Group contribution method: Difference between revisions

Revision as of 16:18, 12 July 2013

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

(\begin{matrix} 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 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & a_{n - 1} & b_{n - 1} \\ 0 & 0 & 0 & \dots & c_{n - 1} & a_{n} \end{matrix}) .

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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+In [[algebra]], the '''continuant''' is a [[multivariate polynomial]] representing the [[determinant]] of a [[tridiagonal matrix]] and having applications in [[generalized continued fraction]]s.
+==Definition==
+The ''n''-th ''continuant'', ''K''(''n''), of a sequence '''a''' = ''a''<sub>1</sub>,...,''a''<sub>''n''</sub>,... is defined recursively by
+:<math> K(0) = 1 ; \, </math>
+:<math> K(1) = a_1 ; \, </math>
+:<math> K(n) = a_n K(n-1) + K(n-2) . \, </math>
+It may also be obtained by taking the sum of all possible products of ''a''<sub>1</sub>,...,''a''<sub>''n''</sub> 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 ''a''<sub>1</sub>,...,''a''<sub>''n''</sub>, ''b''<sub>1</sub>,...,''b''<sub>''n''&minus;1</sub> and ''c''<sub>1</sub>,...,''c''<sub>''n''&minus;1</sub>.  In this case the [[recurrence relation]] becomes
+:<math> K(0) = 1 ; \, </math>
+:<math> K(1) = a_1 ; \, </math>
+:<math> K(n) = a_n K(n-1) - b_{n-1}c_{n-1} K(n-2) . \, </math>
+Since ''b''<sub>''r''</sub> and ''c''<sub>''r''</sub> enter into ''K'' only as a product ''b''<sub>''r''</sub>''c''<sub>''r''</sub> there is no loss of generality in assuming that the ''b''<sub>''r''</sub> are all equal to 1.
+==Applications==
+The simple continuant gives the value of a [[continued fraction]] of the form <math>[a_0;a_1,a_2,\ldots]</math>.  The ''n''-th convergent is
+:<math> \frac{K(n+1,(a_0,\ldots,a_n))}{K(n,(a_1,\ldots,a_n))} . </math>
+The extended continuant is precisely the determinant of the tridiagonal matrix
+:<math> \begin{pmatrix}
+a_1 & b_1 &  0  & \ldots & 0 & 0 \\
+c_1 & a_2 & b_2 & \ldots & 0 & 0 \\
+  & c_2 & a_3 & \ldots & 0 & 0 \\
+ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
+& 0 & 0 & \ldots & a_{n-1} & b_{n-1} \\
+& 0 & 0 & \ldots & c_{n-1} & a_n
+\end{pmatrix} . </math>
+==References==
+* {{cite book | author=Thomas Muir | authorlink=Thomas Muir (mathematician) | title=A treatise on the theory of determinants | date=1960 | publisher=[[Dover Publications]] | pages=516&ndash;525 }}
+* {{cite book | title=The Markoff and Lagrange Spectra | first1=Thomas W. | last1=Cusick | first2=Mary E. | last2=Flahive | publisher=[[American Mathematical Society]] | year=1989 | isbn=0-8218-1531-8 | pages=89 | zbl=0685.10023 | series=Mathematical Surveys and Monographs | volume=30 | location=Providence, RI }}
+* {{cite book | title=Algebra, an Elementary Text-book for the Higher Classes of Secondary Schools and for Colleges: Pt. 1 | author=George Chrystal | authorlink=George Chrystal | publisher=American Mathematical Society | year=1999 | isbn=0-8218-1649-7 | pages=500 }}
+[[Category:Algebra]]
+[[Category:Matrices]]
+[[Category:Polynomials]]
+{{Linear-algebra-stub}}

Group contribution method: Difference between revisions

Revision as of 16:18, 12 July 2013

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