# Shanks Transformation - Generalized Shanks Transformation

Generalized Shanks Transformation

The generalized kth-order Shanks transformation is given as the ratio of the determinants:

$S_k(A_n) = frac{ begin{vmatrix} A_{n-k} & cdots & A_{n-1} & A_n \ Delta A_{n-k} & cdots & Delta A_{n-1} & Delta A_{n} \ Delta A_{n-k+1} & cdots & Delta A_{n} & Delta A_{n+1} \ vdots & & vdots & vdots \ Delta A_{n-1} & cdots & Delta A_{n+k-2} & Delta A_{n+k-1} \ end{vmatrix} }{ begin{vmatrix} 1 & cdots & 1 & 1 \ Delta A_{n-k} & cdots & Delta A_{n-1} & Delta A_{n} \ Delta A_{n-k+1} & cdots & Delta A_{n} & Delta A_{n+1} \ vdots & & vdots & vdots \ Delta A_{n-1} & cdots & Delta A_{n+k-2} & Delta A_{n+k-1} \ end{vmatrix} },$

with It is the solution of a model for the convergence behaviour of the partial sums with distinct transients:

This model for the convergence behaviour contains unknowns. By evaluating the above equation at the elements and solving for the above expression for the kth-order Shanks transformation is obtained. The first-order generalized Shanks transformation is equal to the ordinary Shanks transformation:

The generalized Shanks transformation is closely related to Padé approximants and Padé tables.