22,254 research outputs found
Factorization of Z-homogeneous polynomials in the First (q)-Weyl Algebra
We present algorithms to factorize weighted homogeneous elements in the first
polynomial Weyl algebra and -Weyl algebra, which are both viewed as a
-graded rings. We show, that factorization of homogeneous
polynomials can be almost completely reduced to commutative univariate
factorization over the same base field with some additional uncomplicated
combinatorial steps. This allows to deduce the complexity of our algorithms in
detail. Furthermore, we will show for homogeneous polynomials that
irreducibility in the polynomial first Weyl algebra also implies irreducibility
in the rational one, which is of interest for practical reasons. We report on
our implementation in the computer algebra system \textsc{Singular}. It
outperforms for homogeneous polynomials currently available implementations
dealing with factorization in the first Weyl algebra both in speed and elegancy
of the results.Comment: 26 pages, Singular implementation, 2 algorithms, 1 figure, 2 table
Computing generalized inverses using LU factorization of matrix product
An algorithm for computing {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4} -inverses and
the Moore-Penrose inverse of a given rational matrix A is established. Classes
A(2, 3)s and A(2, 4)s are characterized in terms of matrix products (R*A)+R*
and T*(AT*)+, where R and T are rational matrices with appropriate dimensions
and corresponding rank. The proposed algorithm is based on these general
representations and the Cholesky factorization of symmetric positive matrices.
The algorithm is implemented in programming languages MATHEMATICA and DELPHI,
and illustrated via examples. Numerical results of the algorithm, corresponding
to the Moore-Penrose inverse, are compared with corresponding results obtained
by several known methods for computing the Moore-Penrose inverse
Modular Las Vegas Algorithms for Polynomial Absolute Factorization
Let f(X,Y) \in \ZZ[X,Y] be an irreducible polynomial over \QQ. We give a
Las Vegas absolute irreducibility test based on a property of the Newton
polytope of , or more precisely, of modulo some prime integer . The
same idea of choosing a satisfying some prescribed properties together with
is used to provide a new strategy for absolute factorization of .
We present our approach in the bivariate case but the techniques extend to the
multivariate case. Maple computations show that it is efficient and promising
as we are able to factorize some polynomials of degree up to 400
Differential qd algorithm with shifts for rank-structured matrices
Although QR iterations dominate in eigenvalue computations, there are several
important cases when alternative LR-type algorithms may be preferable. In
particular, in the symmetric tridiagonal case where differential qd algorithm
with shifts (dqds) proposed by Fernando and Parlett enjoys often faster
convergence while preserving high relative accuracy (that is not guaranteed in
QR algorithm). In eigenvalue computations for rank-structured matrices QR
algorithm is also a popular choice since, in the symmetric case, the rank
structure is preserved. In the unsymmetric case, however, QR algorithm destroys
the rank structure and, hence, LR-type algorithms come to play once again. In
the current paper we discover several variants of qd algorithms for
quasiseparable matrices. Remarkably, one of them, when applied to Hessenberg
matrices becomes a direct generalization of dqds algorithm for tridiagonal
matrices. Therefore, it can be applied to such important matrices as companion
and confederate, and provides an alternative algorithm for finding roots of a
polynomial represented in the basis of orthogonal polynomials. Results of
preliminary numerical experiments are presented
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