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The Truncated Fourier Transform and Applications
, 2004
"... In this paper, we present a truncated version of the classical Fast Fourier Transform. When applied to polynomial multiplication, this algorithm has the nice property of eliminating the “jumps ” in the complexity at powers of two. When applied to the multiplication of multivariate polynomials or tru ..."
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Cited by 12 (1 self)
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In this paper, we present a truncated version of the classical Fast Fourier Transform. When applied to polynomial multiplication, this algorithm has the nice property of eliminating the “jumps ” in the complexity at powers of two. When applied to the multiplication of multivariate polynomials or truncated multivariate power series, we gain a logarithmic factor with respect to the best previously known algorithms.
Newton’s method and FFT trading
, 2006
"... Let C[[z]] be the ring of power series over an effective ring C. In Brent and Kung (1978), it was first shown that differential equations over C[[z]] may be solved in an asymptotically efficient way using Newton’s method. More precisely, if M(n) denotes the complexity for multiplying two polynomials ..."
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Cited by 4 (3 self)
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Let C[[z]] be the ring of power series over an effective ring C. In Brent and Kung (1978), it was first shown that differential equations over C[[z]] may be solved in an asymptotically efficient way using Newton’s method. More precisely, if M(n) denotes the complexity for multiplying two polynomials of degree < n over C, then the first n coefficients of the solution can be computed in time O(M(n)). However, this complexity does not take into account the dependency on the order r of the equation, which is exponential for the original method van der Hoeven (2002) and quadratic for a recent improvement Bostan et al. (2007). In this paper, we present a technique to further improve the dependency on r, by applying Newton’s method up to a lower order, such as n/r, and trading the remaining Newton steps against a lazy or relaxed algorithm in a suitable FFT model. The technique leads to improved asymptotic complexities for several basic operations on formal power series, such as division, exponentiation and the resolution of more general linear and nonlinear systems of equations.
From implicit to recursive equations *
"... The technique of relaxed power series expansion provides an efficient way to solve so called recursive equations of the form F = Φ(F ), where the unknown F is a vector of power series, and where the solution can be obtained as the limit of the sequence 0, Φ(0), Φ(Φ(0)), . With respect to other tech ..."
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The technique of relaxed power series expansion provides an efficient way to solve so called recursive equations of the form F = Φ(F ), where the unknown F is a vector of power series, and where the solution can be obtained as the limit of the sequence 0, Φ(0), Φ(Φ(0)), . With respect to other techniques, such as Newton's method, two major advantages are its generality and the fact that it takes advantage of possible sparseness of Φ. In this paper, we consider more general implicit equations of the form Φ(F ) = 0. Under mild assumptions on such an equation, we will show that it can be rewritten as a recursive equation. If we are actually computing with analytic functions, then recursive equations also provide a systematic device for the computation of verified error bounds. We will show how to apply our results in this context.