Brobdingnagian matrices

Robin K. S. Hankin

To cite the Brobdingnag package in publications, please use Hankin (2007). R package Brobdingnag has basic functionality for matrices. It includes matrix multiplication and addition, but determinants and matrix inverses are not implemented. First load the package:

library("Brobdingnag")

The standard way to create a Brobdgingnagian matrix (a brobmat) is to use function brobmat() which takes arguments similar to matrix() and returns a matrix of entries created with brob():

M1 <- brobmat(-10:13,4,6)
colnames(M1) <- state.abb[1:6]
M1
#>      AL        AK       AZ       AR      CA      CO      
#> [1,] +exp(-10) +exp(-6) +exp(-2) +exp(2) +exp(6) +exp(10)
#> [2,] +exp(-9)  +exp(-5) +exp(-1) +exp(3) +exp(7) +exp(11)
#> [3,] +exp(-8)  +exp(-4) +exp(0)  +exp(4) +exp(8) +exp(12)
#> [4,] +exp(-7)  +exp(-3) +exp(1)  +exp(5) +exp(9) +exp(13)

Above, note that all entries of M1 are greater than zero; M1[1,1], for example, is exp(-10), or $e^{-10}\simeq 4.54\times 10^{-5}$. For negative matrix entries, function brobmat() takes a Boolean argument positive that specifies the sign:

M2 <- brobmat(
c(1,104,-66,45,1e40,-2e40,1e-200,232.2),2,4,
positive=c(T,F,T,T,T,F,T,T))
M2
#>      [,1]      [,2]      [,3]         [,4]        
#> [1,] +exp(1)   +exp(-66) +exp(1e+40)  +exp(1e-200)
#> [2,] -exp(104) +exp(45)  -exp(-2e+40) +exp(232.2)

Standard matrix arithmetic is implemented, thus:

rownames(M2) <- c("a","b")
colnames(M2) <- month.abb[1:4]
M2
#>   Jan       Feb       Mar          Apr         
#> a +exp(1)   +exp(-66) +exp(1e+40)  +exp(1e-200)
#> b +exp(104) +exp(45)  +exp(-2e+40) +exp(232.2)
M2[2,3] <- 0
M2
#>   Jan       Feb       Mar         Apr         
#> a +exp(1)   +exp(-66) +exp(1e+40) +exp(1e-200)
#> b +exp(104) +exp(45)  +exp(-Inf)  +exp(232.2)
M2+1000
#>   Jan          Feb          Mar          Apr         
#> a +exp(6.9105) +exp(6.9078) +exp(1e+40)  +exp(6.9088)
#> b +exp(104)    +exp(45)     +exp(6.9078) +exp(232.2)

We can also do matrix multiplication, although it is slow:

M2 %*% M1
#>   AL          AK          AZ          AR          CA          CO         
#> a +exp(1e+40) +exp(1e+40) +exp(1e+40) +exp(1e+40) +exp(1e+40) +exp(1e+40)
#> b +exp(225.2) +exp(229.2) +exp(233.2) +exp(237.2) +exp(241.2) +exp(245.2)

Numerical verification: matrix multiplication

We will verify matrix multiplication by carrying out the same operation in two different ways. First, create two largish Brobdingnagian matrices:

nrows <- 11
ncols <- 18
M3 <- brobmat(rnorm(nrows*ncols),nrows,ncols,positive=sample(c(T,F),nrows*ncols,replace=T))
M4 <- brobmat(rnorm(nrows*ncols),ncols,nrows,positive=sample(c(T,F),nrows*ncols,replace=T))
M3[1:3,1:3]
#>      [,1]          [,2]          [,3]         
#> [1,] +exp(-1.4)    -exp(0.62898) -exp(-1.3045)
#> [2,] -exp(0.25532) +exp(2.065)   -exp(0.73778)
#> [3,] -exp(-2.4373) +exp(-1.631)  +exp(1.8885)

Now calculate the matrix product by coercing to numeric matrices and using base R matrix multiplication:

p1 <- as.matrix(M3) %*% as.matrix(M4)

Secondly, we use Brobdingnagian matrix multiplication, and then coercing to numeric:

p2 <- as.matrix(M3 %*% M4)

The difference:

max(abs(p1-p2))
#> [1] 8.526513e-14

is small. Now the other way:

q1 <- M3 %*% M4
q2 <- as.brobmat(as.matrix(M3) %*% as.matrix(M4))
max(abs(as.brob(q1-q2)))
#> [1] +exp(-30.245)

Above we see that the difference of exp(-30.267) (about $7\times 10^{-14}$), is small.

Numerical verification: integration with the cubature package

The matrix functionality of the Brobdingnag package was originally written to leverage the functionality of the cubature package. Here I give some numerical verification for this.

Suppose we wish to evaluate

$$\int_{x=0}^{x=4}(x^2-4)\,dx$$

using numerical methods. See how the integrand includes positive and negative values; the theoretical value is $\frac{16}{3}=5.33\ldots$. The cubature idiom for this would be

library("cubature")

f.numeric <- function(x){x^2 - 4}

out.num <- cubature::hcubature(f = f.numeric, lowerLimit = 0, upperLimit = 4, vectorInterface = TRUE)
out.num
#> $integral
#> [1] 5.333333
#> 
#> $error
#> [1] 1.763322e-13
#> 
#> $functionEvaluations
#> [1] 15
#> 
#> $returnCode
#> [1] 0

and the Brobdingnagian equivalent would be

f.brob <- function(x) {
    x <- as.brob(x[1, ])
    as.matrix( brobmat(x^2 - 4, ncol = length(x)))
}

out.brob <- cubature::hcubature(f = f.brob, lowerLimit = 0, upperLimit = 4, vectorInterface = TRUE)
out.brob
#> $integral
#> [1] 5.333333
#> 
#> $error
#> [1] 1.763322e-13
#> 
#> $functionEvaluations
#> [1] 15
#> 
#> $returnCode
#> [1] 0

We may compare the two methods:

out.brob$integral - out.num$integral
#> [1] 8.881784e-16

References

Hankin, R. K. S. 2007. “Urn Sampling Without Replacement: Enumerative Combinatorics in .” Journal of Statistical Software 17 (Code Snippet 1).