EXAMPLES OF SYMBOLIC DIFFERENTIATION AND MINIMIZATION IN R.


Taking symbolic derivatives with "D".

> D(quote(a^3),"a")
3 * a^2

> D(quote(b^3*a^3),"a")
b^3 * (3 * a^2)

> D(D(quote(b^3*a^3),"a"),"b")
3 * b^2 * (3 * a^2)


Using "eval" to get the value of a symbolic expression.

> eval(D(D(quote(b^3*a^3),"a"),"b"),list(a=10,b=1))
[1] 900

> k <- function (a) eval(D(quote(a^3),"a"))
> k(100)
[1] 30000


Minimizing a function with "nlm".

> f <- function (x, a) sum((x-a)^2)
> nlm (function (x) f(x,c(1,2,3)), c(10,-10,20))
$minimum
[1] 3.499991e-12

$estimate
[1] 0.9999995 1.9999990 2.9999985

$gradient
[1] -9.529044e-13  3.554712e-12  1.573408e-12

$code
[1] 1

$iterations
[1] 2

> g <- function (d) nlm (function (x) f(x,d), rep(0,length(d)))
> g(10:20)
$minimum
[1] 3.405039e-26

$estimate
 [1] 10 11 12 13 14 15 16 17 18 19 20

$gradient
 [1] -1.598721e-13 -2.842171e-14 -1.065814e-13 -1.882939e-13 -5.329078e-14
 [6]  7.815982e-14  0.000000e+00 -7.815972e-14  4.973801e-14  1.847411e-13
[11]  1.065815e-13

$code
[1] 1

$iterations
[1] 2


Using "deriv" to get a function for evaluating derivatives.

> deriv(quote(x^2),"x",func=T)
function (x) 
{
    .value <- x^2
    .grad <- array(0, c(length(.value), 1), list(NULL, c("x")))
    .grad[, "x"] <- 2 * x
    attr(.value, "gradient") <- .grad
    .value
}

> deriv(quote(x^2),"x",func=T,hessian=T)
function (x) 
{
    .value <- x^2
    .grad <- array(0, c(length(.value), 1), list(NULL, c("x")))
    .hessian <- array(0, c(length(.value), 1, 1), list(NULL, 
        c("x"), c("x")))
    .grad[, "x"] <- 2 * x
    .hessian[, "x", "x"] <- 2
    attr(.value, "gradient") <- .grad
    attr(.value, "hessian") <- .hessian
    .value
}

> nlm(deriv(quote((x-5)^2),"x",func=T,hessian=T),0)
$minimum
[1] 7.888609e-31

$estimate
[1] 5

$gradient
[1] -1.776357e-15

$code
[1] 1

$iterations
[1] 1