MP-Morse Exponent Formula For Extremely Small And Extremely Large Saddle Points

The instability is an important character to a saddle point.For nondegenerate saddle points,the Morse indices can be used to measure their instability.MP-minimax algorithm designed by Li and Zhou in 2001 is a numerical algorithm for calculating saddle points.In 2001 and 2004,Li and Zhou published two papers to establish a Morse index formula for saddle points found by the MP-minimax algorithm.In this thesis,a modified MP-minimax algorithm is presented first.Then,a Morse index formula for saddle points found by this algorithm,i.e.,MP-minimax type saddle points,is verified.For nondegenerate MP-minimax type saddle points,the formula clearly shows what their Morse indices are.Also,a numerical example on the Henon type system of equations is presented to illustrate the application of the formula for computing the Morse index of numerical solution in the calculation of multiple solutions of the system of the semilinear elliptic equations.