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Newton–Cotes Formulas

May 14, 20245 min read

Introduction

Another method to calculate integrals is using approximate numerical methods, which you may see somewhere in Calculus, such as Riemann Sums. Newton-Cotes formulas are a class of methods to achieve that purpose. These include popular rules like Simpson’s rule, midpoint rule, Trapezoidal rule, …

Euclidean Division

Given 2 polynomials and , there exist 2 polynomial (quotient) and (remainder) which statisfy

and

Polynomial Remainder Theorem

Given number and polynomial , by using Euclidean Division theorem for and

Then replacing , we get

which is the remainder. Thus we can rewrite the formula as

Chinese Remainder Theorem

Polynomial Interpolation

Interpolating polynomial is defined to be a polynomial that fits a set of given data points of a function and give the exact value corresponding the value of the function being approximated.

Lagrange Interpolation

Let be the Lagrange Polynomial order of that passes through set of data points and is defined as

where is the Lagrange basis polynomial. Let then the basis can be also represented as

Newton Interpolation

Instead of using Lagrange Polynomial, you can use Newton Polynomial form which is defined as

or by the recursive relation

where is divided difference and with is the Newton basis polynomial.

The forward divided differences can be defined as

It can be also represented by using Newton tableau

By putting the point we get

where

This is also called the remainder of the interpolation polynomial.

Hermite Interpolation Formula

HermiteFormula.excalidraw

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Let be the contour around only one point in a counterclockwise manner then can be rewritten

This can be verified by taking the Residue

Let be the closed contour that contains except the point then from and we have

Now by extending the contour to which contains the points and

or

which is also the remainder term of lagrange polynomial.

So the contour integral representation on contour for lagrange polynomial can be written as

or

This is also known as Hermite interpolation formula

Approximation Error

Consisder remainder term which is the approximation error when approximating function with Interpolation Polynomial

Because the Interpolation Polynomial itself gives the exact value of point thus

Contour integral form of the remainder

Lagrange form of the remainder

Newton form of the remainder

Thus we can observe that has at least distinct zeros at . By Rolle’s theorem between distict and , there exists at least one of . Thus, has at least zeros between and . So on, we can conclude that has at least one zero in the interval

Now by the newton interpolation, we know that Hence, we obtain the mean value theorem for Divided Differences

Finally we can rewrite the remainder as

where

Newton–Cotes Formulas

After constructing the Lagrance polynomial, Newton–Cotes formulas can be defined as follows

where

  • with for closed formula.
  • with for closed formula.

To be able to use we need to calculate the weights

Then can be rewritten as

Error Analysis

The error of the Newton–Cotes formulas is defined as

So using we obtain

where . Thus we obtain

substitute we get the form

For with it is known as the rectangle rule

For with and we get the trapiozal rule

For with points , , and it becomes the 1/3 Simpson’s rule

Oops zero appears!! But it doesnt mean the error is zero. It is just because the cubic term is zero. Instead, we can use the next term of the newton polynomial by adding one more interpolation point to calculate it

References

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