f ( 3 / 8 ) + f ( 5 / 8 ) + f ( 7 / 8 ) ) .
Now calculate T 4 + 2 M 4 . It is convenient for the addition to make sure that M 4 has denominator 8, so write 2 M 4 as 1 8 ( 4 f ( 1 / 8 ) + 4 f ( 3 / 8 ) + 4 f ( 5 / 8 ) + 4 f ( 7 / 8 ) ), and add. Divide by 3 and you will get the expression you would get in S 8 . The same method works in general.

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