6 research outputs found
Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix
Given a nonsingular matrix of univariate polynomials over a
field , we give fast and deterministic algorithms to compute its
determinant and its Hermite normal form. Our algorithms use
operations in ,
where is bounded from above by both the average of the degrees of the rows
and that of the columns of the matrix and is the exponent of matrix
multiplication. The soft- notation indicates that logarithmic factors in the
big- are omitted while the ceiling function indicates that the cost is
when . Our algorithms are based
on a fast and deterministic triangularization method for computing the diagonal
entries of the Hermite form of a nonsingular matrix.Comment: 34 pages, 3 algorithm
Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations
We give a Las Vegas algorithm which computes the shifted Popov form of an nonsingular polynomial matrix of degree in expected
field operations, where is the
exponent of matrix multiplication and
indicates that logarithmic factors are omitted. This is the first algorithm in
for shifted row reduction with arbitrary
shifts.
Using partial linearization, we reduce the problem to the case where is the generic determinant bound, with bounded from above by both the average row degree and the average column
degree of the matrix. The cost above becomes , improving upon the cost of the fastest previously
known algorithm for row reduction, which is deterministic.
Our algorithm first builds a system of modular equations whose solution set
is the row space of the input matrix, and then finds the basis in shifted Popov
form of this set. We give a deterministic algorithm for this second step
supporting arbitrary moduli in
field operations, where is the number of unknowns and is the sum
of the degrees of the moduli. This extends previous results with the same cost
bound in the specific cases of order basis computation and M-Pad\'e
approximation, in which the moduli are products of known linear factors.Comment: 8 pages, sig-alternate class, 5 figures (problems and algorithms
On the complexity of inverting integer and polynomial matrices
Abstract. An algorithm is presented that probabilistically computes the exact inverse of a nonsingular n × n integer matrix A using (n 3 (log ||A|| + log κ(A))) 1+o(1) bit operations. Here, ||A|| = max ij |A ij | denotes the largest entry in absolute value, κ(A) := n||A −1 || ||A|| is the condition number of the input matrix and the "+o(1)" in the exponent indicates a missing factor c 1 (log n) c 2 (loglog ||A||) c 3 for positive real constants c 1 , c 2 , c 3 . A variation of the algorithm is presented for polynomial matrices that computes the inverse of a nonsingular n × n matrix whose entries are polynomials of degree d over a field using (n 3 d) 1+o(1) field operations. Both algorithms are randomized of the Las Vegas type: failure may be reported with probability at most 1/2, and if failure is not reported then the output is certified to be correct in the same running time bound
Smith Normal Form over Local Rings and Related Problems
The Smith normal form is a diagonalization of matrices with many applications in diophantine analysis, graph theory, system control theory, simplicial homology, and more recently, in topological analysis of big data. Efficient computation of Smith normal form is a well-studied area for matrices with integer and polynomial entries. Existing successful algorithms typically rely on elimination for dense matrices and iterative Krylov space methods for sparse matrices.
Our interest lies in computing Smith normal form for sparse matrices over local rings, where traditional iterative methods face challenges due to the lack of unique minimal polynomials. We explore different approaches to tackling this problem for two local rings: the integers modulo a prime power, and the polynomials modulo a power of an irreducible polynomial. Over local polynomial rings, we find success in linearization into larger dimension matrices over the base field. Effectively we transform the problem of computing the Smith normal form into a small number of rank problems over the base field. The latter problem has existing efficient algorithms for sparse and dense matrices.
The problem is harder over local integer rings. We take the approach of hybrid sparse-dense algorithms. We also tackle a restricted version of the problem where we detect only the first non-trivial invariant factor. We also give an algorithm to find the first few invariant factors using iterative rank-1 updates. This method becomes dense when applied to finding all the invariant factors.
We digress slightly into the related problem of preconditioning. We show that linear- time preconditioners are suitable for computing Smith normal form, and computing nullspace samples. For the latter problem we design an algorithm for computing uniform samples from the nullspace.
On a separate track, we focus on the properties of the Smith normal form decomposition. We relate the invariant factors to the eigenvalues. Our ultimate goal is to extend the applications of numerical algorithms for computing eigenvalues to computing the invariant factors of symbolic matrices