Exact solution of linear equations usingP-adic expansions

Abstract

Summary. A method is described for computing the exact rational solution to a regular system Ax=b of linear equations with integer coefficients. The method involves: (i) computing the inverse (modp) of A for some prime p; (ii) using successive refinements to compute an integer vector ~ such that A,2-b (modp") for a suitably large integer m; and (iii) deducing the rational solution x from the p-adic approximation 2. For matrices A and b with entries of bounded size and dimensions n x n and n x 1, this method can be implemented in time O(n3(logn) 2) which is better than methods previously used.

Algorithms for Structured Linear Systems Solving and Their Implementation

“…before reconstructing the solution, making sure to ensure consistency of the modular solutions when ker(T) has dimension more than 1. Other approaches such as Dixon's algorithm [17], Newton iteration or divide-and-conquer algorithms use one prime p, and li the solution modulo powers of p.…”

Computing subfields in algebraic number fields

“…Finally, the technique described, for example, in [5] shows how to recover from a mod-p k approximation of the coefficients of w(X) the rational values of these coefficients.…”

“…for sufficiently large k, and then use the continued fraction technique described in [5] to compute w{X) e Q[X] such that w{X) = w*(X) (modp k ).…”

“…Let a denote a root of f(X). We shall compute a generator for a subfield of Q(a) of degree 4 over QUsing standard algorithms [13] and quadratic factors of f{X) over Z 5 . In particular, this shows that there is at most one block of size 3 which contains the root of the linear factor, and so there is at most one subfield of Q(a) which has degree 4 over Q.…”

“…In particular, this shows that there is at most one block of size 3 which contains the root of the linear factor, and so there is at most one subfield of Q(a) which has degree 4 over Q. The two polynomials (1 3) and (12 3 The product of the roots of the linear and quadratic factors of f(X) in 1 5 [X] is therefore 8 = -5497064 (mod5 10 ).…”