Chapter 1 Regularization and Matrix Computation in Numerical Polynomial Algebra

Numerical polynomial algebra emerges as a growing field of study in recent years with a broad spectrum of applications and many robust algorithms. Among the challenges in solving polynomial algebra problems with floating-point arithmetic, difficulties frequently arise in regularizing ill-posedness and handling large matrices. We elaborate regularization principles for reformulating the illposed algebraic problems, derive matrix computation arising in numerical polynomial algebra, as well as subspace strategies that substantially improve computing efficiency by reducing matrix sizes. Those strategies have been successfully applied to numerical polynomial algebra problems such as GCD, factorization, multiplicity structure and elimination.