By Elise Janvresse, Thierry de la Rue
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0 0 ... 0 1 0 ... 0 , .. .. ... 0 −τ 1 0 1 −t(0) .. 0 ... 0 1 .. ... 0 .. .. 100) 0 0 0 .. 102) The elements of A0 are float numbers, but generally those of A are not. 2 with x replaced by τ (or t(0) ). Paquet shows how we can obtain, ∞ in float arithmetic, a series l=0 xl0 of float numbers convergent to the value of p(τ ). 7. 103) (ii) For j = 0,1,2,... e. 108) where we have set (0) (j) xn+1 = ... 113) 20 1. Evaluation, Convergence, Bounds and with x the exact solution of Ax = p. 113.
5. Detection of Clusters of Zeros (Near-Multiple) 43 ˜ 5 = (x − ζ1 ) Q and we conclude that P1 has one simple root, two of multiplicity 3, and one of multiplicity 5. There are no roots of multiplicity 2 or 4. Note that ˜ 3 will be given as a quadratic, not factorized. We will have to initially Q ˜ i could be of still higher degree. 5 Detection of Clusters of Zeros (Near-Multiple) The methods of the last section break down in the presence of rounding error, as in floating-point computation, since a small change in the coefficients leads to the disintegration of a k-fold zero into a cluster of k distinct (but usually close) zeros.
E. d. d. e. there are no multiple roots. It is recommended that one do this division before applying the Sturm sequence method. Not only is it then easier to apply Sturm, but other methods such as Newton’s, which may be used in conjunction with Sturm, converge faster in the abscence of multiple roots. d. we may obtain a set of polynomials each of which contain only zeros of a specific (known) multiplicity. 13) where Pm+1 is the first Pj to be constant. 4. 15) the i’th factor is replaced by 1. 17 m+1 > all mi and hence m+1 ≥ Max mi +1).