A method of convergence acceleration of some continued fractions II

作者：Nowak Rafal^*

来源：Numerical Algorithms, 2013, 63(4): 573-600.

DOI：10.1007/s11075-012-9642-2

• Convergence acceleration
• Continued fraction
• Tail
• Modified approximant

Most of the methods for convergence acceleration of continued fractions K(a(m)/b(m)) are based on the use of modified approximants S-m(omega(m)) in place of the classical ones S-m(0), where omega(m) are close to the tails f((m)) of the continued fraction. Recently (Nowak, Numer Algorithms 41(3):297-317, 2006), the author proposed an iterative method producing tail approximations whose asymptotic expansion accuracies are being improved in each step. This method can be successfully applied to a convergent continued fraction K(a(m)/b(m)), where a(m) = alpha(-2)m(2) + alpha(-1)m + ..., b(m) = beta(m)(-1) + beta(0) + ... (alpha(-2) not equal = 0, vertical bar beta(-1)vertical bar(2) + |beta(0)vertical bar(2) not equal = 0, i.e. deg a(m) = 2, deg b(m) is an element of {0, 1}). The purpose of this paper is to extend this idea to the class of two-variant continued fractions K(a(n)/b(n) + a'(n)/b'(n)) with a(n), a'(n), b(n), b'(n) being rational in n and deg a(n) = deg a'(n), deg b(n) = deg b'(n). We give examples involving continued fraction expansions of some elementary and special mathematical functions.

• 出版日期2013-8