FunWithNumbers (version 1.1)

bpp: Function which calculates pi, or other irrationals, using the BaileyBorweinPlouffe formula ~~

Description

THe BPP algorithm consists of a double summation over specified fractions. Rather than go into the gory details here, please refer to the link in the References section.

Usage

bpp(k,pdat = c(1,16,8,4,0,0,-2,-1,-1,0,0),mpbits = 400)

Value

A list containing bigq , the gmp fraction calculated, and mdec, the mpfr decimal representation of said fraction.

Arguments

k

The number of terms in the series to calculate. Note that zero is a valid entry.

pdat

The parameter P which is used to define the coefficients used in all fractions in each term of the series. In brief, pdat contains the following BPP parameters: pdat(s,b,m,A) where Acomprises all elements of the vector pdat after the first three. There are strict rules about the length of A; see the Details section.

mpbits

This specifies the number of binary digits of precision to use when the function converts gmp::bigq fractions to mpfr extended precision decimal representation. Failure to use a large enough value may result in a limit to the true output precision.

Author

Carl Witthoft,

Details

The BPP algorithm calculates the sumK=0,k, 1/(b^K) * FracSum , where FracSum is defined by the sum(M=1,m, A[M]/(m*K + M)^s) . This means that the number of elements of A must equal m. Zero values are legal and are used to reject fractions not wanted in the inner sum.

The default values for pdat correspond to the coefficients used to generate pi (the sum to infinity is mathematically equal to pi). Other values have been found to calculate a few other irrationals but there is as yet no known procedure to generate the pdat set for any given number.

References

https://en.wikipedia.org/wiki/Bailey-Borwein-Plouffe_formula and references cited there.

Examples

# Compare the decimal outputs to the first 130 digits of pi, which are:
#  [1] 3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4
#  [26] 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5
#  [51] 1 0 5 8 2 0 9 7 4 9 4 4 5 9 2 3 0 7 8 1 6 4 0 6 2
#  [76] 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 1 1 7 0 6
# [101] 7 9 8 2 1 4 8 0 8 6 5 1 3 2 8 2 3 0 6 6 4 7 0 9 3
# [126] 8 4 4 6 0

# Lots of precision, but most of the digits are inaccurate.
 (bpp(5))
# $bigq
# Big Rational ('bigq') :
# [1] 40413742330349316707/12864093722915635200
# 
# $mdec
# 1 'mpfr' number of precision  400   bits 
# [1] 3.14159265322808753473437803553620446955
# 852801219780193481442230321585101644290504893051
# 16201439239799241252867682875513665

# extend the series.
(bpp(20))
# $bigq
# Big Rational ('bigq') :
# [1] 6978810534836185743790248010742839687036
# 9348283327260905420704007804969465293
# 222142438704194558751308818610402859379
# 08911653929817878006825259792072704000
# 
# $mdec
# 1 'mpfr' number of precision  400   bits 
# [1] 3.1415926535897932384626433832513
# 62615881909316518417908555365030283
# 2142940981052934064597204233958787472300102294523391693

# Accurate but low precision
 bpp(20,mpbits=20)
# $bigq
# Big Rational ('bigq') :
# [1] 6978810534836185743790248010742839
# 6870369348283327260905420704007804969465293
# 2221424387041945587513088186104028593790891
# 1653929817878006825259792072704000
# 
# $mdec
# 1 'mpfr' number of precision  20   bits 
# [1] 3.1415939