From: Russell Martin Subject: Re: Roots bounds for polynomial systems Date: Tue, 20 Apr 1999 00:21:40 GMT Newsgroups: sci.math.num-analysis Simon Langley wrote: > > Given n polynomials in n unknowns with only a finite number of solution > is there any easy way to obtain bounds on the magnitudes of the roots? > > It doesn't have to be very accurate .... > > Pointers to software/papers etc would be most appreciated. > > Thanks > > ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > > Simon Langley > c/o Computer Studies and Mathematics > University of the West of England > Bristol BS8 4TU > UK I'm not sure that I understand your problem. One interpretation of what I think you may be asking suggests to me that you check out the Gerschgorin circle theorem (e.g. _Numerical Analysis_ by Johnson and Riess, Addison-Wesley Publishing, 1977, pages 153-158). Hope that helps. Regards, Russell Martin -- Russell Martin R. L. Martin and Associates russell.martin@mail.wdn.com http://www.rmartin.com ->Insert pithy quotation here.<- ============================================================================== From: Charles Crawford Subject: Re: spectral radius minimization Date: Wed, 21 Apr 1999 09:40:57 -0400 Newsgroups: sci.math.num-analysis Alejandra - Sources for Gerschgorin circles or inclusion/exclusion sets for eigenvalues: Matrix Iterative Analysis by .S. Varga The Theory of Matrices in Numerical Analysis by A.S. Householder The Algebraic Eigenvalue Problem by J.H. Wilkinson These sources might be a bit out of date, but Wilkinson uses the G. c's for estimating perturbations. You might check with G.W. Stewart who I believe still teaches at Univ. of Maryland. -- Chuck Crawford, Toronto ============================================================================== From: Charles Crawford Subject: Re: spectral radius minimization Date: Thu, 22 Apr 1999 09:06:43 -0400 Newsgroups: [missing] To: Dave Rusin Keywords: Gerschgorin circles Dave - Here are some books that mention and use Gerschgorin circles and related ideas. Matrix Iterative Analysis by .S. Varga The Theory of Matrices in Numerical Analysis by A.S. Householder The Algebraic Eigenvalue Problem by J.H. Wilkinson These sources might be a bit out of date, but Wilkinson uses the G. c's for estimating perturbations. The main result is based on the idea that a diagonally-dominant matrix is non-singular. Apply that to A-xI and you get that A-xI is singular only if x lies in at least one of n circles in the complex plane. The centres are the diagonal elements of A and the radii are the sum of the absolute values of the off-diagonals in one row. You could also do it columnwise. To make the result more useful it can be shown that if any circle does not intersect the others, it must contain an eigenvalue. This is the result Wilkinson uses to get perturbation estimates. -- Chuck Crawford, Toronto ==============================================================================