TR-2011002: Symbolic Lifting for Structured Linear Systems of Equations: Numerical Initialization, Nearly Optimal Boolean Cost, Variations, and Extensions

Hensel’s symbolic lifting for a linear system of equations and numerical iterative refinement of its solution have striking similarity. Combining the power of lifting and refinement seems to be a natural resource for further advances, but turns out to be hard to exploit. In this paper, however, we employ iterative refinement to initialize lifting. In the case of Toeplitz, Hankel, and other popular structured inputs our hybrid algorithm supports Boolean (bit operation) time bound that is optimal up to logarithmic factor. The algorithm remains nearly optimal in its extensions to computing polynomial gcds and lcms and Padé approximations, as well as to the Berlekamp–Massey reconstruction of linear recurrences. We also cover Newton’s lifting for matrix inversion, specialize it to the case of structured input, and combine it with Hensel’s to enhance overall efficiency. Our initialization techniques for Hensel’s lifting also work for Newton’s. Furthermore we extend all our lifting algorithms to allow their initialization modulo powers of two, thus implementing them in the binary base.