Chinese Remainder and Interpolation Algorithms.

Lipson, John D.

doi:10.2172/4646840

Cited by 6 publications

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“…Only in the combination phase will the big numbers appear again. In the case of Z as the original domain the well-studied Chinese Remainder Algorithm (CRA) can be used in the combine step [33]. This overall structure of the algorithm is shown in Figure 14.…”

Section: Program Descriptionmentioning

confidence: 99%

Engineering parallel symbolic programs in GPH

Loidl

Trinder

Hammond

et al. 1999

Concurrency: Pract. Exper.

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Section: Program Descriptionmentioning

confidence: 99%

Engineering parallel symbolic programs in GPH

Loidl

Trinder

Hammond

et al. 1999

Concurrency: Pract. Exper.

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“…We note that all ¢~ and all ~ are, in fact, classed togcther as modular homomorplfisms and that the Incremental Interpolation Algorithm is, in fact, an example of a Chinese Rcmainder Algo-570 MICHAEL T. MCCLELLAN rithm (see [20, pp. 394-395], [8], and especially [22] for an in-depth study of thesc concepts).…”

Section: C~mstrucling General Solutions By Using Homomorphismsmentioning

confidence: 99%

The Exact Solution of Systems of Linear Equations with Polynomial Coefficients

McClellan

1973

J. ACM

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An algorithm is presented for computing exactly general solutions for systems of linear equations with integer or polynomial coefficients. The algorithm applies modular homomorphisms-reductions modulo a prime and evaluations eventually applying a Gaussian elimination algorithm to systenis with coefficients in GF(p). Then, by applying interpolation and the Chinese Remainder Algorithm, a general solution is obtained if the system is found to be consistent. Also included is a modular algorithm for matrix multiplication, as required for substitution tests. The computing times of these modular algorithms are analyzed. The computing time bounds, which dominate these times, are obtained as functions of the size of the system and the numeric coefficient and degree sizes of the system coefficients. A comparison with similar bounds for the exact division algorithm reveals the clear superiority of the modular algorithm.previously tractable became so with the discovery of the subresultant polynomial remainder sequence (PRS) algorithm by Collins [9]. Then, after some initial work by Collins [10] and Brown [7], complete modular algorithms for polynomial GCD and resultant calculation were developed, which are far superior [8,13].A parallel development has taken place in the solution of systems of linear equations. Let 9 be an arbitrary integral domain for which a division algorithm exists, and leg A and B be matrices over t~, where A is m X n, nonzero, and of rank r, and B is m X q. Then the matrix equation A X = B represents q systems of linear equations, one for each column of B; but for simplicity we refer to this equation as a system of linear equations. We have mentioned above the inefficiency of exact elimination methods involving rational operations, i.e. operations in the quotient field Q(9), where ~ = I, the ring of integers, or ~ = I [ x l , . . . , x,], the domain of integral polynomials in.s variables. (Let I [ x l , . . . , x,] denote I for s = 0.) The first step in reducing this inefficiency has been to consider only those methods which restrict ~ to be I [ x l , • • • , x,], s > O, and perform the elimination using only arithmetic operations in 9. That is, if C = (A, B) is the augmented matrix of the linear system, then C is transformed only by elementary row operations in ~. One method of this type was devised by Rosser [27] to control integer coefficient growth in inverting an integral matrix. In his method, the elimination involves only subtractions of multiples of rows from other rows, where multiplication by small integers is used almost exclusively; no divisions are performed.Division may be employed in this class of methods if it is assured that the divisions are exact, producing quotients in 9 (i.e. all elements of a row are multiples of the divisor). These methods, referred to as exact division algorithms, by applying predetermined exact divisions, restrict integer coefficient growth and, in the case ~ = I [xl , . . . , x,], s >_ 1, reduce degree growth as well. Brief descriptions of the basic computat...

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“…(See Szabo and Tanaka [1967, pp. 27, 43], Lipson [1971], and Howell and Gregory [1970].) This means that since /?…”

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An Algorithm for the Exact Reduction of a Matrix to Frobenius form Using Modular Arithmetic. II

Howell¹

1973

Mathematics of Computation

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Abstract. Part I contained a description of the single-modulus algorithm for reducing a matrix to Frobenius form, obtaining exact integral factors of the characteristic polynomial. Part II contains a description of the multiple-modulus algorithm. Since different moduli may yield different factorizations, an algorithm is given for determining which factorizations are not correct factorizations over the integers of the characteristic polynomial. Part II also contains a discussion of the selection of the moduli and numerical examples.

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Chinese Remainder and Interpolation Algorithms.

Cited by 6 publications

References 0 publications

Engineering parallel symbolic programs in GPH

Engineering parallel symbolic programs in GPH

The Exact Solution of Systems of Linear Equations with Polynomial Coefficients

An Algorithm for the Exact Reduction of a Matrix to Frobenius form Using Modular Arithmetic. II

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