Timeline for Stability of root-finding near the unit circle
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 5, 2017 at 21:54 | vote | accept | user49097 | ||
| Nov 3, 2017 at 4:49 | comment | added | J. M. isn't a mathematician | @Igor, indeed, it is that the problem is ill-conditioned, and not that the algorithms are unstable. For the "polynomial is given by values at points" case, Trefethen and others have recommended the "barycentric Lagrange form" for good stability. | |
| Nov 3, 2017 at 4:45 | comment | added | Igor Rivin | @J.M.isnotamathematician Thanks! That lends credence to the OP's claim that it is the problem which is unstable (given that the polynomial is presented in the usual way - if the polynomial is given by values at points, presumably there might be a more stable way of finding a chebyshev representation) | |
| Nov 3, 2017 at 4:36 | comment | added | J. M. isn't a mathematician | @user, the references section of that article would be pretty good further reading. | |
| Nov 3, 2017 at 4:36 | comment | added | J. M. isn't a mathematician | @Igor, correct; the map from monomial coefficients to Chebyshev coefficients is a possible source of inaccuracy. I made a Mathematica demo of the conditioning of the basis conversion matrix here. | |
| Nov 3, 2017 at 1:28 | comment | added | user49097 | @Carlo Beenakker: Thanks a lot. Would you know of any references that explain the points in the article in mathematical detail. (Especially why root finding on the unit circle is well-conditioned.) | |
| Nov 2, 2017 at 23:31 | history | edited | Carlo Beenakker | CC BY-SA 3.0 | added 82 characters in body |
| Nov 2, 2017 at 23:27 | comment | added | Igor Rivin | Since the standard basis is not very orthogonal on the interval, then, presumably, rewriting your polynomial in the Chebyshev basis might be unstable in and of itself. | |
| Nov 2, 2017 at 23:27 | history | edited | Carlo Beenakker | CC BY-SA 3.0 | added 285 characters in body |
| Nov 2, 2017 at 23:21 | history | answered | Carlo Beenakker | CC BY-SA 3.0 |