Continued fraction convergents

Description

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

Usage

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

Arguments

Details

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.

Author(s)

Robin K. S. Hankin

References

• 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

See Also

convergents

Examples

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

Approximates a real number in continued fraction form

Description

Approximates a real number in continued fraction form using a standard simple algorithm

Usage

as_cf(x, n = 10)

Arguments

Note

Has difficulties with rational values as expected

Author(s)

Robin K. S. Hankin

See Also

CF,convergents

Examples

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))

Partial convergents of continued fractions

Description

Partial convergents of continued fractions or generalized continued fractions

Usage

convergents(a)
gconvergents(a,b, b0 = 0)

Arguments

Details

Function convergents() returns partial convergents of the continued fraction

a_0+ \frac{1}{a_1+ \frac{1}{a_2+ \frac{1}{a_3+ \frac{1}{a_4+ \frac{1}{a_5+\ddots }}}}}

where a = a_0,a_1,a_2,\ldots (note the off-by-one issue).

Function gconvergents() returns partial convergents of the continued fraction

b_0+ \frac{a_1}{b_1+ \frac{a_2}{b_2+ \frac{a_3}{b_3+ \frac{a_4}{b_4+ \frac{a_5}{b_5+\ddots }}}}}

where a = a_1,a_2,\ldots

Value

Returns a list of two elements, A for the numerators and B for the denominators

Note

This classical algorithm generates very large partial numerators and denominators. To evaluate limits, use functions CF() or GCF().

Author(s)

Robin K. S. Hankin

References

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”

See Also

CF

Examples

# Successive approximations to pi:

jj <- convergents(c(3,7,15,1,292))
jj$A/jj$B - pi     # should get smaller


convergents(rep(1,10))