Algebraic Systems of Equations and Computational Complexity Theory: (269 Mathematics and Its Applications)

One service methematics has rendered 'Et moi, ..., si j'avait su comment en revenir, je n'y serais point alle.' the human race. It has put common sense JulesVerne back where it belongs, on the topmost shelf next to the dusty canister labelled The series is divergent; therefore we may 'discarded nonsecse'. be able to do something with it. Eric T. Bell O.Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlinearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered computer science ...'; 'One service category theory has rendered mathematics ...'. All arguable true. And all statements obtainable this way form part of the raison d'etre of this series.

Table of Contents:
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.