| Title: | Automatic Differentiation of Simple Functions |
|---|---|
| Description: | Implementation of automatically computing derivatives of functions (see Mailund Thomas (2017) <doi:10.1007/978-1-4842-2881-4>). Moreover, calculating gradients, Hessian and Jacobian matrices is possible. |
| Authors: | Thomas Mailund [aut], Konrad Krämer [aut, cre] |
| Maintainer: | Konrad Krämer <[email protected]> |
| License: | GPL-3 |
| Version: | 0.2.0 |
| Built: | 2024-11-04 03:02:47 UTC |
| Source: | https://github.com/konrad1991/dfdr |
Differentiate a function for a single variable.
d(f, x, derivs = NULL)d(f, x, derivs = NULL)
f |
The function to differentiate. |
x |
The variable that f should be differentiated with respect to. |
derivs |
An S4 class of type fcts that defines additional derivatives. See |
The following functions are already supported:
sin, sinh, asin, cos, cosh, acos, tan, tanh, atan, exp, log, sqrt, c, vector, numeric, rep and matrix.
Notably, for the functions: c, vector, numeric, rep and matrix the function is ignored during differentiation.
For example function f and symbol x:
df/dx
library(dfdr) d(sin, x) f <- function(x) -sin(x) d(f, x) # Initialize list lst <- dfdr::fcts() # The function which should be added f <- function(x) x^2 # The dervative function of f f_deriv <- function(x) 2*x # add new entry to list lst <- fcts_add_fct(lst, f, f_deriv) g <- function(z) f(z) d(g, z, lst)library(dfdr) d(sin, x) f <- function(x) -sin(x) d(f, x) # Initialize list lst <- dfdr::fcts() # The function which should be added f <- function(x) x^2 # The dervative function of f f_deriv <- function(x) 2*x # add new entry to list lst <- fcts_add_fct(lst, f, f_deriv) g <- function(z) f(z) d(g, z, lst)
A S4 class containing additional functions which can be used for calculating derivatives with d().
To create a class the function fcts() should be used.
Adding functions is only possible via the function add_fct.
The following functions are already supported:
sin, sinh, asin, cos, cosh, acos, tan, tanh, atan, exp, log, sqrt, c, vector, numeric, rep and matrix.
Notably, for the functions: c, vector, numeric, rep and matrix the function is ignored during differentiation.
funsA list containing the specified functions. This slot should not be accessed and is used only internally.
d()
library(dfdr) # Initialize list lst <- dfdr::fcts() # The function which should be added f <- function(x) x^2 # The dervative function of f f_deriv <- function(x) 2*x # add new entry to list lst <- fcts_add_fct(lst, f, f_deriv) g <- function(z) f(z) df <- d(g, z, lst) dflibrary(dfdr) # Initialize list lst <- dfdr::fcts() # The function which should be added f <- function(x) x^2 # The dervative function of f f_deriv <- function(x) 2*x # add new entry to list lst <- fcts_add_fct(lst, f, f_deriv) g <- function(z) f(z) df <- d(g, z, lst) df
A function which appends a S4 class of type fcts with a new function-derivative pair.
fcts_add_fct(lst, f, f_deriv, keep = FALSE)fcts_add_fct(lst, f, f_deriv, keep = FALSE)
lst |
is the S4 class of type fcts. Newly created by |
f |
is the function which should be differentiated. The argument has to be of type function. |
f_deriv |
is a function defining the derivative of f. The argument has to be of type function. |
keep |
is a logical value. If set to TRUE the function f is ignored of |
The following functions are already supported:
sin, sinh, asin, cos, cosh, acos, tan, tanh, atan, exp, log, sqrt, c, vector, numeric, rep and matrix.
Notably, for the functions: c, vector, numeric, rep and matrix the function is ignored during differentiation.
a S4 class of type fcts extended by the new function-derivative pair.
The body of f and f_deriv have to be defined without curly brackets.
library(dfdr) # Initialize list lst <- dfdr::fcts() # The function which should be added f <- function(x) x^2 # The dervative function of f f_deriv <- function(x) 2*x # add new entry to list lst <- fcts_add_fct(lst, f, f_deriv) g <- function(z) f(z) df <- d(g, z, lst) dflibrary(dfdr) # Initialize list lst <- dfdr::fcts() # The function which should be added f <- function(x) x^2 # The dervative function of f f_deriv <- function(x) 2*x # add new entry to list lst <- fcts_add_fct(lst, f, f_deriv) g <- function(z) f(z) df <- d(g, z, lst) df
Creates a function that computes the derivative of a function with respect to each parameter and return a vector of these.
gradient(f, use_names, ...)gradient(f, use_names, ...)
f |
A function |
use_names |
Should the gradient add variable names to the output of the function? |
... |
The variable names for which gradients should be calculated |
A function that computes the gradient of f at any point.
f <- function(x, y) x^2 + y^2 df <- gradient(f, FALSE, x, y) df(1, 1)f <- function(x, y) x^2 + y^2 df <- gradient(f, FALSE, x, y) df(1, 1)
Creates a function that computes the second-order derivatives of a function with respect to each pair of parameters and return a vector of these.
hessian(f, use_names = FALSE, ...)hessian(f, use_names = FALSE, ...)
f |
A function |
use_names |
Should the gradient add variable names to the output of the function? |
... |
The variable names for which gradients should be calculated |
A function that computes the gradient of f at any point.
f <- function(x, y) x**2 + y**2 h <- hessian(f, FALSE, x, y) h(0, 0)f <- function(x, y) x**2 + y**2 h <- hessian(f, FALSE, x, y) h(0, 0)
Creates a function that computes the jacobi-matrix of a function for one specific variable. Hereinafter the variable is called y. The derivative is calculated with respect to one of the arguments of the function. Subsequently, the variable is called x. The returned function can be called at any possible point of x.
jacobian(f, y, x, derivs = NULL, num_functions = NULL)jacobian(f, y, x, derivs = NULL, num_functions = NULL)
f |
A function |
y |
The variables to compute the derivatives of (the dependent variable). For example: df/dx |
x |
The variables to which respect the variables are calcualted (the independent variable). For example: df/dx |
derivs |
optional input defining own functions which should be used. See |
num_functions |
optional input defining number of functions otherwise a squared matrix form is assumed. |
The function jacobian is intended for using it for functions accepting vectors (in case of x) and returns a vector (for y).
Mentionable, only integers are allowed for indexing the vectors. Moreover, only one element at the time can be changed.
For instance, y[1] is permitted. In contrast, y[1.5] or y[variable] will throw an error.
As usually it is possible to define new variables.
If x and/or y are found at the right side of the assignment operator the variable is replaced in all following lines.
See the example below:
# Old code
a <- x[1]
b <- 3
y[1] <- a*b
# New code
b <- 3
y[1] <- a*3
Furthermore, it is possible to use if, else if, else blocks within the function.
However, the dependent variable have to be located at the left side of the assignment operator.
This restriction is necessary as variables found in previous lines are replaced in the following lines.
# allowed code
f <- function(x, t) {
y <- numeric(2)
y[1] <- 2*x[1]^3
if(t < 3) {
y[2] <- x[2]^2
} else {
y[2] <- x[2]^4
}
return(y)
}
# not allowed code
f <- function(x, t) {
y <- numeric(2)
y[1] <- 2*x[1]^3
a <- 0
if(t < 3) {
a <- x[2]^2
} else {
a <- x[2]^4
}
y[2] <- a
return(y)
}
A function that computes the jacobi-matrix of f. Notably, it expects the dame arguments as the input function f.
f <- function(x) { y <- numeric(2) y[1] <- x[1]^2 + sin(4) y[2] <- x[2]*7 return(y) } jac <- dfdr::jacobian(f, y, x) jac(c(1, 2))f <- function(x) { y <- numeric(2) y[1] <- x[1]^2 + sin(4) y[2] <- x[2]*7 return(y) } jac <- dfdr::jacobian(f, y, x) jac(c(1, 2))
Simplify an expression by computing the values for constant expressions
simplify(expr)simplify(expr)
expr |
An expression |
a simplified expression
ex <- quote(a*0 + b^2 + 0) simplify(ex)ex <- quote(a*0 + b^2 + 0) simplify(ex)