| logcf {DPQ} | R Documentation |
Compute a continued fraction approximation to the series (infinite sum)
sum(k=0,...,Inf; x^k/(i+k*d)) = 1/i + x/(i+d) + x^2/(i+2*d) + x^3/(i+3*d) + ...
Needed as auxiliary function in log1pmx() and lgamma1p().
logcfR (x, i, d, eps, maxit = 10000L, trace = FALSE) logcfR.(x, i, d, eps, maxit = 10000L, trace = FALSE) logcf (x, i, d, eps, trace = FALSE)
x |
numeric vector of values typically less than 1.
"mpfr" (of potentially high precision, package Rmpfr) work in
|
i |
positive numeric |
d |
non-negative numeric |
eps |
positive number, the convergence tolerance. |
maxit |
a positive integer, the maximal number of iterations or terms in the truncated series used. |
trace |
logical (or non-negative integer in the future) indicating if (and how much) diagnostic output should be printed to the console during the computations. |
logcfR.():a pure R version where the iterations happen
vectorized in x, only for those components x[i] they
have not yet converged. This is particularly beneficial for
not-very-short "mpfr" vectors x, and still conceptually
equivalent to the logcfR() version.
logcfR():a pure R version where each x[i] is
treated separately, hence “properly” vectorized, but slowly so.
logcf():only for numeric x, calls
into (a clone of) R's own (non-API currently) logcf() C
Rmathlib function.
a numeric-alike vector with the same attributes as x. For the
logcfR*() versions, an "mpfr" vector if x is one.
Rescaling is done by (namespace hidden) “global”
scalefactor which is 2^{256}, represented exactly (in
double precision).
Martin Maechler, based on R's ‘nmath/pgamma.c’ implementation.
lgamma1p, log1pmx, and
pbeta, whose prinicipal algorithm has evolved from TOMS 708.
x <- (-2:1)/2
logcf (x, 2,3, eps=1e-7, trace=TRUE) # shows iterations for each x[]
logcfR(x, 2,3, eps=1e-7, trace=TRUE) # 1 line per x[]
logcfR(x, 2,3, eps=1e-7, trace= 2 ) # shows iterations for each x[]
n <- 2049; x <- seq(-1,1, length.out = n)[-n] ; stopifnot(diff(x) == 1/1024)
plot(x, (lcf <- logcf(x, 2,3, eps=1e-12)), type="l", col=2)
lcR <- logcfR (x, 2,3, eps=1e-12); all.equal(lcf, lcR , tol=0)
lcR.<- logcfR.(x, 2,3, eps=1e-12); all.equal(lcf, lcR., tol=0)
stopifnot(exprs = {
all.equal(lcf, lcR., tol=1e-14)# seen 0 (!)
all.equal(lcf, lcR, tol=1e-14)# seen 0 (!) -- failed for a while
})
l32 <- curve(logcf(x, 3,2, eps=1e-7), -3, 1)
abline(h=0,v=1, lty=3, col="gray50")
plot(y~x, l32, log="y", type = "o", main = "logcf(*, 3,2) in log-scale")