logcf {DPQ} | R Documentation |
Continued Fraction Approximation of Log-Related Power Series
Description
Compute a continued fraction approximation to the series (infinite sum)
\sum_{k=0}^\infty \frac{x^k}{i +k\cdot d} = \frac{1}{i} + \frac{x}{i+d} +
\frac{x^2}{i+2*d} + \frac{x^3}{i+3*d} + \ldots
Needed as auxiliary function in log1pmx()
and lgamma1p()
.
Usage
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)
Arguments
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. |
Details
logcfR.()
:a pure R version where the iterations happen vectorized in
x
, only for those componentsx[i]
they have not yet converged. This is particularly beneficial for not-very-short"mpfr"
vectorsx
, and still conceptually equivalent to thelogcfR()
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.
Value
a numeric-alike vector with the same attributes as x
. For the
logcfR*()
versions, an "mpfr"
vector if x
is one.
Note
Rescaling is done by (namespace hidden) “global”
scalefactor
which is 2^{256}
, represented exactly (in
double
precision).
Author(s)
Martin Maechler, based on R's ‘nmath/pgamma.c’ implementation.
See Also
lgamma1p
, log1pmx
, and
pbeta
, whose prinicipal algorithm has evolved from TOMS 708.
Examples
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")