Newton-Cotes Formulas

Closed composite Newton-Cotes formulas of degree 2 to 8.

cotes(f, a, b, n, nodes, ...)

Arguments

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.

Details

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.

Value

The integral as a scalar.

Note

It is generally recommended not to apply Newton-Cotes formula of degrees higher than 6, instead increase the number n of subintervals used.

Author

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.

References

Quarteroni, A., R. Sacco, and F. Saleri (2007). Numerical Mathematics. Second Edition, Springer-Verlag, Berlin Heidelberg.

Examples

cotes(sin, 0, pi/2, 20, 2)      # 0.999485905248533
#> [1] 0.9994859
cotes(sin, 0, pi/2, 20, 3)      # 1.000000211546591
#> [1] 1
cotes(sin, 0, pi/2, 20, 4)      # 1.000000391824184
#> [1] 1
cotes(sin, 0, pi/2, 20, 5)      # 0.999999999501637
#> [1] 1
cotes(sin, 0, pi/2, 20, 6)      # 0.999999998927507
#> [1] 1
cotes(sin, 0, pi/2, 20, 7)      # 1.000000000000363  odd degree is better
#> [1] 1
cotes(sin, 0, pi/2, 20, 8)      # 1.000000000002231
#> [1] 1