CF: Continued fraction convergents
In contfrac: Continued Fractions

Description Usage Arguments Details Author(s) References See Also Examples

Returns continued fraction convergent using the modified Lenz's algorithm; function CF() deals with continued fractions and GCF() deals with generalized continued fractions.

1 2	CF(a, finite = FALSE, tol=0) GCF(a,b, b0=0, finite = FALSE, tol=0)

`a,b`	In function `CF()`, the elements of `a` are the partial denominators; in `GCF()` the elements of `a` are the partial numerators and the elements of `b` the partial denominators
`finite`	Boolean, with default `FALSE` meaning to iterate Lenz's algorithm until convergence (a warning is given if the sequence has not converged); and `TRUE` meaning to evaluate the finite continued fraction
`b0`	In function `GCF()`, floor of the continued fraction
`tol`	tolerance, with default `0` silently replaced with `.Machine$double.eps`

Function CF() treats the first element of its argument as the integer part of the convergent.

Function CF() is a wrapper for GCF(); it includes special dispensation for infinite values (in which case the value of the appropriate finite CF is returned).

The implementation is in C; the real and complex cases are treated separately in the interests of efficiency.

The algorithm terminates when the convergence criterion is achieved irrespective of the value of finite.

Robin K. S. Hankin

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling 1992. Numerical recipes 3rd edition: the art of scientific computing. Cambridge University Press; section 5.2 “Evaluation of continued fractions”
W. J. Lenz 1976. Generating Bessel functions in Mie scattering calculations using continued fractions. Applied Optics, 15(3):668-671

convergents

phi <- (sqrt(5)+1)/2
phi_cf <- CF(rep(1,100))     # phi = [1;1,1,1,1,1,...]
phi - phi_cf     # should be small

# The tan function:
"tan_cf" <- function(z,n=20){
     GCF(c(z, rep(-z^2,n-1)), seq(from=1,by=2, len=n))
}

z <- 1+1i
tan(z) - tan_cf(z)   # should be small

# approximate real numbers with continued fraction:
as_cf(pi)

as_cf(exp(1),25)    # OK up to element 21 (which should be 14)

  # Some convergents of pi:
  jj <- convergents(c(3,7,15,1,292))
  jj$A / jj$B - pi


  # An identity of Euler's:
  jj <- GCF(a=seq(from=2,by=2,len=30), b=seq(from=3,by=2,len=30), b0=1) 
  jj - 1/(exp(0.5)-1)   # should be small