numbers (version 0.5-6)

ratFarey: Farey Approximation

Description

Rational approximation of real numbers through Farey fractions.

Usage

ratFarey(x, n, upper = TRUE)

Arguments

x

real number.

n

integer, highest allowed denominator in a rational approximation.

upper

logical; shall the Farey fraction be grater than x.

Value

• Returns a vector with two natural numbers, nominator and denominator.

Details

Rational approximation of real numbers through Farey fractions, i.e. find for x the nearest fraction in the Farey series of rational numbers with denominator not larger than n.

References

Hardy, G. H., and E. M. Wright (1979). An Introduction to the Theory of Numbers. Fifth Edition, Oxford University Press, New York.

See Also

contFrac

Examples

ratFarey(pi, 100)                          # 22/7    0.0013
    ratFarey(pi, 100, upper = FALSE)           # 311/99  0.0002
    ratFarey(-pi, 100)                         # -22/7
    ratFarey(pi - 3, 70)                       # pi ~= 3 + (3/8)^2
    ratFarey(pi, 1000)                         # 355/113
    ratFarey(pi, 10000, upper = FALSE)         # id.
    ratFarey(pi, 1e5, upper = FALSE)           # 312689/99532 - pi ~= 3e-11

    ratFarey(4/5, 5)                           # 4/5
    ratFarey(4/5, 4)                           # 1/1
    ratFarey(4/5, 4, upper = FALSE)            # 3/4