R: double integrals on a function with matrix output - r

Question

I would like to calculate the double integrals on a function with matrix output. My code is here:
foo <- function(x,t){
out1 <- sin(x*t)
out2 <- cos(x*t)
out <- matrix(c(out1,out2),ncol=2)
return(out)
}
library(pracma)
quad2d(foo, xa = 0, xb = 1, ya = 0, yb= 10)
When I run this code, I receive a message:
Error in wx %*% Z : non-conformable arguments
I wonder how I can calculate this double integrals? The 'foo' is just a simple example here; to seperate the 'foo' into two functions is not I want. Thanks in advance!

Answer 1

quad2dapplies Gaussian quadrature internally, but needs the function to be vectorized. Instead, you can apply the quadrature rules explicitly.
Assume the matrix-valued function foo is defined as
foo <- function(x, t)
matrix(c(sin(x*t), cos(x*t), x, t), 2, 2)
Utilize the Gaussian quadrature rules directly like this:
library(pracma)
n <- 32
xa <- 0; xb <- 1; ya <- 0; yb <- 10
cx <- gaussLegendre(n, xa, xb)
x <- cx$x; wx <- cx$w
cy <- gaussLegendre(n, ya, yb)
y <- cy$x; wy <- cy$w
I <- matrix(0, 2, 2)
for (i in 1:n) {
for (j in 1:n) {
I <- I + wx[i] * wy[j] * foo(x[i], y[j])
}
}
I
## [,1] [,2]
## [1,] 2.925257 5
## [2,] 1.658348 50
This may be a bit slower than quad2d as there are two for loops, but may be sufficient if your function is somewhat well behaved.
Of course, all this can be wrapped into a new integration routine like
mquad2d(f, xa, xb, ya, yb, n = 32).

Answer 2

There may be an elegant way to pass a matrix of functions as the first argument of the function quad2d! I don't know that, but I can re-write your code to get the same result:
mquad2d <- function(xa, xb, ya, yb){
fun1 <- function(x,t){
return(sin(x*t))
}
fun2 <- function(x, t){
return(cos(x*t))
}
out1 <- quad2d(fun1, xa, xb, ya, yb)
out2 <- quad2d(fun2, xa, xb, ya, yb)
out <- matrix(c(out1,out2),ncol=2)
return(out)
}
mquad2d(xa = 0, xb = 1, ya = 0, yb= 10)
#output:
# [,1] [,2]
#[1,] 2.925257 1.658348
Hope this helps!