Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Hidden-variable resultant methods are a class of algorithms for solving #R##N#multidimensional polynomial rootfinding problems. In two dimensions, when #R##N#significant care is taken, they are competitive practical rootfinders. Hover, in higher#R##N#dimensions they are known to miss zeros, calculate roots to low precision, #R##N#and introduce spurious solutions.  show that  #R##N#the hidden-variable resultant method based on  #R##N#the Cayley (Dixon or Bezout) resultant is inherently #R##N#and spectacularly numerically unstable by a factor that grows exponentially with #R##N#the dimension.  also show that the Sylvester resultant for solving#R##N#bivariate polynomial systems can square the condition number of the #R##N#problem. #R##N#In other words, two popular hidden-variable resultant methods are numerically #R##N#unstable, and this mathematically explains the difficulties that #R##N#are frequently reported by practitioners. Along the way,  prove that the#R##N#Cayley resultant is a generalization of Cramer's rule for solving linear systems #R##N#and generalize Clenshaw's algorithm to an evaluation scheme for polynomials #R##N#expressed in a degree-graded polynomial basis.
['Vanni Noferini', 'Alex Townsend']
I am writing a research paper on a idea Are resultant methods numerically unstable for multidimensional rootfinding
[{"from": "human", "value": "I am writing a research paper on a idea Are resultant methods numerically unstable for multidimensional rootfinding"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Hidden-variable resultant methods are a class of algorithms for solving #R##N#multidimensional polynomial rootfinding problems. In two dimensions, when #R##N#significant care is taken, they are competitive practical rootfinders. Hover, in higher#R##N#dimensions they are known to miss zeros, calculate roots to low precision, #R##N#and introduce spurious solutions.  show that  #R##N#the hidden-variable resultant method based on  #R##N#the Cayley (Dixon or Bezout) resultant is inherently #R##N#and spectacularly numerically unstable by a factor that grows exponentially with #R##N#the dimension.  also show that the Sylvester resultant for solving#R##N#bivariate polynomial systems can square the condition number of the #R##N#problem. #R##N#In other words, two popular hidden-variable resultant methods are numerically #R##N#unstable, and this mathematically explains the difficulties that #R##N#are frequently reported by practitioners. Along the way,  prove that the#R##N#Cayley resultant is a generalization of Cramer's rule for solving linear systems #R##N#and generalize Clenshaw's algorithm to an evaluation scheme for polynomials #R##N#expressed in a degree-graded polynomial basis."}]