Compute gradient of a multivariate polynomial.

This is a wrapper for deriv.mpoly.

gradient(mpoly)

Arguments

mpoly	an object of class mpoly

Value

An object of class mpoly or mpolyList.

See also

deriv.mpoly()

Examples

m <- mp("x y + y z + z^2")
gradient(m)
#> y
#> x  +  z
#> y  +  2 z


# gradient descent illustration using the symbolically
# computed gradient of the rosenbrock function (shifted)
rosenbrock <- mp("(1 - x)^2 + 100 (y - x^2)^2")
fn <- as.function(rosenbrock)
#> f(.) with . = (x, y)
(rosenbrock_gradient <- gradient(rosenbrock))
#> 400 x^3  -  400 x y  +  2 x  -  2
#> -200 x^2  +  200 y
gr <- as.function(rosenbrock_gradient)
#> f(.) with . = (x, y)

# visualize the function
library(ggplot2)
s <- seq(-5, 5, .05)
df <- expand.grid(x = s, y = s)
df$z <- apply(df, 1, fn)
ggplot(df, aes(x = x, y = y)) +
  geom_raster(aes(fill = z + 1e-10)) +
  scale_fill_continuous(trans = "log10")

# run the gradient descent algorithm using line-search
# step sizes computed with optimize()
current <- steps <- c(-3,-4)
change <- 1
tol <- 1e-5
while(change > tol){
  last  <- current
  delta <- optimize(
    function(delta) fn(current - delta*gr(current)),
    interval = c(1e-10, .1)
  )$minimum
  current <- current - delta*gr(current)
  steps   <- unname(rbind(steps, current))
  change  <- abs(fn(current) - fn(last))
}
steps <- as.data.frame(steps)
names(steps) <- c("x", "y")

# visualize steps, note the optim at c(1,1)
# the routine took 5748 steps
ggplot(df, aes(x = x, y = y)) +
  geom_raster(aes(fill = z + 1e-10)) +
  geom_path(data = steps, color = "red") +
  geom_point(data = steps, color = "red", size = .5) +
  scale_fill_continuous(trans = "log10")

# it gets to the general region of space quickly
# but once it gets on the right arc, it's terrible
# here's what the end game looks like
last_steps <- tail(steps, 100)
rngx <- range(last_steps$x); sx <- seq(rngx[1], rngx[2], length.out = 201)
rngy <- range(last_steps$y); sy <- seq(rngy[1], rngy[2], length.out = 201)
df <- expand.grid(x = sx, y = sy)
df$z <- apply(df, 1, fn)
ggplot(df, aes(x = x, y = y)) +
  geom_raster(aes(fill = z)) +
  geom_path(data = last_steps, color = "red", size = .25) +
  geom_point(data = last_steps, color = "red", size = 1) +
  scale_fill_continuous(trans = "log10")