Xinmao Wang

Acceleration of Euclidean Algorithm and Rational Number Reconstruction

We accelerate the known algorithms for computing a selected entry of the extended Euclidean algorithm for integers and, consequently, for the modular and numerical rational number reconstruction problems. The acceleration is from quadratic to nearly linear time, matching the known complexity bound for the integer gcd, which our algorithm computes as a special case.

Effect of small rank modification on the condition number of a matrix

The condition number of a matrix plays an important role in numerical matrix computations. In this paper, we investigate how much the small rank modification method can reduce the condition number of a matrix.

Inversion of Displacement Operators

We recall briefly the displacement rank approach to the computations with structured matrices, which we trace back to the seminal paper by Kailath, Kung, and Morf [ J. Math. Anal. Appl., 68 (1979), pp. 395--407]. The concluding stage of the computations is the recovery of the output from its compressed representation via the associated displacement operator L. The recovery amounts to the inversion of the operator. That is, one must express a structured matrix M via its image L(M). We show a general method for obtaining such expressions that works for all displacement operators (under only the mildest nonsingularity assumptions) and thus provides the foundation for the displacement rank approach to practical computations with structured matrices. We also apply our techniques to specify the expressions for various important classes of matrices. Besides unified derivation of several known formulae, we obtain some new ones, in particular, for the matrices associated with the tangential Nevanlinna--Pick problems. This enables acceleration of the known solution algorithms. We show several new matrix representations of the problem in the important confluent case. Finally, we substantially improve the known estimates for the norms of the inverse displacement operators, which are critical numerical parameters for computations based on the displacement approach.

Acceleration of Euclidean algorithm and extensions

We accelerate the extended Euclidean algorithm for integers, the rational number reconstruction, and consequently, the stage of the recovery of the solution of a nonsingular integer system of linear equations via Hensels lifting. The acceleration is by the order of magnitude and yields nearly optimal randomized algorithms. In the highly important case of Toeplitz, Hankel, and Toeplitz/Hankel-like linear systems, the accleration is potentially practical.

Iterative inversion of structured matrices

Iterative processes for the inversion of structured matrices can be further improved by using a technique for compression and refinement via the least-squares computation. We review such processes and elaborate upon incorporation of this technique into the known frameworks.

On Rational Number Reconstruction and Approximation

The celebrated LKS algorithm by Lehmer, Knuth, and Schonhage, combined with the product tree technique, enables an equivalently rapid alternative to our recent modification of the extended Euclidean algorithm for the reconstruction of a rational number from its modular as well as numerical approximations.

Root-Finding with Eigen-Solving

We survey and extend the recent progress in polynomial root-finding via eigen-solving for highly structured generalized companion matrices. We cover the selection of eigen-solvers and matrices and show the benefits of exploiting matrix structure. No good estimates for the rate of global convergence of the eigen-solvers are known, but according to ample empirical evidence it is sufficient to use a constant number of iteration steps per eigenvalue. If so, the resulting root-finders are optimal up to a constant factor because they use linear arithmetic time per step and perform with a constant (double) precision. Some by-products of our study are of independent interest. The algorithms can be extended to solving secular equations

Eigen-solving via reduction to DPR1 matrices

Highly effective polynomial root-finders have been recently designed based on eigen-solving for DPR1 (that is diagonal + rank-one) matrices. We extend these algorithms to eigen-solving for the general matrix by reducing the problem to the case of the DPR1 input via intermediate transition to a TPR1 (that is triangular + rank-one) matrix. Our transforms use substantially fewer arithmetic operations than the QR classical algorithms but employ non-unitary similarity transforms of a TPR1 matrix, whose representation tends to be numerically unstable. We, however, operate with TPR1 matrices implicitly, as with the inverses of Hessenberg matrices. In this way our transform of an input matrix into a similar DPR1 matrix partly avoids numerical stability problems and still substantially decreases arithmetic cost versus the QR algorithm.