numbers (version 0.8-5)

cf2num: Generalized Continous Fractions

Description

Evaluate a generalized continuous fraction as an alternating sum.

Usage

cf2num(a, b = 1, a0 = 0, finite = FALSE)

Value

Returns a numerical value, an approximation of the continued fraction.

Arguments

a

numeric vector of length greater than 2.

b

numeric vector of length 1 or the same length as a.

a0

absolute term, integer part of the continuous fraction.

finite

logical; shall Algorithm 1 be applied.

Details

Calculates the numerical value of (simple or generalized) continued fractions of the form $$ a_0 + \frac{b1}{a1+} \frac{b2}{a2+} \frac{b3}{a3+...} $$ by converting it into an alternating sum and then applying the accelleration Algorithm 1 of Cohen et al. (2000).

The argument \(b\) is by default set to \(b = (1, 1, ...)\), that is the continued fraction is treated in its simple form.

With finite=TRUE the accelleration is turned off.

References

H. Cohen, F. R. Villegas, and Don Zagier (2000). Experimental Mathematics, Vol. 9, No. 1, pp. 3-12. <www.emis.de/journals/EM>

See Also

contfrac

Examples

##  Examples from Wolfram Mathworld
print(cf2num(1:25), digits=16)  # 0.6977746579640077, eps()

a = 2*(1:25) + 1; b = 2*(1:25); a0 = 1  # 1/(sqrt(exp(1))-1)
cf2num(a, b, a0)                        # 1.541494082536798

a <- b <- 1:25                          # 1/(exp(1)-1)
cf2num(a, b)                            # 0.5819767068693286

a <- rep(1, 100); b <- 1:100; a0 <- 1   # 1.5251352761609812
cf2num(a, b, a0, finite = FALSE)        # 1.525135276161128
cf2num(a, b, a0, finite = TRUE)         # 1.525135259240266