[理学]《高等数值分析》讲义L21-poly.pdf

Contents Introduction Homotopy Methods Multi-homogeneous Methods Finding the Optimal Partition Bezout Number Computation Discussions 2005-11-28 2 One Dimension Problem Consider the following f(x)=(x-1)(x-2)…….(x-20) Fundamental Theorem in Algebra tells the number of solutions in complex domain Roots for polynomial above clearly are 1 2 3 4 ……… 19 20 Roots for the perturbed problem? 18 f ε(x ) f (x ) +εx 0 2005-11-28 3 ε=10e-5 2005-11-28 4 Polynomial systems The degree of i-th equation is d i 2005-11-28 5 Applications •Chemical reactions •Robot Manipulator •Molecular Chains •Geometric and Solid Modeling •GPS Radar •Discretization of nonlinear DE’s 2005-11-28 6 Molecular Chains 2005-11-28 7 Methods Numerical methods Symbolic methods Wu method 2005-11-28 8 An important problem mathematically Basic problem in Algebraic Geometry A very hot area in pure mathematics Theory is far from complete Real Restricted …………… 2005-11-28 9 Contents Introduction Homotopy Methods Multi-homogeneous Methods Finding the Optimal Partition Bezout Number Computation 2005-11-28 10 Solve It Numerically Homotopy