Applied numerical linear algebra
Applied numerical linear algebra
Complexity and real computation
Complexity and real computation
Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation
ACM Transactions on Mathematical Software (TOMS)
Primary decomposition of lattice basis ideals
Journal of Symbolic Computation - Special issue on symbolic computation in algebra, analysis and geometry
Numerical homotopies to compute generic points on positive dimensional algebraic sets
Journal of Complexity
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
SIAM Journal on Numerical Analysis
Homotopies for Intersecting Solution Components of Polynomial Systems
SIAM Journal on Numerical Analysis
Computing the multiplicity structure in solving polynomial systems
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Newton's method with deflation for isolated singularities of polynomial systems
Theoretical Computer Science
Computing monodromy groups defined by plane algebraic curves
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Adaptive Multiprecision Path Tracking
SIAM Journal on Numerical Analysis
Parallel Implementation of a Subsystem-by-Subsystem Solver
HPCS '08 Proceedings of the 2008 22nd International Symposium on High Performance Computing Systems and Applications
Continuations and monodromy on random riemann surfaces
Proceedings of the 2009 conference on Symbolic numeric computation
An intrinsic homotopy for intersecting algebraic varieties
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Polynomial homotopies on multicore workstations
Proceedings of the 4th International Workshop on Parallel and Symbolic Computation
Polynomial homotopy continuation with PHCpack
ACM Communications in Computer Algebra
Interfacing with the numerical homotopy algorithms in PHCpack
ICMS'06 Proceedings of the Second international conference on Mathematical Software
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Numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complementary dimension. We may represent the linear spaces defined by those planes either by explicit linear equations or in parametric form. These descriptions are respectively called extrinsic and intrinsic representations. While intrinsic representations lower the cost of the linear algebra operations, we observe worse condition numbers. In this paper we describe the local adaptation of intrinsic coordinates to improve the numerical conditioning of sampling algebraic sets. Local intrinsic coordinates also lead to a better step size control. We illustrate our results with Maple experiments and computations with PHCpack on some benchmark polynomial systems.