cfrac: Fractional part of number, continuous fractions

In cwhmisc: Miscellaneous Functions for Math, Plotting, Printing, Statistics, Strings, and Tools

Description

Split off fractional part of a number, compute and evaluate continuous fractions.

Usage

  contfrac( x, depth = 13, f=floor ) 
  evalcfr( cf )
  toCFrac( x, depth=5)
  toCFrac2( x, depth=5)

Arguments

x	Real
f	function to use, normally 'floor', otherwise 'round' or 'trunc'
cf	Vector of integers representing the continued fraction of a real number
depth	Integer

Value

int integer part truncate towards 0.
frac fractional part, if d is missing; else
round(10^d*fractional part), i.e. the fractional part as "integer" (rounded).
contfrac Convert to simple continued fraction representation, cf := a_1 + 1/(a_2 + 1/(a_3 ... )).

evalcfr Evaluate simple continued fraction to corresponding real.
toCFrac Build rational approximation num/den to x using forward continued fraction recursion to a depth of depth. Stopping criterion: either depth is reached, or abs(x - num/den) is increasing again.
toCFrac2 same as toCFrac, but vectors of partial numerators and denominators are returned.

Note

d not missing is practical for use in dc
For confrac see also link[MASS]{fractions}. from Mathematics 5335 Fall 2009 The Euclidean Algorithm and Continued Fractions

Author(s)

Christian W. Hoffmann <[e-mail]>

Examples

(pcf <- contfrac(pi)) # 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, (1)
## last integer incorrect due to rounding errors
evalcfr(pcf)-pi # 0
## To see the first approximants of pi, all of them famous:
for(ii in 1:15) {x<-toCFrac(pi,ii)
print(paste(ii,":",x$num,"/",x$den,"="))
print(paste(formatFix(x$num/x$den,15),", error = ",x$num/x$den-pi))}
# Note how the approximations taper off after depth 5:
# 10 3959189 / 1260249 =  3.141592653515298 -7.44955208631382e-11"
## Same, all at once:
F <- toCFrac2(pi,5)  # $num  3  22 333 355  $den  1   7 106 113
toCFrac( pi, 10 )  #  

Example output

Loading required package: lattice
Loading required package: grid
 [1]   3   7  15   1 292   1   1   1   2   1   3   1  14   3
[1] 0
[1] "1 : 3 / 1 ="
[1] " 3.000000000000000 , error =  -0.141592653589793"
[1] "2 : 22 / 7 ="
[1] " 3.142857142857143 , error =  0.00126448926734968"
[1] "3 : 333 / 106 ="
[1] " 3.141509433962264 , error =  -8.32196275291075e-05"
[1] "4 : 355 / 113 ="
[1] " 3.141592920353982 , error =  2.66764189404967e-07"
[1] "5 : 103993 / 33102 ="
[1] " 3.141592653011902 , error =  -5.77890624242627e-10"
[1] "6 : 104348 / 33215 ="
[1] " 3.141592653921421 , error =  3.31628058347633e-10"
[1] "7 : 208341 / 66317 ="
[1] " 3.141592653467437 , error =  -1.22356347276309e-10"
[1] "8 : 937712 / 298483 ="
[1] " 3.141592653517956 , error =  -7.18372028529757e-11"
[1] "9 : 3021477 / 961766 ="
[1] " 3.141592653514472 , error =  -7.53206386150396e-11"
[1] "10 : 3959189 / 1260249 ="
[1] " 3.141592653515298 , error =  -7.44955208631382e-11"
[1] "11 : 10939855 / 3482264 ="
[1] " 3.141592653515070 , error =  -7.47237827170011e-11"
[1] "12 : 14899044 / 4742513 ="
[1] " 3.141592653515130 , error =  -7.46629424952516e-11"
[1] "13 : 25838899 / 8224777 ="
[1] " 3.141592653515104 , error =  -7.46886996694229e-11"
[1] "14 : 66576842 / 21192067 ="
[1] " 3.141592653515110 , error =  -7.46829265096949e-11"
[1] "15 : 92415741 / 29416844 ="
[1] " 3.141592653515108 , error =  -7.46847028665343e-11"
$num
[1] 3959189

$den
[1] 1260249

$error
[1] 7.449552e-11

cwhmisc documentation built on May 1, 2019, 7:55 p.m.