Michael O. - 1 year ago 76
R Question

# Function for derivaties of polynomials of arbitrary order (symbolic method preferred)

I've found polynomial coefficients from my data:

``````R <- c(0.256,0.512,0.768,1.024,1.28,1.437,1.594,1.72,1.846,1.972,2.098,2.4029)
Ic <- c(1.78,1.71,1.57,1.44,1.25,1.02,0.87,0.68,0.54,0.38,0.26,0.17)
NN <- 3
ft <- lm(Ic ~ poly(R, NN, raw = TRUE))
pc <- coef(ft)
``````

so I can create a polynomial function:

``````f1 <- function(x) pc[1] + pc[2] * x + pc[3] * x ^ 2 + pc[4] * x ^ 3
``````

and for example, take a derivative:

``````g1 <- Deriv(f1)
``````

How to create a universal function so that it doesn't have to be rewritten for every new polynomial degree
`NN`
?

My original answer may not be what you really want, as it was numerical rather symbolic. Here is the symbolic solution.

``````## use `"x"` as variable name
## taking polynomial coefficient vector `pc`
## can return a string, or an expression by further parsing (mandatory for `D`)
f <- function (pc, expr = TRUE) {
stringexpr <- paste("x", seq_along(pc) - 1, sep = " ^ ")
stringexpr <- paste(stringexpr, pc, sep = " * ")
stringexpr <- paste(stringexpr, collapse = " + ")
if (expr) return(parse(text = stringexpr))
else return(stringexpr)
}

## an example cubic polynomial with coefficients 0.1, 0.2, 0.3, 0.4
cubic <- f(pc = 1:4 / 10, TRUE)

## using R base's `D` (requiring expression)
dcubic <- D(cubic, name = "x")
# 0.2 + 2 * x * 0.3 + 3 * x^2 * 0.4

## using `Deriv::Deriv`
library(Deriv)

dcubic <- Deriv(cubic, x = "x", nderiv = 1L)
# expression(0.2 + x * (0.6 + 1.2 * x))

Deriv(f(1:4 / 10, FALSE), x = "x", nderiv = 1L)  ## use string, get string
# [1] "0.2 + x * (0.6 + 1.2 * x)"
``````

Of course, `Deriv` makes higher order derivatives easier to get. We can simply set `nderiv`. For `D` however, we have to use recursion (see examples of `?D`).

``````Deriv(cubic, x = "x", nderiv = 2L)
# expression(0.6 + 2.4 * x)

Deriv(cubic, x = "x", nderiv = 3L)
# expression(2.4)

Deriv(cubic, x = "x", nderiv = 4L)
# expression(0)
``````

If we use expression, we will be able to evaluate the result later. For example,

``````eval(cubic, envir = list(x = 1:4))  ## cubic polynomial
# [1]  1.0  4.9 14.2 31.3

eval(dcubic, envir = list(x = 1:4))  ## its first derivative
# [1]  2.0  6.2 12.8 21.8
``````

The above implies that we can wrap up an expression for a function. Using a function has several advantages, one being that we are able to plot it using `curve` or `plot.function`.

``````fun <- function(x, expr) eval.parent(expr, n = 0L)
``````

Note, the success of `fun` requires `expr` to be an expression in terms of symbol `x`. If `expr` was defined in terms of `y` for example, we need to define `fun` with `function (y, expr)`. Now let's use `curve` to plot `cubic` and `dcubic`, on a range `0 < x < 5`:

``````curve(fun(x, cubic), from = 0, to = 5)  ## colour "black"
curve(fun(x, dcubic), add = TRUE, col = 2)  ## colour "red"
``````

The most convenient way, is of course to define a single function `FUN` rather than doing `f` + `fun` combination. In this way, we also don't need to worry about the consistency on the variable name used by `f` and `fun`.

``````FUN <- function (x, pc, nderiv = 0L) {
## check missing arguments
if (missing(x) || missing(pc)) stop ("arguments missing with no default!")
## expression of polynomial
stringexpr <- paste("x", seq_along(pc) - 1, sep = " ^ ")
stringexpr <- paste(stringexpr, pc, sep = " * ")
stringexpr <- paste(stringexpr, collapse = " + ")
expr <- parse(text = stringexpr)
## taking derivatives
dexpr <- Deriv::Deriv(expr, x = "x", nderiv = nderiv)
## evaluation
val <- eval.parent(dexpr, n = 0L)
## note, if we take to many derivatives so that `dexpr` becomes constant
## `val` is free of `x` so it will only be of length 1
## we need to repeat this constant to match `length(x)`
if (length(val) == 1L) val <- rep.int(val, length(x))
## now we return
val
}
``````

Suppose we want to evaluate a cubic polynomial with coefficients `pc <- c(0.1, 0.2, 0.3, 0.4)` and its derivatives on `x <- seq(0, 1, 0.2)`, we can simply do:

``````FUN(x, pc)
# [1] 0.1000 0.1552 0.2536 0.4144 0.6568 1.0000

FUN(x, pc, nderiv = 1L)
# [1] 0.200 0.368 0.632 0.992 1.448 2.000

FUN(x, pc, nderiv = 2L)
# [1] 0.60 1.08 1.56 2.04 2.52 3.00

FUN(x, pc, nderiv = 3L)
# [1] 2.4 2.4 2.4 2.4 2.4 2.4

FUN(x, pc, nderiv = 4L)
# [1] 0 0 0 0 0 0
``````

Now plotting is also easy:

``````curve(FUN(x, pc), from = 0, to = 5)
curve(FUN(x, pc, 1), from = 0, to = 5, add = TRUE, col = 2)
curve(FUN(x, pc, 2), from = 0, to = 5, add = TRUE, col = 3)
curve(FUN(x, pc, 3), from = 0, to = 5, add = TRUE, col = 4)
``````

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