Algebraic Systems of Equations and Computational Complexity Theory

Chpater 1 Kuhn’s algorithm for algebraic equations.- §1. Triangulation and labelling.- §2. Complementary pivoting algorithm.- §3. Convergence, I.- §4. Convergence, II.- 2 Efficiency of Kuhn’s algorithm.- §1. Error estimate.- §2. Cost estimate.- §3. Monotonicity problem.- §4. Results on monotonicity.- 3 Newton method and approximate zeros.- §1. Approximate zeros.- §2. Coefficients of polynomials.- §3. One step of Newton iteration.- §4. Conditions for approximate zeros.- 4 A complexity comparison of Kuhn’s algorithm and Newton method.- §1. Smale’s work on the complexity of Newton method.- §2. Set of bad polynomials and its volume estimate.- §3. Locate approximate zeros by Kuhn’s algorithm.- §4. Some remarks.- 5 Incremental algorithms and cost theory.- §1. Incremental algorithms Ih,f.- §2. Euler’s algorithm is of efficiency k.- §3. Generalized approximate zeros.- §4. Ek iteration.- §5. Cost theory of Ek as an Euler’s algorithm.- §6. Incremental algorithms of efficiency k.- 6 Homotopy algorithms.- §1. Homotopies and Index Theorem.- §2. Degree and its invariance.- §3. Jacobian of polynomial mappings.- §4. Conditions for boundedness of solutions.- 7 Probabilistic discussion on zeros of polynomial mappings.- §1. Number of zeros of polynomial mappings.- §2. Isolated zeros.- §3. Locating zeros of analytic functions in bounded regions.- 8 Piecewise linear algorithms.- §1. Zeros of PL mapping and their indexes.- §2. PL approximations.- §3. PL homotopy algorithms work with probability one.- References.- Acknowledgments.