## Accurate solution of polynomial equations using Macaulay resultant matrices

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- by Guðbjörn F. Jónsson and Stephen A. Vavasis PDF
- Math. Comp.
**74**(2005), 221-262 Request permission

## Abstract:

We propose an algorithm for solving two polynomial equations in two variables. Our algorithm is based on the Macaulay resultant approach combined with new techniques, including randomization, to make the algorithm accurate in the presence of roundoff error. The ultimate computation is the solution of a generalized eigenvalue problem via the QZ method. We analyze the error due to roundoff of the method, showing that with high probability the roots are computed accurately, assuming that the input data (that is, the two polynomials) are well conditioned. Our analysis requires a novel combination of algebraic and numerical techniques.## References

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## Additional Information

**Guðbjörn F. Jónsson**- Affiliation: Center for Applied Mathematics, Rhodes Hall, Cornell University, Ithaca, New York 14853
- Address at time of publication: deCODE Genetics, Lynghals 1, IS-110 Reykjavik, Iceland
- Email: gfj@decode.is
**Stephen A. Vavasis**- Affiliation: Department of Computer Science, Upson Hall, Cornell University, Ithaca, New York 14853
- Email: vavasis@cs.cornell.edu
- Received by editor(s): March 7, 2001
- Received by editor(s) in revised form: December 30, 2002
- Published electronically: July 22, 2004
- Additional Notes: This work was supported in part by NSF grants CCR-9619489 and EIA-9726388. Research also supported in part by NSF through grant DMS-9505155 and ONR through grant N00014-96-1-0050.
- © Copyright 2004 American Mathematical Society
- Journal: Math. Comp.
**74**(2005), 221-262 - MSC (2000): Primary :, 13P10, 65F15; Secondary :, 68W30
- DOI: https://doi.org/10.1090/S0025-5718-04-01722-3
- MathSciNet review: 2085409