Generalized Continous Fractions
Evaluate a generalized continuous fraction as an alternating sum.
cf2num(a, b = 1, a0 = 0, finite = FALSE)
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. |
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.
Returns a numerical value, an approximation of the continued fraction.
This function is not vectorized.
H. Cohen, F. R. Villegas, and Don Zagier (2000). Experimental Mathematics, Vol. 9, No. 1, pp. 3-12. <www.emis.de/journals/EM>
## 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
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