Newton-Cotes Formulas
Closed composite Newton-Cotes formulas of degree 2 to 8.
cotes(f, a, b, n, nodes, ...)
f |
the integrand as function of two variables. |
a, b |
lower and upper limit of the integral. |
n |
number of subintervals (grid points). |
nodes |
number of nodes in the Newton-Cotes formula. |
... |
additional parameters to be passed to the function. |
2 to 8 point closed and summed Newton-Cotes numerical integration formulas.
These formulas are called ‘closed’ as they include the endpoints. They are called ‘composite’ insofar as they are combined with a Lagrange interpolation over subintervals.
The integral as a scalar.
It is generally recommended not to apply Newton-Cotes formula of degrees
higher than 6, instead increase the number n of subintervals used.
Standard Newton-Cotes formulas can be found in every textbook. Copyright (c) 2005 Greg von Winckel of nicely vectorized Matlab code, available from MatlabCentral, for 2 to 11 grid points. R version by Hans W Borchers, with permission.
Quarteroni, A., R. Sacco, and F. Saleri (2007). Numerical Mathematics. Second Edition, Springer-Verlag, Berlin Heidelberg.
cotes(sin, 0, pi/2, 20, 2) # 0.999485905248533 cotes(sin, 0, pi/2, 20, 3) # 1.000000211546591 cotes(sin, 0, pi/2, 20, 4) # 1.000000391824184 cotes(sin, 0, pi/2, 20, 5) # 0.999999999501637 cotes(sin, 0, pi/2, 20, 6) # 0.999999998927507 cotes(sin, 0, pi/2, 20, 7) # 1.000000000000363 odd degree is better cotes(sin, 0, pi/2, 20, 8) # 1.000000000002231
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