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Function Repository Resource:

 NewtonCotesWeights

Source Notebook

Get the abscissa-weight pairs of the elementary Newton-Cotes formula

Contributed by: Jan Mangaldan

ResourceFunction["NewtonCotesWeights"][n,a,b]

gives a list of the n pairs {xi,wi} of the elementary n-point Newton-Cotes formula for quadrature on the interval a to b, where wi is the weight of the abscissa xi.

Details and Options

Newton-Cotes quadrature approximates the value of an integral as a linear combination of values of the integrand evaluated at equally spaced points: .
The abscissas may or may not include the endpoints a and b of the interval. The option "QuadratureType" is used to control whether endpoints are included as abscissas.
With the default setting "QuadratureType"→Closed, the endpoints are included as abscissas.
With the setting "QuadratureType"→Open, the endpoints are not included as abscissas. In this case, the initial abscissa is chosen to be a half step above the lower endpoint and the final is a half step below the upper endpoint.

Examples

Basic Examples (1) 

Find the Newton-Cotes value weight:

In[1]:=
ResourceFunction["NewtonCotesWeights"][10, 0, 9]
Out[1]=

Options (1) 

Set "QuadratureType"→Open to exclude the endpoints:

In[2]:=
ResourceFunction["NewtonCotesWeights"][10, 0, 9, "QuadratureType" -> Open]
Out[2]=

Applications (5) 

Approximate the area below a curve f(x)

Define the function:

In[3]:=
f[x_] := x^3 - 5 x^2 + 3 x + 9

Visualize the area between 0 and 5:

In[4]:=
Plot[f[x], {x, 0, 5}, Filling -> Axis]
Out[4]=

Determine the Newton-Cotes weights:

In[5]:=
weights = ResourceFunction["NewtonCotesWeights"][6, 0, 5]
Out[5]=

Multiply the value at each x-value by its weight:

In[6]:=
Total[#[[2]]*f[#[[1]]] & /@ weights] // N
Out[6]=

Compare to the solution of NIntegrate:

In[7]:=
NIntegrate[f[x], {x, 0, 5}]
Out[7]=

Properties and Relations (1) 

The resource function NewtonCotesError can be used to determine the error in the Newton–Cotes approximation:

In[8]:=
ResourceFunction["NewtonCotesError"][6, f, 0, 5]
Out[8]=

Version History

  • 1.1.0 – 04 January 2021
  • 1.0.0 – 20 November 2019

Source Metadata

  • Citation:
    • Hildebrand, F. B., Introduction to Numerical Analysis. New York: McGraw-Hill, 160–161, 1956.

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