

         Notes on the Inverse Iteration Scheme in the xxx.f file.


        In the current code in MOPAC 5.0, the eigenvalues and eigenvectors
    are found using the following sequence;

            1) TRED3 - reduce the FOCK matrix to a similar tri-diagonal form.

            2) TQLRAT - determine the eigenvalues using a rational QL method.

            3) TQL2   - Find the eigenvectors associated with the eigenvalues, 
                        of the tri-diagonal matrix.          

            4) TRBAK3 - Back transform the eigenvectors of the tri-diagonal
                        matrix to the eigenvectors of the original matrix
                        using the transformations from TRED3.



       This scheme uses a slower, safer QL method where the eigenvalues are 
    determined by a repeated applications of o particular unitary 
    transformation to the tridiagonal matrix.  


       In the other method, found in rsp.f, the eigenvalues and eigenvectors
    are found using the following sequence;

           1) TRED3  - reduce the FOCK matrix to a similar tridiagonal form.

           2) IMTQLV - using an implicit QL method, find the eigenvalues of the
                       tridiagonal matrix.

           3) TINVIT - the eigenvectors are found using an inverse iterative
                       scheme.

           4) TRBAK3 - Back transform the eigenvectors of the tridiagonal 
                       matrix to obtain the eigenvectors of the original matrix
                       using the transformations from TRED3.


        This scheme uses inverse iteration, which is performed by solving the
     relevant equations using Gaussian elimination with partial pivoting.  
     For coincident or close eigenvalues, orthogonality of the eigenvectors
     is generally poorer than TRED3 or TQL2 methods.  This scheme is roughly
     a factor of 2 faster than the QL method, however can be less stable.  


        Both methods are included and have been checked out using all the 
     MOPAC 5.0 test cases.  The original code is in a file called xxx.f and 
     the inverse iterative scheme is in a file called rsp.f.  To test the 
     method, a driver is supplied in the file rsp.f.  Further information can 
     be found in "Linear Algebra", by J.H Wilkinson and C. Reinsch. and the 
     comments in the file rsp.f. The original code is in the file xxx.f

     Code was supplied by Cleve Moler at Ardent Computers.              
