A fast algorithm for computing the Smith normal form with multipliers for a nonsingular integer matrix
نویسندگان
چکیده
A Las Vegas randomized algorithm is given to compute the Smith multipliers for a nonsingular integer matrix A, that is, unimodular matrices U and V such AV=US, with S normal form of A. The expected running time about same as required multiply together two dimension size entries Explicit bounds are in both multipliers. main tool used by massager, relaxed version V, specifying column operations computation. From perspective efficiency, tools fast linear system solving partial linearization matrices. As an application algorithm, find fractional part inverse input matrix.
منابع مشابه
A Fast Las Vegas Algorithm for Computing the Smith Normal Form of a Polynomial Matrix
A Las Vegas probabilistic algorithm is presented that finds the Smith normal form S ∈ Q[x] of a nonsingular input matrix A ∈ Z [x]. The algorithm requires an expected number of O (̃nd(d + n log ||A||)) bit operations (where ||A|| bounds the magnitude of all integer coefficients appearing in A and d bounds the degrees of entries of A). In practice, the main cost of the computation is obtaining a ...
متن کاملThe Smith Normal Form Distribution of A Random Integer Matrix
We show that the density μ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities μps of SNF over Z/p sZ with p a prime and s some positive integer. Our approach is to connect the SNF of a matrix with the greatest common divisors (gcds) of certain polynomials of matrix entries, and develop the theory of multi-gcd distribution of polynomial values at a...
متن کاملComputing the Determinant and Smith Form of an Integer Matrix
A probabilistic algorithm is presented to find the determinant of a nonsingular, integer matrix. For a matrix A n n the algorithm requires O n3 5 logn 4 5 bit operations (assuming for now that entries in A have constant size) using standard matrix and integer arithmetic. Using asymptotically fast matrix arithmetic, a variant is described which requires O n2 2 log2 n loglogn bit operations, wher...
متن کاملA fast, deterministic algorithm for computing a Hermite Normal Form of a polynomial matrix
Given a square, nonsingular matrix of univariate polynomials F ∈ K[x] over a field K, we give a fast, deterministic algorithm for finding the Hermite normal form of F with complexity O (nωd) where d is the degree of F. Here soft-O notation is Big-O with log factors removed and ω is the exponent of matrix multiplication. The method relies of a fast algorithm for determining the diagonal entries ...
متن کاملThe Smith Normal Form of a Matrix
In this note we will discuss the structure theorem for finitely generated modules over a principal ideal domain from the point of view of matrices. We will then give a matrixtheoretic proof of the structure theorem from the point of view of the Smith normal form of a matrix over a principal ideal domain. One benefit from this method is that there are algorithms for finding the Smith normal form...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Symbolic Computation
سال: 2023
ISSN: ['1095-855X', '0747-7171']
DOI: https://doi.org/10.1016/j.jsc.2022.09.002