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38
Factoring polynomials with rational coefficients
 MATH. ANN
, 1982
"... In this paper we present a polynomialtime algorithm to solve the following problem: given a nonzero polynomial fe Q[X] in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q[X]. It is well known that this is equivalent to factoring primitive polynomia ..."
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Cited by 982 (11 self)
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In this paper we present a polynomialtime algorithm to solve the following problem: given a nonzero polynomial fe Q[X] in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q[X]. It is well known that this is equivalent to factoring primitive polynomials feZ[X] into irreducible factors in Z[X]. Here we call f ~ Z[X] primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. [8]. Its running time, measured in bit operations, is O(nl2+n9(log[fD3). Here f~Tl[X] is the polynomial to be factored, n = deg(f) is the degree of f, and for a polynomial ~ a ~ i with real coefficients a i. i An outline of the algorithm is as follows. First we find, for a suitable small prime number p, a padic irreducible factor h of f, to a certain precision. This is done with Berlekamp's algorithm for factoring polynomials over small finite fields, combined with Hensel's lemma. Next we look for the irreducible factor h o of f in
Analysis of PSLQ, An Integer Relation Finding Algorithm
 Mathematics of Computation
, 1999
"... Let K be either the real, complex, or quaternion number system and let O(K) be the corresponding integers. Let × = (Xl, • • • , ×n) be a vector in K n. The vector × has an integer relation if there exists a vector m = (ml,..., mn) E O(K) n, m = _ O, such that mlx I + m2x 2 +... + mnXn = O. In th ..."
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Cited by 94 (29 self)
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Let K be either the real, complex, or quaternion number system and let O(K) be the corresponding integers. Let × = (Xl, • • • , ×n) be a vector in K n. The vector × has an integer relation if there exists a vector m = (ml,..., mn) E O(K) n, m = _ O, such that mlx I + m2x 2 +... + mnXn = O. In this paper we define the parameterized integer relation construction algorithm PSLQ(r), where the parameter rcan be freely chosen in a certain interval. Beginning with an arbitrary vector X = (Xl,..., Xn) _ K n, iterations of PSLQ(r) will produce lower bounds on the norm of any possible relation for X. Thus PS/Q(r) can be used to prove that there are no relations for × of norm less than a given size. Let M x be the smallest norm of any relation for ×. For the real and complex case and each fixed parameter rin a certain interval, we prove that PSLQ(r) constructs a relation in less than O(fl 3 + n 2 log Mx) iterations.
Experimental Evaluation of Euler Sums
, 1993
"... In response to a letter from Goldbach, Euler considered sums of the form 1 X k=1 ` 1 + 1 2 m + \Delta \Delta \Delta + 1 k m ' (k + 1) \Gamman for positive integers m and n. Euler was able to give explicit values for certain of these sums in terms of the Riemann zeta function. In a r ..."
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Cited by 64 (11 self)
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In response to a letter from Goldbach, Euler considered sums of the form 1 X k=1 ` 1 + 1 2 m + \Delta \Delta \Delta + 1 k m ' (k + 1) \Gamman for positive integers m and n. Euler was able to give explicit values for certain of these sums in terms of the Riemann zeta function. In a recent companion paper, Euler's results were extended to a significantly larger class of sums of this type, including sums with alternating signs. This research was facilitated by numerical computations using an algorithm that can determine, with high confidence, whether or not a particular numerical value can be expressed as a rational linear combination of several given constants. The present paper presents the numerical techniques used in these computations and lists many of the experimental results that have been obtained.
Parallel Integer Relation Detection: Techniques and Applications
 Mathematics of Computation
, 2000
"... Let {x1,x2, ···,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 +a2x2 +···+ anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and phy ..."
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Cited by 55 (35 self)
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Let {x1,x2, ···,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 +a2x2 +···+ anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, due in part to the requirement for multiprecision arithmetic, yet are poorly suited for parallel processing. This paper presents a new integer relation algorithm designed for parallel computer systems, but as a bonus it also gives superior results on single processor systems. Singleand multilevel implementations of this algorithm are described, together with performance results on a parallel computer system. Several applications of these programs are discussed, including some new results in mathematical number theory, quantum field theory and chaos theory.
MPFUN: A Portable High Performance Multiprecision Package
, 1990
"... The author has written a package of Fortran routines that perform a variety of arithmetic operations and transcendental functions on floating point numbers of arbitrarily high precision, including large integers. This package features (1) virtually universal portability, (2) high performance, especi ..."
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Cited by 51 (4 self)
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The author has written a package of Fortran routines that perform a variety of arithmetic operations and transcendental functions on floating point numbers of arbitrarily high precision, including large integers. This package features (1) virtually universal portability, (2) high performance, especially on vector supercomputers, (3) advanced algorithms, including FFTbased multiplication and quadratically convergent algorithms for π and transcendental functions, and (4) extensive selfchecking and debug facilities that permit the package to be used as a rigorous system integrity test. Converting application programs to run with these routines is facilitated by an automatic translator program. This paper describes the routines in the package and includes discussion of the algorithms employed, the implementation techniques, performance results and some applications. Notable among the performance results is that this package runs up to 40 times faster than another widely used package on a RISC workstation, and it runs up to 400 times faster than the other package on a Cray supercomputer.
A Polynomial Time, Numerically Stable Integer Relation Algorithm
, 1991
"... Let x =(x1,x2, ···,xn) be a vector of real numbers. x is said to possess an integer relation if there exist integers ai not all zero such that a1x1 + a2x2 + ·· · + anxn =0. Beginning in 1977 several algorithms (with proofs) have been discovered to recover the ai given x. The most efficient of these ..."
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Cited by 47 (6 self)
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Let x =(x1,x2, ···,xn) be a vector of real numbers. x is said to possess an integer relation if there exist integers ai not all zero such that a1x1 + a2x2 + ·· · + anxn =0. Beginning in 1977 several algorithms (with proofs) have been discovered to recover the ai given x. The most efficient of these existing integer relation algorithms (in terms of run time and the precision required of the input) has the drawback of being very unstable numerically. It often requires a numeric precision level in the thousands of digits to reliably recover relations in modestsized test problems. We present here a new algorithm for finding integer relations, which we have named the “PSLQ ” algorithm. It is proved in this paper that the PSLQ algorithm terminates with a relation in a number of iterations that is bounded by a polynomial in n. Because this algorithm employs a numerically stable matrix reduction procedure, it is free from the numerical difficulties that plague other integer relation algorithms. Furthermore, its stability admits an efficient implementation with lower run times on average than other algorithms currently in use. Finally, this stability can be used to prove that relation bounds obtained from computer runs using this algorithm are numerically accurate.
Theory and applications of the doublebase number system
 IEEE Transactions on Computers
, 1999
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Multiple Polylogarithms: A Brief Survey
"... . We survey various results and conjectures concerning multiple polylogarithms and the multiple zeta function. Among the results, we announce our resolution of several conjectures on multiple zeta values. We also provide a new integral representation for the general multiple polylogarithm, and devel ..."
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Cited by 29 (8 self)
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. We survey various results and conjectures concerning multiple polylogarithms and the multiple zeta function. Among the results, we announce our resolution of several conjectures on multiple zeta values. We also provide a new integral representation for the general multiple polylogarithm, and develop a qanalogue of the shuffle product. 1.
Experimental Mathematics: Recent Developments and Future Outlook
 CECM PREPRINT 99:143] FFL J.M. BORWEIN AND P.B. BORWEIN, &QUOT;CHALLENGES FOR MATHEMATICAL COMPUTING,&QUOT; COMPUTING IN SCIENCE & ENGINEERING, 2001. [CECM PREPRINT 01:160
, 2000
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Geodesic multidimensional continued fractions
 Proc. London Math. Soc
, 1994
"... A multidimensional continued fraction expansion is given which finds provably good Diophantine approximations for all 6 e U'1. For any Q> 1 it finds some approximation (p, q) e Z ' with 1 s £ q = £ Q satisfying \\q % p  = £ V j + 1 Q~xld. This expansion consists of a sequence of redu ..."
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Cited by 25 (1 self)
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A multidimensional continued fraction expansion is given which finds provably good Diophantine approximations for all 6 e U'1. For any Q> 1 it finds some approximation (p, q) e Z ' with 1 s £ q = £ Q satisfying \\q % p  = £ V j + 1 Q~xld. This expansion consists of a sequence of reduced lattice bases for a parametrized series of lattice bases B,(8) (of different lattices) in GL(d + 1,0?), where the positive real parameter / varies. This parametrized family B,(&) forms a geodesic in GL(d + 1,0?), and also projects to a geodesic ge in the Riemannian symmetric space 9>dJrX of all positive definite symmetric matrices. The multidimensional continued fraction expansion is a 'cutting sequence ' expansion for g0 using a Minkowski fundamental domain of GL(d + 1, 2) \ ^ r f + 1. This method generalizes to give continued fraction expansions finding good Diophantine approximations to an arbitrary set of linear forms. 1.