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Removing redundancy in highprecision Newton iteration
, 2004
"... This paper speeds up Brent's algorithms for various highprecision computations in the power series ring C[[t]]. If it takes time 3 to compute a product then it takes time roughly 5:6 to compute a reciprocal; roughly 8:2 to compute a quotient or a logarithm; roughly 6:5 to compute a square ro ..."
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Cited by 31 (6 self)
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This paper speeds up Brent's algorithms for various highprecision computations in the power series ring C[[t]]. If it takes time 3 to compute a product then it takes time roughly 5:6 to compute a reciprocal; roughly 8:2 to compute a quotient or a logarithm; roughly 6:5 to compute a square root; roughly 9 to compute both a square root and a reciprocal square root; and roughly 10:4 to compute an exponential. The same ideas apply to approximate computations in R, Q p, etc.
Arbitrarily Tight Bounds On The Distribution Of Smooth Integers
 Proceedings of the Millennial Conference on Number Theory
, 2002
"... This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve ..."
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Cited by 3 (1 self)
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This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.
Aimed at Math. Comp. I need to do some software verification first. REMOVING REDUNDANCY IN HIGHPRECISION NEWTON ITERATION
"... Abstract. This paper presents new algorithms for several highprecision operations in the power series ring C[[x]]. Compared to computing n coefficients of a product in C[[x]], computing n coefficients of a reciprocal in C[[x]] takes 1.5+o(1) times longer; a quotient or logarithm, 2.16666...+o(1) ti ..."
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Abstract. This paper presents new algorithms for several highprecision operations in the power series ring C[[x]]. Compared to computing n coefficients of a product in C[[x]], computing n coefficients of a reciprocal in C[[x]] takes 1.5+o(1) times longer; a quotient or logarithm, 2.16666...+o(1) times longer; a square root, 1.83333...+o(1) times longer; an exponential, 2.83333...+o(1) times longer. Previous algorithms had worse constants. The same ideas apply to highprecision computations in R, Q p, etc.