pracma (version 1.1.6)

quad2d: 2-d Gaussian Quadrature

Description

Two-dimensional Gaussian Quadrature.

Usage

quad2d(f, xa, xb, ya, yb, n = 32, ...)

Arguments

f

function of two variables; needs to be vectorized.

xa, ya

lower limits of integration; must be finite.

xb, yb

upper limits of integration; must be finite.

n

number of nodes used per direction.

...

additional arguments to be passed to f.

Value

• A single numerical value, the computed integral.

Details

Extends the Gaussian quadrature to two dimensions by computing two sets of nodes and weights (in x- and y-direction), evaluating the function on this grid and multiplying weights appropriately.

The function f needs to be vectorized in both variables such that f(X, Y) returns a matrix when X an Y are matrices (of the same size).

quad is not suitable for functions with singularities.

References

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

See Also

quad, cubature::adaptIntegrate

Examples

##  Example:  f(x, y) = (y+1)*exp(x)*sin(16*y-4*(x+1)^2)
f <- function(x, y)
        (y+1) * exp(x) * sin(16*y-4*(x+1)^2)
# this is even faster than cubature::adaptIntegral():
quad2d(f, -1, 1, -1, 1)
# 0.0179515583236958  # true value 0.01795155832370

##  Volume of the sphere: use polar coordinates
f0 <- function(x, y) sqrt(1 - x^2 - y^2)  # for x^2 + y^2 <= 1
fp <- function(x, y) y * f0(y*cos(x), y*sin(x))
quad2d(fp, 0, 2*pi, 0, 1, n = 101)  # 2.09439597740074
2/3 * pi                            # 2.0943951023932