Could the version for complex numbers use polar coordinates? Bill.
On 25 Sep, 18:55, cwitty <[EMAIL PROTECTED]> wrote: > On Sep 25, 9:28 am, "John Cremona" <[EMAIL PROTECTED]> wrote: > > > I thought this had been solved some time ago, and was implemented in > > pari. Or is that only for real roots of real polynomials? > > > John > > > On 9/25/07, cwitty <[EMAIL PROTECTED]> wrote: > > > The biggest obstacle to handling Qbar directly is that I haven't found > > > a good way of isolating the roots of a complex polynomial (that is, > > > finding the roots with a GUARANTEED error bound) and then refining a > > > root to arbitrary precision. > > Maybe so, but the documentation doesn't give enough detail for me to > be confident. The documentation for Pari's polroots function says, > "... it is guaranteed to converge and to give the roots to the > required accuracy." But I don't know what that means. Am I supposed > to trust every bit of the returned value (that is, the error is at > most a half-ULP)? I'm fairly sure I could construct examples where > determining a 100-bit result to within a half-ULP would require > intermediate values that were a million bits. Does Pari go ahead and > use the million-bit numbers here, or does it allow errors more than a > half-ULP? If it allows errors more than a half-ULP, how large are the > errors it does allow? > > Worse, I need to work with polynomials with inexact (interval) > coefficients, and polynomial root-finding is inherently unstable > (small changes in the coefficients can make large changes in the > locations of the roots); even if Pari gives exactly-correct (within a > half-ULP) answers for the polynomials you give it, how much do you > increase the size of the interval outputs to account for the > uncertainty in the inputs? > > All of this is to support a constructor that takes a polynomial with > Qbar coefficients, and a complex interval (a rectangle in the complex > plane) enclosing exactly one root, and return an element of Qbar. > Instead, there could be a constructor that takes a polynomial with > Qbar coefficients, and a complex interval enclosing exactly one root > tightly enough that it can be arbitrarily refined using Newton- > Raphson; and then it would be the responsibility of the caller to find > such a complex interval. But how would the caller do that? > > Carl --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
