[摘要] This thesis presents techniques for accurately computing a number of fundamental operations on approximate polynomials. The general goal is to determine nearby polynomialswhich have a non-trivial result for the operation. We proceed by first translating each of the polynomial operations to a particular structured matrix system, constructed to represent dependencies in the polynomial coefficients.Perturbing this matrix system to a nearby system of reduced rank yields the nearby polynomials that have a non-trivial result. The translation from polynomial operation to matrix system permits the use of emergingmethods for solving sophisticated least squares problems. These methods introduce therequired dependencies in the system in a structured way, ensuring a certain minimization ismet. This minimization ensures the determined polynomials are close to the original input. We present translations for the following operations on approximate polynomials:
- Division
- Greatest Common Divisor (GCD)
- Bivariate Factorization
- Decomposition
The Least Squares problems considered include classical Least Squares (LS), Total LeastSquares (TLS) and Structured Total Least Squares (STLS). In particular, we make useof some recent developments in formulation of STLS, to perturb the matrix system, whilemaintaining the structure of the original matrix. This allows reconstruction of the resulting polynomials without applying any heuristics or iterative refinements, and guarantees aresult for the operation with zero residual. Underlying the methods for the LS, TLS and STLS problems are varying uses of the Singular Value Decomposition (SVD). This decomposition is also a vital tool for deter-mining appropriate matrix rank, and we spend some time establishing the accuracy of theSVD. We present an algorithm for
relatively accurate SVD recently introduced in [8], thenused to solve LS and TLS problems. The result is confidence in the use of LS and TLS forthe polynomial operations, to provide a fair contrast with STLS. The SVD is also used toprovide the starting point for our STLS algorithm, with the prescribed guaranteed accuracy. Finally, we present a generalized implementation of the Riemannian SVD (RiSVD), whichcan be applied on any structured matrix to determine the result for STLS. This has theadvantage of being applicable to all of our polynomial operations, with the penalty ofdecreased efficiency. We also include a novel, yet naive, improvement that relies on ran-domization to increase the efficiency, by converting a rectangular system to one that issquare. The results for each of the polynomial operations are presented in detail, and the benefitsof each of the Least Squares solutions are considered. We also present distance boundsthat confirm our solutions are within an acceptable tolerance.
