An effective implementation of a symbolic-numeric cylindrical algebraic decomposition for quantifier elimination

Recently quantifier elimination (QE) has been of great interest in many fields of science and engineering. In this paper an effective symbolic-numeric cylindrical algebraic decomposition (SNCAD) algorithm and its variant specially designed for QE are proposed based on the authors' previous work and our implementation of those is reported. Based on analysing experimental performances, we are improving our design/synthesis of the SNCAD for its practical realization with existing efficient computational techniques and several newly introduced ones. The practicality of the SNCAD is now examined by a number of experimental results including practical engineering problems, which also reveals the quality of the implementation.

[1]  Thomas Sturm,et al.  Real Quantifier Elimination in Practice , 1997, Algorithmic Algebra and Number Theory.

[2]  Lihong Zhi,et al.  Global optimization of polynomials using generalized critical values and sums of squares , 2010, ISSAC.

[3]  Hirokazu Anai,et al.  Development of SyNRAC-Formula Description and New Functions , 2004, International Conference on Computational Science.

[4]  Fabrice Rouillier,et al.  Classification of the perspective-three-point problem, discriminant variety and real solving polynomial systems of inequalities , 2008, ISSAC '08.

[5]  Christopher W. Brown,et al.  Solution formula construction for truth invariant cad's , 1999 .

[6]  Siegfried M. Rump.  10. Computer-Assisted Proofs and Self-Validating Methods , 2005, Accuracy and Reliability in Scientific Computing.

[7]  Dominique Duval,et al.  About a New Method for Computing in Algebraic Number Fields , 1985, European Conference on Computer Algebra.

[8]  Pablo A. Parrilo,et al.  Semidefinite Programming Relaxations and Algebraic Optimization in Control , 2003, Eur. J. Control.

[9]  Adam W. Strzebonski,et al.  Cylindrical Algebraic Decomposition using validated numerics , 2006, J. Symb. Comput..

[10]  Bin Li,et al.  Exact certification of global optimality of approximate factorizations via rationalizing sums-of-squares with floating point scalars , 2008, ISSAC '08.

[11]  H. Yanami,et al.  Fixed-structure robust controller synthesis based on symbolic-numeric computation: design algorithms with a CACSD toolbox , 2004, Proceedings of the 2004 IEEE International Conference on Control Applications, 2004..

[12]  Hirokazu Anai,et al.  Cylindrical Algebraic Decomposition via Numerical Computation with Validated Symbolic Reconstruction , 2005, Algorithmic Algebra and Logic.

[13]  Christopher W. Brown.  QEPCAD B: a program for computing with semi-algebraic sets using CADs , 2003, SIGS.

[14]  Volker Weispfenning,et al.  Simulation and Optimization by Quantifier Elimination , 1997, J. Symb. Comput..

[15]  Scott McCallum,et al.  On projection in CAD-based quantifier elimination with equational constraint , 1999, ISSAC '99.

[16]  Adam W. Strzebonski,et al.  Solving Systems of Strict Polynomial Inequalities , 2000, J. Symb. Comput..

[17]  Rudolf Krawczyk,et al.  Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken , 1969, Computing.

[18]  Thomas Sturm,et al.  Rounding and Blending of Solids by a Real Elimination Method , 1997 .

[19]  Siegfried M. Rump,et al.  Algebraic Computation, Numerical Computation and Verified Inclusions , 1988, Trends in Computer Algebra.

[20]  Adam W. Strzebonski.  A Real Polynomial Decision Algorithm Using Arbitrary-Precision Floating Point Arithmetic , 1999, Reliab. Comput..

[21]  Andreas Seidl,et al.  Efficient projection orders for CAD , 2004, ISSAC '04.

[22]  Pablo A. Parrilo,et al.  Minimizing Polynomial Functions , 2001, Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science.

[23]  Christopher W. Brown.  Improved Projection for Cylindrical Algebraic Decomposition , 2001, J. Symb. Comput..

[24]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition--preliminary report , 1974, SIGS.

[25]  Hirokazu Anai,et al.  The Maple package SyNRAC and its application to robust control design , 2007, Future Gener. Comput. Syst..

[26]  George E. Collins,et al.  Partial Cylindrical Algebraic Decomposition for Quantifier Elimination , 1991, J. Symb. Comput..

[27]  T. Simpson,et al.  Efficient Pareto Frontier Exploration using Surrogate Approximations , 2000 .

[28]  H. Hong.  An improvement of the projection operator in cylindrical algebraic decomposition , 1990, ISSAC '90.

[29]  Mohab Safey El Din,et al.  Computing the global optimum of a multivariate polynomial over the reals , 2008, ISSAC '08.

[30]  Ramon E. Moore,et al.  SAFE STARTING REGIONS FOR ITERATIVE METHODS , 1977 .

[31]  Hirokazu Anai,et al.  A Practical Implementation of a Symbolic-Numeric Cylindrical Algebraic Decomposition for Quantifier Elimination , 2009 .

[32]  Ramon E. Moore.  A Test for Existence of Solutions to Nonlinear Systems , 1977 .

[33]  E. Pistikopoulos,et al.  Multi-parametric programming : theory, algorithms and applications , 2007 .

[34]  Volker Weispfenning,et al.  Quantifier Elimination for Real Algebra — the Quadratic Case and Beyond , 1997, Applicable Algebra in Engineering, Communication and Computing.

[35]  Adam Strzebonski.  A Real Polynomial Decision Algorithm Using Arbitrary-Precision Floating Point Arithmetic , 1998, SCAN.

[36]  Hoon Hong,et al.  Quantifier elimination for formulas constrained by quadratic equations , 1993, ISSAC '93.

[37]  H. Hong.  An efficient method for analyzing the topology of plane real algebraic curves , 1996 .

[38]  S. Basu,et al.  Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics) , 2006 .

[39]  Volker Weispfenning,et al.  The Complexity of Linear Problems in Fields , 1988, Journal of symbolic computation.

[40]  Christopher W. Brown.  Improved projection for CAD's of R3 , 2000, ISSAC.

[41]  Scott McCallum.  Solving Polynomial Strict Inequalities Using Cylindrical Algebraic Decomposition , 1993, Comput. J..

[42]  S. Hara,et al.  Fixed-structure robust controller synthesis based on sign definite condition by a special quantifier elimination , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[43]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[44]  Dominique Duval,et al.  Algebraic Numbers: An Example of Dynamic Evaluation , 1994, J. Symb. Comput..

[45]  L. González-Vega.  A Combinatorial Algorithm Solving Some Quantifier Elimination Problems , 1998 .

[46]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[47]  Pablo A. Parrilo,et al.  Convex Quantifier Elimination for Semidefinite Programming , 2003 .

[48]  Rüdiger Loos,et al.  Applying Linear Quantifier Elimination , 1993, Comput. J..

[49]  Laureano González-Vega,et al.  Applying Quantifier Elimination to the Birkhoff Interpolation Problem , 1996, J. Symb. Comput..

[50]  Markus Schweighofer.  Global Optimization of Polynomials Using Gradient Tentacles and Sums of Squares , 2006, SIAM J. Optim..

[51]  Stefan Ratschan,et al.  Approximate Quantified Constraint Solving by Cylindrical Box Decomposition , 2002, Reliab. Comput..

[52]  Scott McCallum.  On propagation of equational constraints in CAD-based quantifier elimination , 2001, ISSAC '01.

[53]  George E. Collins,et al.  Interval Arithmetic in Cylindrical Algebraic Decomposition , 2002, J. Symb. Comput..

[54]  B. F. Caviness,et al.  Quantifier Elimination and Cylindrical Algebraic Decomposition , 2004, Texts and Monographs in Symbolic Computation.

[55]  James H. Davenport,et al.  Real Quantifier Elimination is Doubly Exponential , 1988, J. Symb. Comput..