| Title: | Continued Fractions |
|---|---|
| Description: | Various utilities for evaluating continued fractions. |
| Authors: | Robin K. S. Hankin [aut, cre] |
| Maintainer: | Robin K. S. Hankin <[email protected]> |
| License: | GPL-2 |
| Version: | 1.1-13 |
| Built: | 2024-12-25 05:12:06 UTC |
| Source: | https://github.com/robinhankin/contfrac |
Various utilities for evaluating continued fractions.
The DESCRIPTION file:
| Package: | contfrac |
| Title: | Continued Fractions |
| Version: | 1.1-13 |
| Authors@R: | person(given=c("Robin", "K. S."), family="Hankin", role = c("aut","cre"), email="[email protected]", comment = c(ORCID = "0000-0001-5982-0415")) |
| Description: | Various utilities for evaluating continued fractions. |
| Maintainer: | Robin K. S. Hankin <[email protected]> |
| License: | GPL-2 |
| URL: | https://github.com/RobinHankin/contfrac.git |
| Suggests: | testthat (>= 3.0.0) |
| Config/testthat/edition: | 3 |
| Repository: | https://robinhankin.r-universe.dev |
| RemoteUrl: | https://github.com/robinhankin/contfrac |
| RemoteRef: | HEAD |
| RemoteSha: | ef04e939db5c479e2dd354ed19d419e7260ada35 |
| Author: | Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>) |
Index of help topics:
CF Continued fraction convergents
as_cf Approximates a real number in continued
fraction form
contfrac-package Continued Fractions
convergents Partial convergents of continued fractions
Robin K. S. Hankin [aut, cre] (<https://orcid.org/0000-0001-5982-0415>)
## CF() takes an integer sequence and returns the value of its continued fraction: phi <- (sqrt(5)+1)/2 phi_cf <- CF(rep(1,100)) # phi = [1;1,1,1,1,1,...] phi - phi_cf # should be small ## as_cf() takes a real and returns its continued fraction representation: as_cf(phi) as_cf(pi) as_cf(exp(1),25) # OK up to element 21 (which should be 14) ## GCF() is a generalized continued fraction: GCF(a=2:100,b=2:100,b0=1,finite=FALSE) # This due to Euler ## convergents() gives a sequence of partial convergents: convergents(rep(1,10))## CF() takes an integer sequence and returns the value of its continued fraction: phi <- (sqrt(5)+1)/2 phi_cf <- CF(rep(1,100)) # phi = [1;1,1,1,1,1,...] phi - phi_cf # should be small ## as_cf() takes a real and returns its continued fraction representation: as_cf(phi) as_cf(pi) as_cf(exp(1),25) # OK up to element 21 (which should be 14) ## GCF() is a generalized continued fraction: GCF(a=2:100,b=2:100,b0=1,finite=FALSE) # This due to Euler ## convergents() gives a sequence of partial convergents: convergents(rep(1,10))
Approximates a real number in continued fraction form using a standard simple algorithm
as_cf(x, n = 10)as_cf(x, n = 10)
x |
real number to be approximated in continued fraction form |
n |
Number of partial denominators to evaluate; see Notes |
Has difficulties with rational values as expected
Robin K. S. Hankin
phi <- (sqrt(5)+1)/2 as_cf(phi,50) # loses it after about 38 iterations ... not bad ... as_cf(pi) # looks about right as_cf(exp(1),20) f <- function(x){CF(as_cf(x,30),TRUE) - x} x <- runif(40) plot(sapply(x,f))phi <- (sqrt(5)+1)/2 as_cf(phi,50) # loses it after about 38 iterations ... not bad ... as_cf(pi) # looks about right as_cf(exp(1),20) f <- function(x){CF(as_cf(x,30),TRUE) - x} x <- runif(40) plot(sapply(x,f))
Returns continued fraction convergent using the modified Lentz's
algorithm; function CF() deals with continued fractions and
GCF() deals with generalized continued fractions.
CF(a, finite = FALSE, tol=0) GCF(a,b, b0=0, finite = FALSE, tol=0)CF(a, finite = FALSE, tol=0) GCF(a,b, b0=0, finite = FALSE, tol=0)
a, b
|
In function |
finite |
Boolean, with default |
b0 |
In function |
tol |
tolerance, with default |
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. Lentz 1976. Generating Bessel functions in Mie scattering calculations using continued fractions. Applied Optics, 15(3):668-671
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 smallphi <- (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
Partial convergents of continued fractions or generalized continued fractions
convergents(a) gconvergents(a,b, b0 = 0) nconv(a, give=FALSE) ngconv(a, b, b0 = 0, give=FALSE)convergents(a) gconvergents(a,b, b0 = 0) nconv(a, give=FALSE) ngconv(a, b, b0 = 0, give=FALSE)
a, b
|
In function |
b0 |
The floor of the fraction |
give |
Boolean, with |
Function convergents() returns partial convergents of the continued fraction
where a = (note the
off-by-one issue).
Function gconvergents() returns partial convergents of the continued fraction
where a =
Functions nconv() and ngconv() are convenience wrappers
that return the numerical values of the convergents.
Returns a list of two elements, A for the numerators and
B for the denominators
This classical algorithm generates very large partial numerators and denominators.
To evaluate limits, use functions CF() or GCF().
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”
# Successive approximations to pi: jj <- convergents(c(3,7,15,1,292)) jj$A/jj$B - pi # should get smaller convergents(rep(1,10)) nconv(1:6,give=TRUE)# Successive approximations to pi: jj <- convergents(c(3,7,15,1,292)) jj$A/jj$B - pi # should get smaller convergents(rep(1,10)) nconv(1:6,give=TRUE)