BellBruno

Compute Bell polynomials for Faà di Bruno's formula.

Basic usage

N_der = 10;              # Maximum order of Bell polynomial
bp = bell_poly(N_der);   # Create bell polynomials
bc = bell_coeff(bp);     # Compute bell coefficients

Computing and saving Bell polynomials

If you want to compute and save the Bell polynomials you may use

N_der = 20;
path_2_folder = "my_folder_path"
bp = bell_poly( N_der; 
                save_on_disk=true, 
                path_to_folder=path_2_folder, 
                print_iteration=true)

or simply

bp = bell_poly(N_der, save_on_disk = true)

to save the files in folder "bell_results/".

Reading the Bell polynomials

The Bell polynomials can be read with

bp_new = read_bell_poly(path_to_folder=path_2_folder)

or if the standard folder "bell_results/" is used

bp_new = read_bell_poly()

Quick Tutorial

We are wish to find the derivatives of function composition

$$h(t)=f(g(t))=\exp (p\sin (t))$$

up to the order of $N=10$. We assume $p=0.1$.

The derivative of the outer function is noted as

$$\frac{d^n}{d{x}^n}f(x)=p^n\exp (px)$$

and the derivative of the inner function is noted as

$$\frac{d^n}{d{t}^n}g(t)=\sin (t+n\pi /2).$$

Derivative of outer function

p = 0.1;
f_der(x,n) = p^n * exp(p*x) 

Inner function and its derivatives

g(t) = sin(t)
g_der(t,n) = sin(t+n*π/2)

Bell polynomial and coefficients data

using BellBruno
bp = bell_poly(10)
bc = bell_coeff(bp)

Sampling points and Faà di Bruno's formula

tgrid = -π:0.01:2π # Sampling points
diff_data = faa_di_bruno(f_der, g, g_der, tgrid,bp, bc)

The full listing tutorial_example.jl can be found here.

Derivatives up to order 5

Derivatives up to order 10