Solving Diophantine Quadratic Equations and Computing Smith Normal Form Using General ABS Algorithms

We present a new class of algorithms for computing unimodular matrices V and U so that T V AU is a Smith normal form of an integer matrix A with arbitrary rank, making use of scaled extended integer ABS algorithms. We propose a new class of extended integer ABS (EIABS) algorithms generating an integer basis for the integer null space of an integer matrix. Using the algorithms, we simultaneously construct the columns of V and U inductively. In our proposed approach, we need to solve quadratic Diophantine equations of the form x T Ay 􀀠 b . We present two algorithms for solving such equations