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《Algebraic Systems of Equati*** and Computational Complexity Theory》:simplicial and continuation methods for approximating fixed points and soluti*** to systems of equati*** as well as their computational complexity problems are active topics ofrecent research．the starting point ofthe complexity theory in numerical methods in the elegant paper entitled "the fundamental theorem of algebra and complexity theory" by s．smale．since that，among others there are two main developments：complexity theories of simplical (or piecewise linear) homotopy methods and complexity theories of incremental algorithms or global newton methods．this monograph provides a systemstical and self-contained presentation of the developments．extra efforts have been paid upon its clarity． researchers and postgraduate students interested in computational mathematics and computer science．

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《Algebraic Systems of Equati*** and Computational Complexity Theory》:simplicial and continuation methods for approximating fixed points and soluti*** to systems of equati*** as well as their computational complexity problems are active topics ofrecent research．the starting point ofthe complexity theory in numerical methods in the elegant paper entitled "the fundamental theorem of algebra and complexity theory" by s．smale．since that，among others there are two main developments：complexity theories of simplical (or piecewise linear) homotopy methods and complexity theories of incremental algorithms or global newton methods．this monograph provides a systemstical and self-contained presentation of the developments．extra efforts have been paid upon its clarity． researchers and postgraduate students interested in computational mathematics and computer science．