Eig.java

package jmathlib.toolbox.jmathlib.matrix._private.Jampack;

/**
   Eig implements the eigenvalue-vector decomposition of
   of a square matrix.  Specifically given a diagonalizable
   matrix A, there is a matrix nonsingular matrix X such that
<pre>
*      D = X<sup>-1</sup> AX
</pre>
   is diagonal.  The columns of X are eigenvectors of A corresponding
   to the diagonal elements of D.  Eig implements X as a Zmat and
   D as a Zdiagmat.
<p>
   Warning: if A is defective rounding error will allow Eig to
   compute a set of eigevectors.  However, the matrix X will
   be ill conditioned.
@version Pre-alpha
@author G. W. Stewart
*/

public class Eig{

/** The matrix of eigevectors */
    public Zmat X;

/** The diagonal matrix of eigenvalues */
    public Zdiagmat D;

/**
   Creates an eigenvalue-vector decomposition of a square matrix A.

   @param       A The matrix whose decomposition is to be
                  computed
   @exception   JampackException
                  Thrown if A is not square. <br>
                  Passed from below.
*/                

   public Eig(Zmat A)
   throws JampackException{

      int i, j, k;
      double norm, scale;
      Z z, d;

      A.getProperties();

      if (A.nr != A.nc){
         throw new JampackException
            ("Matrix not square.");
      }

      int n = A.nr;

     /* Compute the Schur decomposition of $A$ and set up T and D. */

      Schur S = new Schur(A);


      Zmat T = S.T;

      D = new Zdiagmat(T);

      norm = Norm.fro(A);

      X = new Zmat(n, n);

      /* Compute the eigevectors of T */

      for (k=n-1; k>=0; k--){

         d = T.get0(k, k);

         X.re[k][k] = 1.0;
         X.im[k][k] = 0.0;

         for (i=k-1; i>=0; i--){

            X.re[i][k] = -T.re[i][k];
            X.im[i][k] = -T.im[i][k];

            for(j=i+1; j<k; j++){

               X.re[i][k] = X.re[i][k] - T.re[i][j]*X.re[j][k]
                                       + T.im[i][j]*X.im[j][k];
               X.im[i][k] = X.im[i][k] - T.re[i][j]*X.im[j][k]
                                       - T.im[i][j]*X.re[j][k];
            }

            z = T.get0(i,i);
            z.Minus(z, d);
            if (z.re==0.0 && z.im==0.0){ // perturb zero diagonal
               z.re = 1.0e-16*norm;      // to avoid division by zero
            }
            z.Div(X.get0(i,k), z);
            X.put0(i, k, z);
         }

        /* Scale the vector so its norm is one. */

         scale = 1.0/Norm.fro(X, X.bx, X.rx, X.bx+k, X.bx+k);
         for (i=0; i<X.nr; i++){
            X.re[i][k] = scale*X.re[i][k];
            X.im[i][k] = scale*X.im[i][k];
         }
      }         
      X = Times.o(S.U, X);
   }
}