deriv                  package: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
     expression({
              .value <- x^2
              .grad <- array(0, c(length(.value), 1), list(NULL, c("x")))
              .grad[, "x"] <- 2 * x
              attr(.value, "gradient") <- .grad
              .value
     })
     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()" :

     -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)

