laBeta: Accurate Approximation of 'log(a * beta(a,b))' for small 'a'

In DPQ: Density, Probability, Quantile ('DPQ') Computations

Accurate Approximation of log(a * beta(a,b)) for small a

Description

Uses a Taylor series to compute log(a * beta(a,b)) = % no \Beta, use B \log(a B(a,b)) = \log a + \log B(a,b) = log(a) + lbeta(a,b), where both direct formulas suffer from cancellation for small a, i.e., already for a < \frac 1 2 (slightly)

Usage

laBeta(a, b, nT)

Arguments

a, b	numeric(-alike) vectors of positive numbers. Needs a < 1 for convergence.
nT	an integer \ge 1, the number of Taylor series terms to use for the approximation. Currently without a default; typically nT = 17 is enough to get a more values than with the direct formula.

Value

a numeric vector (of “confirming length”) with an approximation to log(a * beta(a,b)).

Author(s)

Martin Maechler, January 2022, then October 2025

References

The author derived the Taylor series by integrating Abramowitz & Stegun, p.259, (6.3.14) series for \psi(1+z) to get a series for log \Gamma(z+1) = \log(z \Gamma(z)) = \log z + \log\Gamma(z). A. & St.: see reference e.g., in lbeta_asy.

See Also

R's beta function and its logarithm beta() and lbeta().

Examples

b <- c(.25, .5, 1:8, 12, 20, 49)
laB0 <- function(a,b) log(a * beta(a,b))
laB1 <- function(a,b) log(a) + lbeta(a,b)
a <- 1/32
labMat <- cbind(laB0= laB0(a, b), laB1 = laB1(a, b),
                laBe05 = laBeta(a=a, b = b, nT=5),
                laBe10 = laBeta(a=a, b = b, nT=10),
                laBe15 = laBeta(a=a, b = b, nT=15),
                laBe17 = laBeta(a=a, b = b, nT=17))
cbind(b, labMat)
stopifnot(exprs = {
    all.equal(labMat[,1], labMat[,"laB1"],   tolerance = 4e-14) # see 1.777e-14
    all.equal(labMat[,2], labMat[,"laBe05"], tolerance = 1e-06) # see 6.387e-7
    all.equal(labMat[,2], labMat[,"laBe10"], tolerance = 2e-11) # 1.037e-11
    all.equal(labMat[,2], labMat[,"laBe15"], tolerance = 8e-15) # 2.52e-15
})
cbind(b, laB1 = laB1(2,b), laBe1 = laBeta(2,b, nT=1), laBe2 = laBeta(2,b, nT=2),
      laBe20 = laBeta(2,b, nT=20))
if(requireNamespace("Rmpfr")) withAutoprint({
    asNumeric <- Rmpfr::asNumeric
    mpfr <- Rmpfr::mpfr
    beta <- Rmpfr::beta
  (laBM <- asNumeric(log(a * beta(mpfr(a,128),b)))) # the "true" value
  stopifnot(identical(laBM, asNumeric(laB0(mpfr(a,128), b))))
  relErr <- sfsmisc::relErr
  relErr(laBM, log(a * beta(a,b))) # 1.79e-14
  cbind(apply(labMat, 2, relErr, target = laBM))
  ## laB0   1.795634e-14
  ## laB1   2.189151e-15
  ## laBe05 6.387111e-07
  ## laBe10 1.037178e-11
  ## laBe15 3.787459e-16
  ## laBe17 1.666482e-16

  a <- 0.001; aM <- mpfr(a, 128) ## -- smaller a -- losing 3 decimals w/ direct formula:
  relErr(asNumeric(laB0(aM, b)) -> laBM, laB0(a,b))   # 5.55e-13
  relErr(laBM,                           laB1(a,b))   # 1.49e-13 slightly better
  relErr(laBM, laBeta(a,b, nT =  5)) # 2.33e-14 (better!)
  relErr(laBM, laBeta(a,b, nT = 10)) # 1.60e-16
  relErr(laBM, laBeta(a,b, nT = 20)) #  ditto, i.e., not better
})

DPQ documentation built on Aug. 21, 2026, 3 p.m.