optimizeR {Rmpfr} | R Documentation |
optimizeR
searches the intervalfrom
lower
to upper
for a minimum
of the function f
with respect to its first argument.
optimizeR(f, lower, upper, ..., tol = 1e-20, method = c("Brent", "GoldenRatio"), maximum = FALSE, precFactor = 2.0, precBits = -log2(tol) * precFactor, maxiter = 1000, trace = FALSE)
f |
the function to be optimized. |
... |
additional named or unnamed arguments to be passed
to |
lower |
the lower end point of the interval to be searched. |
upper |
the upper end point of the interval to be searched. |
tol |
the desired accuracy, typically higher than double
precision, i.e., |
method |
|
maximum |
logical indicating if f() should be maximized or minimized (the default). |
precFactor |
only for default |
precBits |
number of bits to be used for
|
maxiter |
maximal number of iterations to be used. |
trace |
integer or logical indicating if and how iterations should be monitored; if an integer k, print every k-th iteration. |
"Brent"
:Brent(1973)'s simple and robust algorithm
is a hybrid, using a combination of the golden ratio and local
quadratic (“parabolic”) interpolation. This is the same
algorithm as standard R's optimize()
, adapted to
high precision numbers.
In smooth cases, the convergence is considerably faster than the golden section or Fibonacci ratio algorithms.
"GoldenRatio"
:The golden ratio method works as follows: from a given interval containing the solution, it constructs the next point in the golden ratio between the interval boundaries.
A list
with components minimum
(or maximum
)
and objective
which give the location of the minimum (or maximum)
and the value of the function at that point;
iter
specifiying the number of iterations, the logical
convergence
indicating if the iterations converged and
estim.prec
which is an estimate or an upper bound of the final
precision (in x).
method
the string of the method used.
"GoldenRatio"
is based on Hans W Borchert's
golden_ratio
;
modifications and "Brent"
by Martin Maechler.
R's standard optimize
; Rmpfr's unirootR
.
iG5 <- function(x) -exp(-(x-5)^2/2) curve(iG5, 0, 10, 200) o.dp <- optimize (iG5, c(0, 10)) #-> 5 of course oM.gs <- optimizeR(iG5, 0, 10, method="Golden") oM.Br <- optimizeR(iG5, 0, 10, method="Brent", trace=TRUE) oM.gs$min ; oM.gs$iter oM.Br$min ; oM.Br$iter (doExtras <- Rmpfr:::doExtras()) if(doExtras) {## more accuracy {takes a few seconds} oM.gs <- optimizeR(iG5, 0, 10, method="Golden", tol = 1e-70) oM.Br <- optimizeR(iG5, 0, 10, tol = 1e-70) } rbind(Golden = c(err = as.numeric(oM.gs$min -5), iter = oM.gs$iter), Brent = c(err = as.numeric(oM.Br$min -5), iter = oM.Br$iter)) ## ==> Brent is orders of magnitude more efficient ! ## Testing on the sine curve with 40 correct digits: sol <- optimizeR(sin, 2, 6, tol = 1e-40) str(sol) sol <- optimizeR(sin, 2, 6, tol = 1e-50, precFactor = 3.0, trace = TRUE) pi.. <- 2*sol$min/3 print(pi.., digits=51) stopifnot(all.equal(pi.., Const("pi", 256), tolerance = 10*1e-50)) if(doExtras) { # considerably more expensive ## a harder one: f.sq <- function(x) sin(x-2)^4 + sqrt(pmax(0,(x-1)*(x-4)))*(x-2)^2 curve(f.sq, 0, 4.5, n=1000) msq <- optimizeR(f.sq, 0, 5, tol = 1e-50, trace=5) str(msq) # ok stopifnot(abs(msq$minimum - 2) < 1e-49) ## find the other local minimum: -- non-smooth ==> Golden-section is used msq2 <- optimizeR(f.sq, 3.5, 5, tol = 1e-50, trace=10) stopifnot(abs(msq2$minimum - 4) < 1e-49) ## and a local maximum: msq3 <- optimizeR(f.sq, 3, 4, maximum=TRUE, trace=2) stopifnot(abs(msq3$maximum - 3.57) < 1e-2) }#end {doExtras} ##----- "impossible" one to get precisely ------------------------ ff <- function(x) exp(-1/(x-8)^2) curve(exp(-1/(x-8)^2), -3, 13, n=1001) (opt. <- optimizeR(function(x) exp(-1/(x-8)^2), -3, 13, trace = 5)) ## -> close to 8 {but not very close!} ff(opt.$minimum) # gives 0 if(doExtras) { ## try harder ... in vain .. str(opt1 <- optimizeR(ff, -3,13, tol = 1e-60, precFactor = 4)) print(opt1$minimum, digits=20) ## still just 7.99998038 or 8.000036655 {depending on method} }