R: Continued Fraction Approximation of Log-Related Power Series

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,...,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()`.

### 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, "mpfr" now works, too.
`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.()`:

  the first pure R version where the iterations happen vectorized in `x`, i.e., relatively fast; however convergence and rescaling are a “group decision” which is really suboptimal for reproducibility or careful comparisons.

`logcfR()`:

  a pure R version where each `x[i]` is treated separately, hence “properly” vectorized, but slowly so. Now recommended when `x` is an "mpfr"-number vector.

`logcf()`:

  only for `numeric` `x`, calls into (a clone of) R's own (non-API currently) `logcf()` C Mathlib function.

### Value

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

### 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

    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")

[Package DPQ version 0.4-4 Index]