Computational Solutions of Matrix Problems Over an Integral Domain

Bareiss, E.H.

doi:10.1093/imamat/10.1.68

Cited by 58 publications

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“…Each considered all integers occurring in the computations as fixed precision. Bareiss again considered the integer case in [2] along with a modular algorithm, this time allowing multiple-precision integers. He also studied these methods again for the related problem of determinant calculation in [3], where three-step and general k-step algorithms are described.…”

Section: Analytical Computing Timesmentioning

confidence: 99%

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A Comparison of Algorithms for the Exact Solution of Linear Equations

McClellan

1977

ACM Trans. Math. Softw.

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A computing-time study is presented of several algorithms for the exact solution of dense systems of linear equations with integer or dense polynomial coefficients. The analytical computing times for rational Gauss elimination, exact division elimination (one step and two step), and the modular algorithm are summarized and supplemented. Extensive empirical studies illustrate the superiority of the modular algorithm for completely dense problems, in agreement with the analytical results. All algorithms were programmed in Fortran IV for the SAC-1 System, and all cases were run on a Univac 1108.

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Section: Analytical Computing Timesmentioning

confidence: 99%

“…For algorithm M assumptions concerning the nonoccurrence of certain highly improbable cases were made [19,20]. = m(n-}-q)r2'+SL(rd)2K2 + n(n-t-q-r-t-1), (2) ti+(m,n, q, r, d, ml, . .…”

Section: • 149mentioning

confidence: 99%

A Comparison of Algorithms for the Exact Solution of Linear Equations

McClellan

1977

ACM Trans. Math. Softw.

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“…Afterwards, efficient methods for alleviating the expression swell problems from Gaussian elimination are given in Refs. [3,[8][9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Fraction-free matrix factors: new forms for LU and QR factors

Zhou

Jeffrey

2008

Front. Comput. Sci. China

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Gaussian elimination and LU factoring have been greatly studied from the algorithmic point of view, but much less from the point view of the best output format. In this paper, we give new output formats for fraction free LU factoring and for QR factoring. The formats and the algorithms used to obtain them are valid for any matrix system in which the entries are taken from an integral domain, not just for integer matrix systems. After discussing the new output format of LU factoring, the complexity analysis for the fraction free algorithm and fraction free output is given. Our new output format contains smaller entries than previously suggested forms, and it avoids the gcd computations required by some other partially fraction free computations. As applications of our fraction free algorithm and format, we demonstrate how to construct a fraction free QR factorization and how to solve linear systems within a given domain.

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“…A deterministic O B ðn oþ1 dÞ algorithm is given in [29,Section 2]. This algorithm is a fraction-free version over K½x (Bareiss' approach [1]) of the recursive inversion algorithms over K cited above. We see that none of these methods seems to reduce the complexity estimate over K below the order of n oþ1 d: With classical matrix multiplication ðo ¼ 3Þ the cost of inversion was still about n times higher than the typical size of the inverse.…”

Section: Introductionmentioning

confidence: 99%

Essentially optimal computation of the inverse of generic polynomial matrices

Jeannerod¹,

Villard²

2005

Journal of Complexity

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We present an inversion algorithm for nonsingular n Â n matrices whose entries are degree d polynomials over a field. The algorithm is deterministic and, when n is a power of two, requires O B ðn 3 dÞ field operations for a generic input; the soft-O notation O B indicates some missing logðndÞ factors. Up to such logarithmic factors, this asymptotic complexity is of the same order as the number of distinct field elements necessary to represent the inverse matrix. r 2004 Elsevier Inc. All rights reserved.

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Computational Solutions of Matrix Problems Over an Integral Domain

Cited by 58 publications

References 0 publications

A Comparison of Algorithms for the Exact Solution of Linear Equations

A Comparison of Algorithms for the Exact Solution of Linear Equations

Fraction-free matrix factors: new forms for LU and QR factors

Essentially optimal computation of the inverse of generic polynomial matrices

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