numeric-symbolic exact rational linear system solver
An iterative refinement approach is taken to rational linear system solving. Such methods produce, for each entry of the solution vector, a rational approximation with denominator a power of 2. From this the correct rational entry can be reconstructed. Our iteration is a numeric-symbolic hybrid in that it uses an approximate numeric solver at each step together with a symbolic (exact arithmetic) residual computation and symbolic rational reconstruction. The rational solution may be checked symbolically (exactly). However, there is some possibility of failure of convergence, usually due to numeric ill-conditioning. Alternatively, the algorithm may be used to obtain an extended precision floating point approximation of any specified precision. In this case we cannot guarantee the result by rational reconstruction and an exact solution check, but the approach gives evidence (not proof) that the probability of error is extremely small. The chief contributions of the method and implementation are (1) confirmed continuation, (2) improved rational reconstruction, and (3) faster and more robust performance.
rational linear system iterative refinement rational reconstruction