DPQ (version 0.6-0)

logcf: 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

logcf (x, i, d, eps, maxit = 10000L, trace = FALSE)
logcfR(x, i, d, eps, maxit = 10000L, trace = FALSE)
logcfR_vec(x, i, d, eps, maxit = 10000L, trace = FALSE)

Value

a numeric-alike vector with the same attributes as x. For the

logcfR*() versions, an "mpfr" vector if x is one.

Arguments

x

numeric vector of values less than 1. "mpfr"-numbers (of potentially high precision, package Rmpfr) work in logcfR*(x,*).

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.

Author

Martin Maechler, based on R's nmath/pgamma.c implementation.

Details

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_vec() version.

logcfR_vec():

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.

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_vec(x, 2,3, eps=1e-7, trace=TRUE) # 1 line per x[]
logcfR_vec(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_vec (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)
all.equal(lcR, lcR., tol=0) # TRUE
all.equal(lcf, lcR., tol=0) # TRUE (x86_64, Lnx)
stopifnot(exprs = {
  all.equal(lcf, lcR., tol=1e-14)# seen 0 (!)
  all.equal(lcR, lcR., tol=5e-16)# seen 0 above
})

l32 <- curve(logcf(x, 3,2, eps=1e-7), -3, 1, n = 1000)
abline(h=0,v=1, lty=3, col="gray50")
##
plot(y~x, l32, log="y", type="l", main= "logcf(x, i, d)  in log-scale",
     ylim = c(.01, 10), col=2); abline(v=1, lty=3, col="gray50")
## other (i,d) than the (3,2) needed for log1pmx():
curve(logcfR(x,  5,  4,  eps = 1e-5), n=1000, add=TRUE, col = 4)
curve(logcfR(x, 20,  1,  eps = 1e-5), n=1000, add=TRUE, col = 5)
curve(logcfR(x, 1/4,1/2, eps = 1e-5), n=1000, add=TRUE, col = 6)
i.d <- cbind(c(2,3), c(5,4), c(20,1), c(1/4, 1/2))
legend("topleft", apply(i.d, 2, \(k) paste0("(i=",k[1],", d=", k[2],")")),
       lwd = 2, col = c(2,4:6), bty="n")