A General Convergence Criterion for Continued Fractions K(a n /b n )

1965 ◽  
Vol 16 (6) ◽  
pp. 1256 ◽  
Author(s):  
K. L. Hillam ◽  
W. J. Thron
Keyword(s):  
Continued Fractions ◽  
Convergence Criterion ◽  
General Convergence

scholarly journals On the convergence criterion for branched continued fractions with independent variables

2018 ◽  
Vol 9 (2) ◽  
pp. 120-127 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Special Form ◽  
Absolute Convergence ◽  
Convergence Criterion ◽  
Complex Numbers ◽  
Independent Variables ◽  
Nonnegative Coefficients ◽  
Multiple Power ◽  
Multiple Power Series

In this paper, we consider the problem of convergence of an important type of multidimensional generalization of continued fractions, the branched continued fractions with independent variables. These fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. We have established the effective criterion of absolute convergence of branched continued fractions of the special form in the case when the partial numerators are complex numbers and partial denominators are equal to one. This result is a multidimensional analog of the Worpitzky's criterion for continued fractions. We have investigated the polycircular domain of uniform convergence for multidimensional C-fractions with independent variables in the case of nonnegative coefficients of this fraction.

scholarly journals Generalized continued fractions: a unified definition and a Pringsheim-type convergence criterion

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hendrik Baumann
Keyword(s):  
Continued Fractions ◽  
Convergence Theory ◽  
Convergence Criterion ◽  
Convergence Results ◽  
Starting Point ◽  
Special Cases ◽  
Definition Of ◽  
Generalized Continued Fractions

Abstract In the literature, many generalizations of continued fractions have been introduced, and for each of them, convergence results have been proved. In this paper, we suggest a definition of generalized continued fractions which covers a great variety of former generalizations as special cases. As a starting point for a convergence theory, we prove a Pringsheim-type convergence criterion which includes criteria for the aforementioned special cases. Furthermore, we address several fields in which our definition may be applied.

scholarly journals Convergence criterion for branched contіnued fractions of the special form with positive elements

2017 ◽  
Vol 9 (1) ◽  
pp. 13-21 ◽  
Author(s):  
D.I. Bodnar ◽  
I.B. Bilanyk
Keyword(s):  
Continued Fractions ◽  
Special Form ◽  
Sufficient Conditions ◽  
Convergence Criterion ◽  
Sufficient Condition ◽  
Independent Variables ◽  
Positive Elements ◽  
Multiple Power ◽  
Necessary And Sufficient ◽  
Multiple Power Series

In this paper the problem of convergence of the important type of a multidimensional generalization of continued fractions, the branched continued fractions with independent variables, is considered. This fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. When variables are fixed these fractions are called the branched continued fractions of the special form. Their structure is much simpler then the structure of general branched continued fractions. It has given a possibility to establish the necessary and sufficient conditions of convergence of branched continued fractions of the special form with the positive elements. The received result is the multidimensional analog of Seidel's criterion for the continued fractions. The condition of convergence of investigated fractions is the divergence of series, whose elements are continued fractions. Therefore, the sufficient condition of the convergence of this fraction which has been formulated by the divergence of series composed of partial denominators of this fraction, is established. Using the established criterion and Stieltjes-Vitali Theorem the parabolic theorems of branched continued fractions of the special form with complex elements convergence, is investigated. The sufficient conditions gave a possibility to make the condition of convergence of the branched continued fractions of the special form, whose elements lie in parabolic domains.

scholarly journals On a general convergence for Broyden like update method

1991 ◽  
Vol 14 (2) ◽  
pp. 349-361
Author(s):  
Rabindranath Sen ◽  
Rini Chattopadhyay ◽  
Tripti Saha
Keyword(s):  
Optimization Problems ◽  
Convergence Criterion ◽  
Algebraic Equations ◽  
Broyden’S Method ◽  
Unconstrained Optimization Problems ◽  
Nonlinear Algebraic Equations ◽  
Broyden's Method ◽  
Quasi Newton ◽  
General Convergence

The role of Broyden's method as a powerful quasi-Newton method for solving unconstrained optimization problems or a system of nonlinear algebraic equations is well known. We offer here a general convergence criterion for a method akin to Broyden's method inRn. The approach is different from those of other convergence proofs which are available only for the direct prediction methods.

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