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THE COMPLEXITY OF MULTIPLEPRECISION ARITHMETIC
, 1976
"... In studying the complexity of iterative processes it is usually assumed that the arithmetic operations of addition, multiplication, and division can be performed in certain constant times. This assumption is invalid if the precision required increases as the computation proceeds. We give upper and l ..."
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In studying the complexity of iterative processes it is usually assumed that the arithmetic operations of addition, multiplication, and division can be performed in certain constant times. This assumption is invalid if the precision required increases as the computation proceeds. We give upper and lower bounds on the number of singleprecision operations required to perform various multipleprecision operations, and deduce some interesting consequences concerning the relative efficiencies of methods for solving nonlinear equations using variablelength multipleprecision arithmetic.
JOURNAL OF COMPUTER AND SYSTEM SCIENCES 7, 334342 (1973) A Bound on the Multiplicative Efficiency of Iteration*
, 1972
"... For a convergent sequence {xi} generated by xi+l = ~(x~, xt _ l,..., Xi_d+l) , define the multiplicative efficiency measure E to be (log~p)/M, where p is the order of convergence and M is the number of multiplications or divisions needed to compute rp. Then, if 9 is any multivariate rational funct ..."
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For a convergent sequence {xi} generated by xi+l = ~(x~, xt _ l,..., Xi_d+l) , define the multiplicative efficiency measure E to be (log~p)/M, where p is the order of convergence and M is the number of multiplications or divisions needed to compute rp. Then, if 9 is any multivariate rational function, E < 1. Since E = 1 for the sequence {xi} generated by x~+l = x ~ ~ + xi ~ with the limit1/2, the bound on E is sharp. Let PM denote the maximal order for a sequence generated by an iteration with M multiplications. Then PM < 2 M for all positive integers M. Moreover this bound is sharp. 1.
Opt ima l Order o f OnePo in t and Muhipoint Iteration
"... ABSTRACT. The problem is to calculate a simple zero of a nonlinear function f by iteration. There is exhibited a family of iterations of order 2 "1 which use n evaluations of f and no derivative valuations, as well as a second family of iterations of order 2 "1 based on n 1 evaluations ..."
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ABSTRACT. The problem is to calculate a simple zero of a nonlinear function f by iteration. There is exhibited a family of iterations of order 2 "1 which use n evaluations of f and no derivative valuations, as well as a second family of iterations of order 2 "1 based on n 1 evaluations of f and one of f'. In particular, with four evaluations an iteration of eighth order is constructed. The best previous result for four evaluations was fifth order. It is proved that the optimal order of one general class of multipoint iterations is 2 "t and that an upper bound on the order of a multipoint iteration based on n evaluations of] (no derivatives) is 2 n. It is conjectured that a multipoint iteration without memory based on n evaluations has optimal order 2 "1.