Automatic Differentiation with Dual Numbers

Automatic differentiation is achieved by using dual numbers without providing hand-coded gradient functions. The output value of a mathematical function is returned with the values of its exact first derivative (or gradient). For more details see Baydin, Pearlmutter, Radul, and Siskind (2018) < https://jmlr.org/papers/volume18/17-468/17-468.pdf>.

dual: an R package for dual numbers

Authors

Luca Sartore

Maintainer: Luca Sartore

Features of the package

Dual numbers are mainly used to implement automatic differentiation. The dual package provides mathematical functions that are able to handle computations with dual numbers. The package is useful to calculate exact derivatives in R without providing self-coded functions.

For a complete list of exported functions, use library(help = "dual") once the dual package is installed (see the inst/INSTALL.md file for a detailed description of the setup process).

Example

library(dual)
x <- dual(f = 1.5, grad = c(1:0, 0))
y <- dual(f = 0.5, grad = c(0:1, 0))
z <- dual(f = 1.0, grad = c(0, 0:1))
exp(z - x) * sin(x)^y / x

a <- dual(f = 1.1, grad = c(1.2, 2.3, 3.4, 4.5, 5.6))
0.5 * a^2 - 0.1

lambertW <- function(x) {
  w0 <- 1
  w1 <- w0 - (w0*exp(w0)-x)/((w0+1)*exp(w0)-(w0+2)*(w0*exp(w0)-x)/(2*w0+2))
  while(abs(w1-w0) > 1e-15) {
    w0 <- w1
    w1 <- w0 - (w0*exp(w0)-x)/((w0+1)*exp(w0)-(w0+2)*(w0*exp(w0)-x)/(2*w0+2))
  }
  return(w1)
}
lambertW(dual(1, 1))

References

Baydin, A. G., Pearlmutter, B. A., Radul, A. A., & Siskind, J. M. (2018). Automatic differentiation in machine learning: a survey. Journal of Machine Learning Research, 18, 1-43.

Cheng, H. H. (1994). Programming with dual numbers and its applications in mechanisms design. Engineering with Computers, 10(4), 212-229.

Kisil, V. V. (2007). Erlangen program at large-2: inventing a wheel. The parabolic one. arXiv: 0707.4020.