dual: Automatic Differentiation with Dual Numbers

Description

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.

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.

Details

For a complete list of exported functions, use library(help = "dual").

Author(s)

Maintainer: Luca Sartore drwolf85@gmail.com (ORCID = "0000-0002-0446-1328")

Authors:

• Luca Sartore drwolf85@gmail.com (ORCID = "0000-0002-0446-1328")

Luca Sartore drwolf85@gmail.com

Maintainer: Luca Sartore drwolf85@gmail.com

References

Baydin, A. G., Pearlmutter, B. A., Radul, A. A., & Siskind, J. M. (2018). Automatic differentiation in machine learning: a survey. Journal of Marchine 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.

Examples

library(dual)

# Initilizing variables of the function
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))
# Computing the function and its gradient
exp(z - x) * sin(x)^y / x

# General use for computations with dual numbers
a <- dual(1.1, grad = c(1.2, 2.3, 3.4, 4.5, 5.6))
0.5 * a^2 - 0.1

# Johann Heinrich Lambert's W-function
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))

Arithmetic Operators

Description

These unary and binary operators perform arithmetic on dual objects.

Usage

## S4 method for signature 'dual,missing'
e1 + e2

## S4 method for signature 'dual,numeric'
e1 + e2

## S4 method for signature 'numeric,dual'
e1 + e2

## S4 method for signature 'dual,dual'
e1 + e2

## S4 method for signature 'dual,missing'
e1 - e2

## S4 method for signature 'dual,numeric'
e1 - e2

## S4 method for signature 'numeric,dual'
e1 - e2

## S4 method for signature 'dual,dual'
e1 - e2

## S4 method for signature 'dual,numeric'
e1 * e2

## S4 method for signature 'numeric,dual'
e1 * e2

## S4 method for signature 'dual,dual'
e1 * e2

## S4 method for signature 'dual,numeric'
e1 / e2

## S4 method for signature 'numeric,dual'
e1 / e2

## S4 method for signature 'dual,dual'
e1 / e2

## S4 method for signature 'dual,numeric'
e1 ^ e2

## S4 method for signature 'numeric,dual'
e1 ^ e2

## S4 method for signature 'dual,dual'
e1 ^ e2

Arguments

Value

The correspondent values of the arithmetic operation on e1 and e2 is returned.

Examples

x <- dual(1.5, 1:0)
y <- dual(2.6, 0:1)
+x
-x
x - y
x * y
x / y
x ^ y
x + y

Special Functions of Mathematics

Description

Special mathematical functions related to the error function.

The function erfc(x) is a variant of the cumulative normal (or Gaussian) distribution funciton.

The functions erfinv(x) and erfcinv(x) respectively implement the inverse functions of erf(x) and erfc(x).

Usage

erf(x)

## S4 method for signature 'dual'
erf(x)

erfinv(x)

## S4 method for signature 'dual'
erfinv(x)

erfc(x)

## S4 method for signature 'dual'
erfc(x)

erfcinv(x)

## S4 method for signature 'dual'
erfcinv(x)

Arguments

Value

A dual object containing the transformed values according to the chosen function.

Examples

x <- dual(0.5, 1)
erf(x)
erfc(x)
erfinv(x)
erfcinv(x)

Hyperbolic Functions

Description

These functions provide the obvious hyperbolic functions. They respectively compute the hyperbolic cosine, sine, tangent, and their inverses, arc-cosine, arc-sine, arc-tangent.

Usage

## S4 method for signature 'dual'
cosh(x)

## S4 method for signature 'dual'
sinh(x)

## S4 method for signature 'dual'
tanh(x)

## S4 method for signature 'dual'
acosh(x)

## S4 method for signature 'dual'
asinh(x)

## S4 method for signature 'dual'
atanh(x)

Arguments

Value

A dual object containing the transformed values according to the chosen function.

Examples

x <- dual(0.5, 1)
cosh(x)
sinh(x)
tanh(x)
acosh(1 + x)
asinh(x)
atanh(x)

Logic Operators for Comparing Dual Numbers

Description

These functions provide the operators for logical comparisons between dual numbers. These operators are designed to test equality and inequalities (such as "not equal", "greater", "less", "greater or equal", "less or equal").

Usage

## S4 method for signature 'dual,numeric'
e1 == e2

## S4 method for signature 'numeric,dual'
e1 == e2

## S4 method for signature 'dual,dual'
e1 == e2

## S4 method for signature 'dual,numeric'
e1 != e2

## S4 method for signature 'numeric,dual'
e1 != e2

## S4 method for signature 'dual,dual'
e1 != e2

## S4 method for signature 'dual,numeric'
e1 > e2

## S4 method for signature 'numeric,dual'
e1 > e2

## S4 method for signature 'dual,dual'
e1 > e2

## S4 method for signature 'dual,numeric'
e1 >= e2

## S4 method for signature 'numeric,dual'
e1 >= e2

## S4 method for signature 'dual,dual'
e1 >= e2

## S4 method for signature 'dual,numeric'
e1 < e2

## S4 method for signature 'numeric,dual'
e1 < e2

## S4 method for signature 'dual,dual'
e1 < e2

## S4 method for signature 'dual,numeric'
e1 <= e2

## S4 method for signature 'numeric,dual'
e1 <= e2

## S4 method for signature 'dual,dual'
e1 <= e2

Arguments

Value

The correspondent boolean/logical value of the comparison between e1 and e2 is returned.

Examples

x <- dual(1.5, 1:0)
y <- dual(2.6, 0:1)
x == y
x != y
x > y
x >= y
x < y
x <= y

Miscellaneous Mathematical Functions

Description

The function abs(x) computes the absolute value of x, while sqrt(x) computes the square root of x.

Usage

## S4 method for signature 'dual'
sqrt(x)

## S4 method for signature 'dual'
abs(x)

Arguments

Value

A dual object containing the transformed values according to the chosen function.

Examples

x <- dual(4.3, 1:0)
y <- dual(7.6, 0:1)
abs(-2.2 * x + 0.321 * y)
sqrt(y - x)

Special Functions of Mathematics

Description

Special mathematical functions related to the beta and gamma.

Usage

## S4 method for signature 'dual,dual'
beta(a, b)

## S4 method for signature 'dual,numeric'
beta(a, b)

## S4 method for signature 'numeric,dual'
beta(a, b)

## S4 method for signature 'dual,dual'
lbeta(a, b)

## S4 method for signature 'dual,numeric'
lbeta(a, b)

## S4 method for signature 'numeric,dual'
lbeta(a, b)

## S4 method for signature 'dual'
gamma(x)

## S4 method for signature 'dual'
lgamma(x)

## S4 method for signature 'dual'
psigamma(x, deriv = 0L)

## S4 method for signature 'dual'
digamma(x)

## S4 method for signature 'dual'
trigamma(x)

## S4 method for signature 'dual,dual'
choose(n, k)

## S4 method for signature 'numeric,dual'
choose(n, k)

## S4 method for signature 'dual,numeric'
choose(n, k)

## S4 method for signature 'dual,dual'
lchoose(n, k)

## S4 method for signature 'numeric,dual'
lchoose(n, k)

## S4 method for signature 'dual,numeric'
lchoose(n, k)

## S4 method for signature 'dual'
factorial(x)

## S4 method for signature 'dual'
lfactorial(x)

Arguments

Value

A dual object containing the transformed values according to the chosen function.

Examples

x <- dual(0.5, 1)
a <- dual(1.2, 1:0)
b <- dual(2.1, 0:1)

beta(a, b)
beta(1, b)
beta(a, 1)
lbeta(a, b)
lbeta(1, b)
lbeta(a, 1)

gamma(x)
lgamma(x)
psigamma(x, deriv = 0)
digamma(x)
trigamma(x)
psigamma(x, 2)
psigamma(x, 3)


n <- 7.8 + a
k <- 5.6 + b
choose(n, k)
choose(5, k)
choose(n, 2)

lchoose(n, k)
lchoose(5, k)
lchoose(n, 2)

factorial(x)
lfactorial(x)

Trigonometric Functions

Description

These functions give the obvious trigonometric functions. They respectively compute the cosine, sine, tangent, arc-cosine, arc-sine, arc-tangent, and the two-argument arc-tangent.

cospi(x), sinpi(x), and tanpi(x), compute cos(pi*x), sin(pi*x), and tan(pi*x).

Usage

## S4 method for signature 'dual'
cos(x)

## S4 method for signature 'dual'
sin(x)

## S4 method for signature 'dual'
tan(x)

## S4 method for signature 'dual'
acos(x)

## S4 method for signature 'dual'
asin(x)

## S4 method for signature 'dual'
atan(x)

## S4 method for signature 'dual,numeric'
atan2(y, x)

## S4 method for signature 'numeric,dual'
atan2(y, x)

## S4 method for signature 'dual,dual'
atan2(y, x)

## S4 method for signature 'dual'
cospi(x)

## S4 method for signature 'dual'
sinpi(x)

## S4 method for signature 'dual'
tanpi(x)

Arguments

Value

A dual object containing the transformed values according to the chosen function.

Examples

x <- dual(1, 1:0)
y <- dual(1, 0:1)

cos(x)
sin(x)
tan(x)
acos(x - 0.5)
asin(x - 0.5)
atan(x - 0.5)
atan2(x, y)
atan2(2.4, y)
atan2(x, 1.2)
cospi(1.2 * x)
sinpi(3.4 * x)
tanpi(5.6 * x)

Dual object class An S4 Class for dual numbers

Description

The method initialize sets the initial values of a new object of the class dual.

The function dual generates an object of class dual for the representation of dual numbers.

The function is.dual returns TRUE if x is of the class dual. It retuns FALSE otherwise.

The method show shows the content of a dual object.

Usage

dual(f, grad)

## S4 method for signature 'dual'
initialize(.Object, f = numeric(0), grad = numeric(0))

dual(f, grad)

is.dual(x)

## S4 method for signature 'dual'
show(object)

Arguments

Value

an object of the class dual.

a logical value indicating if the object is of the class dual or not.

Slots

f

a single numeric value denoting the "Real" component of the dual number

grad

a numeric vector rappresenting the "Dual" components of the dual number

Examples

x <- dual(3, 0:1)
library(dual)
x <- new("dual", f = 1, grad = 1)
is.dual(3)
is.dual(x)

Logarithms and Exponentials

Description

Logarithms and Exponentials

Usage

## S4 method for signature 'dual'
log(x)

## S4 method for signature 'dual,numeric'
logb(x, base = exp(1))

## S4 method for signature 'numeric,dual'
logb(x, base = exp(1))

## S4 method for signature 'dual,dual'
logb(x, base = exp(1))

## S4 method for signature 'dual'
log10(x)

## S4 method for signature 'dual'
log2(x)

## S4 method for signature 'dual'
log1p(x)

## S4 method for signature 'dual'
exp(x)

## S4 method for signature 'dual'
expm1(x)

Arguments

Value

A dual object containing the transformed values according to the chosen function.

Examples

x <- dual(sqrt(pi), 1:0)
y <- dual(pi * .75, 0:1)
log(x)
logb(x, base = 1.1)
logb(3.1, base = x)

logb(x, y)
log10(x)
log2(x)

log1p(x)

exp(2*x)
expm1(2*x)