Algorithms for computer algebra
Algorithms for computer algebra
Recursiveness in matrix rational interpolation problems
Journal of Computational and Applied Mathematics - Special issue: ROLLS symposium
A modular algorithm for computing greatest common right divisors of Ore polynomials
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Modern computer algebra
On the genericity of the modular polynomial GCD algorithm
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Shifted normal forms of polynomial matrices
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors
Journal of the ACM (JACM)
On solutions of linear functional systems
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Fraction-Free Computation of Matrix Rational Interpolants and Matrix GCDs
SIAM Journal on Matrix Analysis and Applications
Fraction-free row reduction of matrices of skew polynomials
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Exact solution of linear equations
SYMSAC '71 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation
On the complexity of polynomial matrix computations
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
On lattice reduction for polynomial matrices
Journal of Symbolic Computation
Algorithms for normal forms for matrices of polynomials and ore polynomials
Algorithms for normal forms for matrices of polynomials and ore polynomials
Fraction-free row reduction of matrices of Ore polynomials
Journal of Symbolic Computation
Normal forms for general polynomial matrices
Journal of Symbolic Computation
On computing polynomial GCDs in alternate bases
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Computing polynomial LCM and GCD in lagrange basis
ACM Communications in Computer Algebra
Submersive rational difference systems and their accessibility
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
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We give modular algorithms to compute row-reduced forms, weak Popov forms, and Popov forms of polynomial matrices, as well as the corresponding unimodular transformation matrices. Our algorithms improve on existing fraction-free algorithms. In each case, we define lucky homomorphisms, determine the appropriate normalization, as well as bound the number of homomorphic images required. The algorithms have the advantage that they are output-sensitive; that is, the number of homomorphic images required depends on the size of the output. Furthermore, there is no need to verify the result by trial division or multiplication. Our algorithms can be used to compute normalized one-sided greatest common divisors and least common multiples of polynomial matrices, along with irreducible matrix-fraction descriptions of matrix rational functions. When our algorithm is used to compute polynomial greatest common divisors, we obtain a new output-sensitive modular algorithm.