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A Local Construction of the Smith Normal Form of a Matrix Polynomial
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"... We present an algorithm for computing a Smith form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and then combines them into a global Smith form, ..."
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We present an algorithm for computing a Smith form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and then combines them into a global Smith form, whereas other algorithms apply a sequence of unimodular operations to the original matrix row by row (or column by column). The performance of the algorithm in exact arithmetic is reported for several test cases.
BANDED MATRIX SOLVERS AND POLYNOMIAL DIOPHANTINE EQUATIONS ∗
"... Numerical procedures and codes for linear Diophantine polynomial equations are proposed in this paper based on the banded matrix algorithms and solvers. Both the scalar and matrix cases are covered. The algorithms and programs developed are based on the observation that a set of constant linear eq ..."
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Numerical procedures and codes for linear Diophantine polynomial equations are proposed in this paper based on the banded matrix algorithms and solvers. Both the scalar and matrix cases are covered. The algorithms and programs developed are based on the observation that a set of constant linear equations resulting from the polynomial problem features a special structure. This structure, known as Sylvester, or block Syelvester in the matrix case, can in turn be accommodated in the banded matrix framework. Reliable numerical algorithms and programs for banded matrices are readily available at present, for instance in the well known LAPACK package. Software routines based on dedicated LAPACK band matrix solvers were programmed in the C language and linked to MATLAB, mainly for two reasons: to provide prospective users with an environment most of them are familiar with, and to gain the possibility of direct comparison with related functions of the Polynomial Toolbox for MATLAB. Performance of the codes was evaluated by extensive numerical experiments and also in a reallife audio application. 1