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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 NewtonCotesError

Source Notebook

Find the error in the Newton–Cotes quadrature formula

Contributed by: Jan Mangaldan

ResourceFunction["NewtonCotesError"][n,f,a,b]

gives the error in the elementary n-point Newton–Cotes quadrature formula for the function f on an interval from a to b.

Details and Options

The function f may be a pure function or a named function defined using Set or SetDelayed.
The error is given as a function of n, a and b multiplied by an mth derivative of the function f. The size of the error is bounded by the maximum of this expression over the interval from a to b.
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.

Examples

Basic Examples (1) 

Find the error in the five-point Newton–Cotes quadrature of a function:

In[1]:=
ResourceFunction["NewtonCotesError"][5, f, -1, 1]
Out[1]=

Scope (2) 

Determine the error in the Newton–Cotes approximation with four steps between 2 and 7 for a pure function:

In[2]:=
ResourceFunction["NewtonCotesError"][4, Cos[#1] + 4 #1^7 &, 2, 7]
Out[2]=

The error for a function defined using SetDelayed:

In[3]:=
ResourceFunction["NewtonCotesError"][4, h, 2, 7]
Out[3]=

Properties and Relations (2) 

The Newton–Cotes error involves a derivative of f at an unknown point, so you do not really know what the error itself is:

In[4]:=
ResourceFunction["NewtonCotesError"][5, f, a, b]
Out[4]=

However, the error decreases rapidly with the length of the interval:

In[5]:=
ResourceFunction["NewtonCotesError"][5, f, a, a + h]
Out[5]=

For a polynomial of order n, the n-point Newton–Cotes formula gives zero error, expressed through a zero-valued function 0&:

In[6]:=
ResourceFunction["NewtonCotesError"][3, f, 0, 5]
Out[6]=

The Newton–Cotes result is the same as the one returned by Integrate:

In[7]:=
weights = ResourceFunction["NewtonCotesWeights"][3, 0, 5]
Out[7]=
In[8]:=
Total[#[[2]]*f[#[[1]]] & /@ weights]
Out[8]=
In[9]:=
Integrate[f[x], {x, 0, 5}]
Out[9]=

Version History

  • 1.1.0 – 04 January 2021
  • 1.0.0 – 15 November 2019

Source Metadata

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

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