deriv {stats}R Documentation

Symbolic and Algorithmic Derivatives of Simple Expressions


Compute derivatives of simple expressions, symbolically.


    D (expr, name)
 deriv(expr, ...)
deriv3(expr, ...)

 ## Default S3 method:
 deriv(expr, namevec, function.arg = NULL, tag = ".expr",
       hessian = FALSE, ...)
 ## S3 method for class 'formula':
 deriv(expr, namevec, function.arg = NULL, tag = ".expr",
       hessian = FALSE, ...)

## Default S3 method:
deriv3(expr, namevec, function.arg = NULL, tag = ".expr",
       hessian = TRUE, ...)
## S3 method for class 'formula':
deriv3(expr, namevec, function.arg = NULL, tag = ".expr",
       hessian = TRUE, ...)


expr A expression or call or (except D) a formula with no lhs.
name,namevec character vector, giving the variable names (only one for D()) with respect to which derivatives will be computed.
function.arg If specified and non-NULL, a character vector of arguments for a function return, or a function (with empty body) or TRUE, the latter indicating that a function with argument names namevec should be used.
tag character; the prefix to be used for the locally created variables in result.
hessian a logical value indicating whether the second derivatives should be calculated and incorporated in the return value.
... arguments to tbe passed to or from methods.


D is modelled after its S namesake for taking simple symbolic derivatives.

deriv is a generic function with a default and a formula method. It returns a call for computing the expr and its (partial) derivatives, simultaneously. It uses so-called “algorithmic derivatives”. If function.arg is a function, its arguments can have default values, see the fx example below.

Currently, deriv.formula just calls deriv.default after extracting the expression to the right of ~.

deriv3 and its methods are equivalent to deriv and its methods except that hessian defaults to TRUE for deriv3.

The internal code knows about the arithmetic operators +, -, *, / and ^, and the single-variable functions exp, log, sin, cos, tan, sinh, cosh, sqrt, pnorm, dnorm, asin, acos, atan, gamma and lgamma. (Note that only the standard normal distribution is considered.)


D returns a call and therefore can easily be iterated for higher derivatives.
deriv and deriv3 normally return an expression object whose evaluation returns the function values with a "gradient" attribute containing the gradient matrix. If hessian is TRUE the evaluation also returns a "hessian" attribute containing the Hessian array.
If function.arg is not NULL, deriv and deriv3 return a function with those arguments rather than an expression.


Griewank, A. and Corliss, G. F. (1991) Automatic Differentiation of Algorithms: Theory, Implementation, and Application. SIAM proceedings, Philadelphia.

Bates, D. M. and Chambers, J. M. (1992) Nonlinear models. Chapter 10 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

See Also

nlm and optim for numeric minimization which could make use of derivatives,


## formula argument :
dx2x <- deriv(~ x^2, "x") ; dx2x
## Not run: 
         .value <- x^2
         .grad <- array(0, c(length(.value), 1), list(NULL, c("x")))
         .grad[, "x"] <- 2 * x
         attr(.value, "gradient") <- .grad
## End(Not run)
x <- -1:2

## Something 'tougher':
trig.exp <- expression(sin(cos(x + y^2)))
( <- D(trig.exp, "x") )
all.equal(D(trig.exp[[1]], "x"),

( dxy <- deriv(trig.exp, c("x", "y")) )
y <- 1

## function returned:
deriv((y ~ sin(cos(x) * y)), c("x","y"), func = TRUE)

## function with defaulted arguments:
(fx <- deriv(y ~ b0 + b1 * 2^(-x/th), c("b0", "b1", "th"),
             function(b0, b1, th, x = 1:7){} ) )

## Higher derivatives
deriv3(y ~ b0 + b1 * 2^(-x/th), c("b0", "b1", "th"),
     c("b0", "b1", "th", "x") )

## Higher derivatives:
DD <- function(expr,name, order = 1) {
   if(order < 1) stop("'order' must be >= 1")
   if(order == 1) D(expr,name)
   else DD(D(expr, name), name, order - 1)
DD(expression(sin(x^2)), "x", 3)
## showing the limits of the internal "simplify()" :
## Not run: 
-sin(x^2) * (2 * x) * 2 + ((cos(x^2) * (2 * x) * (2 * x) + sin(x^2) *
    2) * (2 * x) + sin(x^2) * (2 * x) * 2)
## End(Not run)

[Package stats version 2.5.0 Index]