Group contribution method

In algebra, the continuant is a multivariate polynomial representing the determinant of a tridiagonal matrix and having applications in generalized continued fractions.

Definition

The n-th continuant, K(n), of a sequence a = a₁,...,a_n,... is defined recursively by

$K(0)=1;$

$K(1)=a_1;$

$K(n)=a_{n}K(n-1)+K(n-2).$

It may also be obtained by taking the sum of all possible products of a₁,...,a_n in which any pairs of consecutive terms are deleted.

An extended definition takes the continuant with respect to three sequences a, b and c, so that K(n) is a polynomial of a₁,...,a_n, b₁,...,b_n−1 and c₁,...,c_n−1. In this case the recurrence relation becomes

$K(0)=1;$

$K(1)=a_1;$

$K(n)=a_{n}K(n-1)-b_{n-1}{c}_{n-1}K(n-2).$

Since b_r and c_r enter into K only as a product b_rc_r there is no loss of generality in assuming that the b_r are all equal to 1.

Applications

The simple continuant gives the value of a continued fraction of the form $[a_0;a_1,a_2,\dots ]$. The n-th convergent is

$\frac{K(n+1,(a_0,\dots ,a_n))}{K(n,(a_1,\dots ,a_n))}.$

The extended continuant is precisely the determinant of the tridiagonal matrix

$\left(\begin{array}{cccccc}{a}_1& b_1& 0& \dots & 0& 0\\ c_1& a_2& b_2& \dots & 0& 0\\ 0& c_2& a_3& \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & a_{n-1}& b_{n-1}\\ 0& 0& 0& \dots & c_{n-1}& a_n\end{array}\right).$

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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