periodicTrapRule1D {sdetorus}

Quadrature rules in 1D, 2D and 3D

Description

Quadrature rules for definite integrals over intervals in 1D, \int_{x_1}^{x_2} f(x)dx, rectangles in 2D,
\int_{x_1}^{x_2}\int_{y_1}^{y_2} f(x,y)dydx and cubes in 3D, \int_{x_1}^{x_2}\int_{y_1}^{y_2}\int_{z_1}^{z_2} f(x,y,z)dzdydx. The trapezoidal rules assume that the function is periodic, whereas the Simpson rules work for arbitrary functions.

Usage

periodicTrapRule1D(fx, endsMatch = FALSE, na.rm = TRUE,
  lengthInterval = 2 * pi)

periodicTrapRule2D(fxy, endsMatch = FALSE, na.rm = TRUE,
  lengthInterval = rep(2 * pi, 2))

periodicTrapRule3D(fxyz, endsMatch = FALSE, na.rm = TRUE,
  lengthInterval = rep(2 * pi, 3))

integrateSimp1D(fx, lengthInterval = 2 * pi, na.rm = TRUE)

integrateSimp2D(fxy, lengthInterval = rep(2 * pi, 2), na.rm = TRUE)

integrateSimp3D(fxyz, lengthInterval = rep(2 * pi, 3), na.rm = TRUE)

Arguments

fx

vector containing the evaluation of the function to integrate over a uniform grid in [x_1,x_2].

endsMatch

flag to indicate whether the values of the last entries of fx, fxy or fxyz are the ones in the first entries (elements, rows, columns, slices). See examples for usage.

na.rm

logical. Should missing values (including NaN) be removed?

lengthInterval

vector containing the lengths of the intervals of integration.

fxy

matrix containing the evaluation of the function to integrate over a uniform grid in [x_1,x_2]\times[y_1,y_2].

fxyz

three dimensional array containing the evaluation of the function to integrate over a uniform grid in [x_1,x_2]\times[y_1,y_2]\times[z_1,z_2].

Details

The simple trapezoidal rule has a very good performance for periodic functions in 1D and 2D(order of error ). The higher dimensional extensions are obtained by iterative usage of the 1D rules.

Value

The value of the integral.

References

Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P. (1996). Numerical Recipes in Fortran 77: The Art of Scientific Computing (Vol. 1 of Fortran Numerical Recipes). Cambridge University Press, Cambridge.

Examples

# In 1D. True value: 3.55099937
N <- 21
grid <- seq(-pi, pi, l = N)
fx <- sin(grid)^2 * exp(cos(grid))
periodicTrapRule1D(fx = fx, endsMatch = TRUE)
periodicTrapRule1D(fx = fx[-N], endsMatch = FALSE)
integrateSimp1D(fx = fx, lengthInterval = 2 * pi)
integrateSimp1D(fx = fx[-N]) # Worse, of course

# In 2D. True value: 22.31159
fxy <- outer(grid, grid, function(x, y) (sin(x)^2 * exp(cos(x)) +
                                         sin(y)^2 * exp(cos(y))) / 2)
periodicTrapRule2D(fxy = fxy, endsMatch = TRUE)
periodicTrapRule2D(fxy = fxy[-N, -N], endsMatch = FALSE)
periodicTrapRule1D(apply(fxy[-N, -N], 1, periodicTrapRule1D))
integrateSimp2D(fxy = fxy)
integrateSimp1D(apply(fxy, 1, integrateSimp1D))

# In 3D. True value: 140.1878
fxyz <- array(fxy, dim = c(N, N, N))
for (i in 1:N) fxyz[i, , ] <- fxy
periodicTrapRule3D(fxyz = fxyz, endsMatch = TRUE)
integrateSimp3D(fxyz = fxyz)