Exponential growth rate from the dominant singularity

Coefficients of a rational function grow like \(C \rho^{-n}\) where \(\rho\) is the denominator root closest to the origin (Flajolet-Sedgewick Theorem IV.7). Found by bisection on the positive real axis – the Pringsheim case, where non-negative coefficients put a singularity there. If coefficients are supplied, the measured ratio of the last two is reported against the predicted \(1/\rho\).

Usage

morie_dominant_singularity_growth(denominator, coefficients = NULL)

Arguments

denominator

Coefficient vector, constant first.

coefficients

Optional sequence to check the prediction on.

Value

A list with radius, growth_rate, measured_ratio, relative_gap.

Examples

morie_dominant_singularity_growth(c(1, -1, -1))
#> $radius
#> [1] 0.618034
#> 
#> $growth_rate
#> [1] 1.618034
#> 
#> $estimate
#> [1] 1.618034
#> 
#> $measured_ratio
#> NULL
#> 
#> $relative_gap
#> NULL
#> 
#> $n
#> [1] 2
#> 
#> $method
#> [1] "Analytic combinatorics (Flajolet and Sedgewick 2009)"
#>