deriv {base}

R Documentation

Symbolic and Algorithmic Derivatives of Simple Expressions

Description

Compute derivatives of simple expressions, symbolically.

Usage

    D (expr, name)
 deriv(expr, namevec, function.arg, tag = ".expr", hessian = FALSE)
deriv3(expr, namevec, function.arg, tag = ".expr", hessian = TRUE)

Arguments

expr	expression or call to be differentiated.
name, namevec	character vector, giving the variable names (only one for D()) with respect to which derivatives will be computed.
function.arg	If specified, 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.

Details

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.

Value

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 specified, deriv and deriv3 return a function with those arguments rather than an expression.

References

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

See Also

nlm and optim for numeric minimization which could make use of derivatives, nls in package nls.

Examples

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

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

( dxy <- deriv(trig.exp, c("x", "y")) )
y <- 1
eval(dxy)
eval(D.sc)
stopifnot(eval(D.sc) ==
          attr(eval(dxy),"gradient")[,"x"])

## function returned:
deriv((ff <- y ~ sin(cos(x) * y)), c("x","y"), func = TRUE)
stopifnot(all.equal(deriv(ff, c("x","y"), func = TRUE ),
                    deriv(ff, c("x","y"), func = function(x,y){ } )))
## function with defaulted arguments:
(fx <- deriv(y ~ b0 + b1 * 2^(-x/th), c("b0", "b1", "th"),
             function(b0, b1, th, x = 1:7){} ) )
fx(2,3,4)

## 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)

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