quad2d: 2-d Gaussian Quadrature

In pracma: Practical Numerical Math Functions

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.

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.

Value

A single numerical value, the computed integral.

Note

The extension of Gaussian quadrature to two dimensions is obvious, but see also the example ‘integral2d.m’ at Nick Trefethens “10 digits 1 page”.

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

pracma documentation built on Nov. 10, 2023, 1:14 a.m.