Integer extended ABS algorithms and possible control of intermediate results for linear Diophantine systems

Khorramizadeh, Mostafa; Mahdavi–Amiri, Nezam

doi:10.1007/s10288-008-0082-8

Cited by 11 publications

(4 citation statements)

References 14 publications

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“…Suppose that A has full row rank and the Diophantine system (1) has an integer solution. It can be verified that s (i) is a particular solution and N (i) is a basis for the integer null space of the first i equations, see [6] for the proof. Here, to consider the sparsity issues, in every iteration, among all rows not considered so far, we choose the row with smallest numer of nonzeros for computing s i .…”

Section: Sparsity In Linear Diophantine Systemsmentioning

confidence: 99%

Preserving sparity for general solution of linear Diophantine systems

Khorramizadeh¹

2014

ijcms

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Section: Sparsity In Linear Diophantine Systemsmentioning

confidence: 99%

Preserving sparity for general solution of linear Diophantine systems

Khorramizadeh¹

2014

ijcms

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“…In [12], we have recently presented two algorithms for solving such equations. The first algorithm, solving the quadratic equation by the divisibility sequence approach, makes use of a divisibility sequence basis for the row space of A, and the second algorithm, solving the quadratic equation by integer ABS (QEIABS) algorithms with the intention of controlling the growth of intermediate results, is based on a recently proposed integer ABS algorithm in [17]. In [13], we showed that the integer biconjugation process is an special case of the scaled extended integer ABS algorithm [7,17,18].…”

Section: Downloaded By [University Of Otago] At 04:08 03 October 2015mentioning

confidence: 99%

Real and integer Wedderburn rank reduction formulas for matrix decompositions

Mahdavi–Amiri

Golpar-Raboky

2015

Optimization Methods and Software

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The Wedderburn rank reduction formula is a powerful method for developing matrix factorizations and many fundamental numerical linear algebra processes. We present a new interpretation of the Wedderburn rank reduction formula and its associated biconjugation process and see a more extensive result of the formula. In doing this, we propose a new formulation based on the null space transformations on rows and columns of A simultaneously, and show several matrix factorizations that can be derived from the Wedderburn rank reduction formula. We also present a generalization of the biconjugation process and compute banded and Hessenberg factorizations. Using the new formulation, we compute the WZ and ZW factorizations of a nonsingular matrix as well as the Z T Z and the W T W factorizations of a symmetric positive definite matrix. Then, we develop the integer Wedderburn rank reduction formula and its integer biconjugation process and present a class of algorithms for computing all integer matrix factorizations such as the integer LU factorization and the Smith normal form of an arbitrary integer matrix.

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“…ABS class of algorithms was constructed for the solution of linear systems utilizing some basic ideas such as vector projection and rank one update techniques [1,3]. The ABS class was later extended to solve optimization problems [3] and systems of linear Diaphantine equations (see [5,6,17,18]). Reviews of ABS methods and their extensions can be found in [21,22].…”

Section: Introductionmentioning

confidence: 99%

A new interpretation of the WZ factorization using block scaled ABS algorithms

Golpar-Raboky¹,

Mahdavi–Amiri

2014

Stat., optim. inf. comput.

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The WZ factorization suitable for parallel computing was introduced by Evans. Here, a block generalization of the ABS class of methods for the solution of linear system of equations is given and it is shown that it covers the whole class of conjugate direction methods defined by Stewart. The methods produce a factorization of the coefficient matrix implicitly, generating well-known matrix factorizations. We show how to set the parameters of a block ABS algorithm to compute the WZ and ZW factorizations of a nonsingular matrix as well as the W T W and Z T Z factorizations of a symmetric positive definite matrix. We also provide a new interpretation of the necessary and sufficient conditions for the existence of the WZ and ZW factorizations of a nonsingular matrix. We show how to appropriate the parameters to construct algorithms for computing the QZ and the QW factorizations, where Q T Q is an X-matrix. We also provide the conditions for the existence of the integer WZ and the integer ZW factorizations of an integer matrix and compute the factorizations by the integer ABS algorithm.

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Integer extended ABS algorithms and possible control of intermediate results for linear Diophantine systems

Cited by 11 publications

References 14 publications

Preserving sparity for general solution of linear Diophantine systems

Preserving sparity for general solution of linear Diophantine systems

Real and integer Wedderburn rank reduction formulas for matrix decompositions

A new interpretation of the WZ factorization using block scaled ABS algorithms

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