Binary Scindage

The binary scindage (in English binary splitting ) is a method of acceleration of the calculation of sums with terms rational S.

The binary scindage for example is used for the evaluation of hypergeometric series in rational points.

The basic principle of the binary scindage consists in recursively dividing the field of summation into small groups of rational to add, to simplify the sums of small groups (reduction with the same denominator, elimination of the common factors) and to reiterate on larger groups.

Operation

That is to say the sum

$S\; (has,\; b)\; =\; \backslash \; sum\_\; \{n=a\}\; ^b\; \backslash \; frac\; \{p\_n\}\; \{q\_n\}$,

where p_n , q_n , has and B is entireties. The binary scindage makes it possible to calculate the entireties P ( has , B ) and Q ( has , B ) such as

$S\; (has,\; b)\; =\; \backslash \; frac\; \{P\; (has,\; b)\}\; \{Q\; (has,\; b)\}.$

The scindage consists in dividing the interval '' B '' into two equal intervals: '' m '' and '' B '' (where m = is the medium of the segment) and to calculate recursively P ( has , B ) and Q ( has , B ) starting from P ( has , m ), P ( m , B ), Q ( has , m ) and Q ( m , B ).

When the two terminals of the interval of the subdivision '' J '' are sufficiently close, the calculation of P ( I , J ) and Q ( I , J ) is p_j carried out directly starting from the p_i… p_j and q_i… q_j .

Note

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