An analysis of cancellation error in multivariate Hensel construction with floating-point number arithmetic
Tateaki Sasaki, Satoshi Yamaguchi · 1998
Algebraic computation with oating-point numbers often causes large numerical errors due to cancellation of almost the same numbers.In this paper, we take up multivariate Hensel construction F(x; u) G (k) (x; u)H (k) (x; u) (mod (u , s) k+1 ), where (u) denotes (u1; : : : ; u r ) and (s) = ( s 1 ; : : : ; s r ) is a set of numbers, and analyze the cancellation errors contained in the coecients of G (k) and H (k) , in the case that the expansion point (i.e.(u) = ( s )) is near a singular point.Let be the distance between the expansion point and the nearest singular point, hence 1.We i n v estigate the -dependencies of norms kG (k) k, kH (k) k and the terms appearing in the computation, and show that the initial errors are magni ed by about O( , k ), where is a nonnegative n umber.We nd = 0 in some cases but 1 in other cases; the value of depends on the type of singularity and choice of initial factors G (0) and H (0) .The theory developed is con rmed by several numerical examples.