Numerical Instability of Resultant Methods for Multidimensional Rootfinding

Hidden-variable resultant methods are a class of algorithms for solving multidimensional polynomial rootfinding problems. In two dimensions, when significant care is taken, they are competitive practical rootfinders. However, in higher dimensions they are known to miss zeros, calculate roots to low precision, and introduce spurious solutions. We show that the hidden variable resultant method based on the Cayley (Dixon or Bézout) matrix is inherently and spectacularly numerically unstable by a factor that grows exponentially with the dimension. We also show that the Sylvester matrix for solving bivariate polynomial systems can square the condition number of the problem. In other words, two popular hidden variable resultant methods are numerically unstable, and this mathematically explains the difficulties that are frequently reported by practitioners. Regardless of how the constructed polynomial eigenvalue problem is solved, severe numerical difficulties will be present. Along the way, we prove that the Cayley resultant is a generalization of Cramer's rule for solving linear systems and generalize Clenshaw's algorithm to an evaluation scheme for polynomials expressed in a degree-graded polynomial basis

Solving singular generalized eigenvalue problems. Part II: projection and augmentation

Generalized eigenvalue problems involving a singular pencil may be very challenging to solve, both with respect to accuracy and efficiency. While Part I presented a rank-completing addition to a singular pencil, we now develop two alternative methods. The first technique is based on a projection onto subspaces with dimension equal to the normal rank of the pencil while the second approach exploits an augmented matrix pencil. The projection approach seems to be the most attractive version for generic singular pencils because of its efficiency, while the augmented pencil approach may be suitable for applications where a linear system with the augmented pencil can be solved efficiently