Singular or Nonpositive Definite Hessian Matrix

Many devices have been proposed to solve problems arising when the Hessian is nonpositive definite. Only the main devices are considered below  

1) Greenstadt (1967) proposed evaluating the eigenvalues and eigenvectors of G. As the matrix is symmetric, there are techniques that allow these values to be obtained in a safe and accurate way (Buzzi-Ferraris and Manenti, 2010a). 

Given the matrix eigenvalues and eigenvectors, the axes of the quadratic function that approximates the original function are also known. These axes can be adopted as preferred search directions. 

The GetEigenvaluesAndEigenvectors function returns the vector d, matrix eigenvalues, and eigenvectors in the columns of the matrix U. 

The calculation of eigenvalues and eigenvectors requires about 20-40 times more computational time than the Cholesky factorization. 

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