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Algorithm for Nonrecursive Calculation of Bernoulli Numbers

IP.com Disclosure Number: IPCOM000076208D
Original Publication Date: 1972-Jan-01
Included in the Prior Art Database: 2005-Feb-24
Document File: 2 page(s) / 35K

Publishing Venue

IBM

Related People

Stacy, EW: AUTHOR

Abstract

The algorithm calculates the value of any Bernoulli number B1 selected by the user from among B1, B2, .... . Knowledge of Bernoulli numbers other than B1 is not employed in calculating B1. A terminating series yields a rational number which is the exact value of B1.

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Algorithm for Nonrecursive Calculation of Bernoulli Numbers

The algorithm calculates the value of any Bernoulli number B1 selected by the user from among B1, B2, .... . Knowledge of Bernoulli numbers other than
B1 is not employed in calculating B1. A terminating series yields a rational number which is the exact value of B1.

Notation for calculating BI: A, an integer index over 1, 2, ..., I; sA, the Ath power of (-1); CAI, the number of combinations of I-1 things taken I-A at a time;

J, an integer index over 1, 2, ..., CAI; VAIJ, the Jth of some ordered sequence of all distinct vectors which can be characterized by having A components whose sum is I-A; FAIJ, a vector obtained from VAIJ by replacing the respective elements by factorials of numbers obtained by adding 2 to each of the original elements; RAIJ, the reciprocal of the product of all elements FAIJ; SAI, the product of sA and the sum of all RAIJ values for a particular value of A. Steps for calculating B1 A. If the value of I is odd and exceeds 2, set B1=0. B. Otherwise,
1) For A=1, ..., I a) generate and store sA CAI VAI1, VAI2,..., VAIK (where K=CAI) b) generate and store RAI1, RAI2, ..., RAIK from the above V's (and the intermediate F's, e.g.) c) generate and store the product of sA and the sum of the above R's denoting the result by SAI 2) Set B1 equal to the product of factorial I,
i.e. I!, times the sum of the numbers S1I, S2I, ..., S1I. Flow Chart.

Implementation is shown by the flow chart. After ST...