package beast.base.math.matrixalgebra;


import beast.base.math.MathUtils;
import cern.colt.matrix.DoubleFactory1D;
import cern.colt.matrix.DoubleFactory2D;
import cern.colt.matrix.DoubleMatrix1D;
import cern.colt.matrix.DoubleMatrix2D;
import cern.colt.matrix.linalg.Property;

/**
 * Copyright ? 1999 CERN - European Organization for Nuclear Research.
 *
 * This code is adapted from COLT 1.2; Eigendecomposition now throws an ArithmeticException if calculations
 * do not converge within 'maxIterations'
 *
 * @author Marc A. Suchard
 */

public class RobustEigenDecomposition implements java.io.Serializable {
	static final long serialVersionUID = 1020;
	/** Row and column dimension (square matrix).
	@serial matrix dimension.
	*/
	private int n;

	/** Symmetry flag.
	@serial internal symmetry flag.
	*/
	private boolean issymmetric;

	/** Arrays for internal storage of eigenvalues.
	@serial internal storage of eigenvalues.
	*/
	private double[] d, e;

	/** Array for internal storage of eigenvectors.
	@serial internal storage of eigenvectors.
	*/
	private double[][] V;

	/** Array for internal storage of nonsymmetric Hessenberg form.
	@serial internal storage of nonsymmetric Hessenberg form.
	*/
	private double[][] H;

	/** Working storage for nonsymmetric algorithm.
	@serial working storage for nonsymmetric algorithm.
	*/
	private double[] ort;

	// Complex scalar division.

	private transient double cdivr, cdivi;


        private int maxIterations;

        static private int maxIterationsDefault = 100000;
        static private String ERROR_STRING = "Eigendecomposition is not converged.";

/**
Constructs and returns a new eigenvalue decomposition object;
The decomposed matrices can be retrieved via instance methods of the returned decomposition object.
Checks for symmetry, then constructs the eigenvalue decomposition.
@param A    A square matrix.
@return     A decomposition object to access <tt>D</tt> and <tt>V</tt>.
@throws IllegalArgumentException if <tt>A</tt> is not square.
*/
public RobustEigenDecomposition(DoubleMatrix2D A) throws ArithmeticException {
    this(A,maxIterationsDefault);
}


public RobustEigenDecomposition(DoubleMatrix2D A, int maxIterations) throws ArithmeticException {

	Property.DEFAULT.checkSquare(A);

    this.maxIterations = maxIterations;

	n = A.columns();
	V = new double[n][n];
	d = new double[n];
	e = new double[n];

	issymmetric = Property.DEFAULT.isSymmetric(A);

	if (issymmetric) {
		for (int i = 0; i < n; i++) {
			for (int j = 0; j < n; j++) {
				V[i][j] = A.getQuick(i,j);
			}
		}

		// Tridiagonalize.
		tred2();

		// Diagonalize.
		tql2();

	}
	else {
		H = new double[n][n];
		ort = new double[n];

		for (int j = 0; j < n; j++) {
			for (int i = 0; i < n; i++) {
				H[i][j] = A.getQuick(i,j);
			}
		}

		// Reduce to Hessenberg form.
		orthes();

		// Reduce Hessenberg to real Schur form.
		hqr2();
	}
}
private void cdiv(double xr, double xi, double yr, double yi) {
	double r,d;
	if (Math.abs(yr) > Math.abs(yi)) {
		r = yi/yr;
		d = yr + r*yi;
		cdivr = (xr + r*xi)/d;
		cdivi = (xi - r*xr)/d;
	}
	else {
		r = yr/yi;
		d = yi + r*yr;
		cdivr = (r*xr + xi)/d;
		cdivi = (r*xi - xr)/d;
	}
}
/**
Returns the block diagonal eigenvalue matrix, <tt>D</tt>.
@return     <tt>D</tt>
*/
public DoubleMatrix2D getD() {
	double[][] D = new double[n][n];
	for (int i = 0; i < n; i++) {
		for (int j = 0; j < n; j++) {
			D[i][j] = 0.0;
		}
		D[i][i] = d[i];
		if (e[i] > 0) {
			D[i][i+1] = e[i];
		}
		else if (e[i] < 0) {
			D[i][i-1] = e[i];
		}
	}
	return DoubleFactory2D.dense.make(D);
}
/**
Returns the imaginary parts of the eigenvalues.
@return     imag(diag(D))
*/
public DoubleMatrix1D getImagEigenvalues () {
	return DoubleFactory1D.dense.make(e);
}
/**
Returns the real parts of the eigenvalues.
@return     real(diag(D))
*/
public DoubleMatrix1D getRealEigenvalues () {
	return DoubleFactory1D.dense.make(d);
}
/**
Returns the eigenvector matrix, <tt>V</tt>
@return     <tt>V</tt>
*/
public DoubleMatrix2D getV () {
	return DoubleFactory2D.dense.make(V);
}
/**
Nonsymmetric reduction from Hessenberg to real Schur form.
*/
private void hqr2()  throws ArithmeticException {
	  //  This is derived from the Algol procedure hqr2,
	  //  by Martin and Wilkinson, Handbook for Auto. Comp.,
	  //  Vol.ii-Linear Algebra, and the corresponding
	  //  Fortran subroutine in EISPACK.

	  // Initialize

	  int nn = this.n;
	  int n = nn-1;
	  int low = 0;
	  int high = nn-1;
	  double eps = Math.pow(2.0,-52.0);
	  double exshift = 0.0;
	  double p=0,q=0,r=0,s=0,z=0,t,w,x,y;

	  // Store roots isolated by balanc and compute matrix norm

	  double norm = 0.0;
	  for (int i = 0; i < nn; i++) {
		 if (i < low | i > high) {
			d[i] = H[i][i];
			e[i] = 0.0;
		 }
		 for (int j = Math.max(i-1,0); j < nn; j++) {
			norm = norm + Math.abs(H[i][j]);
		 }
	  }

	  // Outer loop over eigenvalue index

	  int iter = 0;
	  while (n >= low) {

		 // Look for single small sub-diagonal element

		 int l = n;
		 while (l > low) {
			s = Math.abs(H[l-1][l-1]) + Math.abs(H[l][l]);
			if (s == 0.0) {
			   s = norm;
			}
			if (Math.abs(H[l][l-1]) < eps * s) {
			   break;
			}
			l--;
		 }

		 // Check for convergence
		 // One root found

		 if (l == n) {
			H[n][n] = H[n][n] + exshift;
			d[n] = H[n][n];
			e[n] = 0.0;
			n--;
			iter = 0;

		 // Two roots found

		 } else if (l == n-1) {
			w = H[n][n-1] * H[n-1][n];
			p = (H[n-1][n-1] - H[n][n]) / 2.0;
			q = p * p + w;
			z = Math.sqrt(Math.abs(q));
			H[n][n] = H[n][n] + exshift;
			H[n-1][n-1] = H[n-1][n-1] + exshift;
			x = H[n][n];

			// Real pair

			if (q >= 0) {
			   if (p >= 0) {
				  z = p + z;
			   } else {
				  z = p - z;
			   }
			   d[n-1] = x + z;
			   d[n] = d[n-1];
			   if (z != 0.0) {
				  d[n] = x - w / z;
			   }
			   e[n-1] = 0.0;
			   e[n] = 0.0;
			   x = H[n][n-1];
			   s = Math.abs(x) + Math.abs(z);
			   p = x / s;
			   q = z / s;
			   r = Math.sqrt(p * p+q * q);
			   p = p / r;
			   q = q / r;

			   // Row modification

			   for (int j = n-1; j < nn; j++) {
				  z = H[n-1][j];
				  H[n-1][j] = q * z + p * H[n][j];
				  H[n][j] = q * H[n][j] - p * z;
			   }

			   // Column modification

			   for (int i = 0; i <= n; i++) {
				  z = H[i][n-1];
				  H[i][n-1] = q * z + p * H[i][n];
				  H[i][n] = q * H[i][n] - p * z;
			   }

			   // Accumulate transformations

			   for (int i = low; i <= high; i++) {
				  z = V[i][n-1];
				  V[i][n-1] = q * z + p * V[i][n];
				  V[i][n] = q * V[i][n] - p * z;
			   }

			// Complex pair

			} else {
			   d[n-1] = x + p;
			   d[n] = x + p;
			   e[n-1] = z;
			   e[n] = -z;
			}
			n = n - 2;
			iter = 0;

		 // No convergence yet

		 } else {

			// Form shift

			x = H[n][n];
			y = 0.0;
			w = 0.0;
			if (l < n) {
			   y = H[n-1][n-1];
			   w = H[n][n-1] * H[n-1][n];
			}

			// Wilkinson's original ad hoc shift

			if (iter == 10) {
			   exshift += x;
			   for (int i = low; i <= n; i++) {
				  H[i][i] -= x;
			   }
			   s = Math.abs(H[n][n-1]) + Math.abs(H[n-1][n-2]);
			   x = y = 0.75 * s;
			   w = -0.4375 * s * s;
			}

			// MATLAB's new ad hoc shift

			if (iter == 30) {
				s = (y - x) / 2.0;
				s = s * s + w;
				if (s > 0) {
					s = Math.sqrt(s);
					if (y < x) {
					   s = -s;
					}
					s = x - w / ((y - x) / 2.0 + s);
					for (int i = low; i <= n; i++) {
					   H[i][i] -= s;
					}
					exshift += s;
					x = y = w = 0.964;
				}
			}

			iter = iter + 1;   // (Could check iteration count here.)
                         if(iter > maxIterations)
                                 throw new ArithmeticException(ERROR_STRING);

			// Look for two consecutive small sub-diagonal elements

			int m = n-2;
			while (m >= l) {
			   z = H[m][m];
			   r = x - z;
			   s = y - z;
			   p = (r * s - w) / H[m+1][m] + H[m][m+1];
			   q = H[m+1][m+1] - z - r - s;
			   r = H[m+2][m+1];
			   s = Math.abs(p) + Math.abs(q) + Math.abs(r);
			   p = p / s;
			   q = q / s;
			   r = r / s;
			   if (m == l) {
				  break;
			   }
			   if (Math.abs(H[m][m-1]) * (Math.abs(q) + Math.abs(r)) <
				  eps * (Math.abs(p) * (Math.abs(H[m-1][m-1]) + Math.abs(z) +
				  Math.abs(H[m+1][m+1])))) {
					 break;
			   }
			   m--;
			}

			for (int i = m+2; i <= n; i++) {
			   H[i][i-2] = 0.0;
			   if (i > m+2) {
				  H[i][i-3] = 0.0;
			   }
			}

			// Double QR step involving rows l:n and columns m:n

			for (int k = m; k <= n-1; k++) {
			   boolean notlast = (k != n-1);
			   if (k != m) {
				  p = H[k][k-1];
				  q = H[k+1][k-1];
				  r = (notlast ? H[k+2][k-1] : 0.0);
				  x = Math.abs(p) + Math.abs(q) + Math.abs(r);
				  if (x != 0.0) {
					 p = p / x;
					 q = q / x;
					 r = r / x;
				  }
			   }
			   if (x == 0.0) {
				  break;
			   }
			   s = Math.sqrt(p * p + q * q + r * r);
			   if (p < 0) {
				  s = -s;
			   }
			   if (s != 0) {
				  if (k != m) {
					 H[k][k-1] = -s * x;
				  } else if (l != m) {
					 H[k][k-1] = -H[k][k-1];
				  }
				  p = p + s;
				  x = p / s;
				  y = q / s;
				  z = r / s;
				  q = q / p;
				  r = r / p;

				  // Row modification

				  for (int j = k; j < nn; j++) {
					 p = H[k][j] + q * H[k+1][j];
					 if (notlast) {
						p = p + r * H[k+2][j];
						H[k+2][j] = H[k+2][j] - p * z;
					 }
					 H[k][j] = H[k][j] - p * x;
					 H[k+1][j] = H[k+1][j] - p * y;
				  }

				  // Column modification

				  for (int i = 0; i <= Math.min(n,k+3); i++) {
					 p = x * H[i][k] + y * H[i][k+1];
					 if (notlast) {
						p = p + z * H[i][k+2];
						H[i][k+2] = H[i][k+2] - p * r;
					 }
					 H[i][k] = H[i][k] - p;
					 H[i][k+1] = H[i][k+1] - p * q;
				  }

				  // Accumulate transformations

				  for (int i = low; i <= high; i++) {
					 p = x * V[i][k] + y * V[i][k+1];
					 if (notlast) {
						p = p + z * V[i][k+2];
						V[i][k+2] = V[i][k+2] - p * r;
					 }
					 V[i][k] = V[i][k] - p;
					 V[i][k+1] = V[i][k+1] - p * q;
				  }
			   }  // (s != 0)
			}  // k loop
		 }  // check convergence
	  }  // while (n >= low)

	  // Backsubstitute to find vectors of upper triangular form

	  if (norm == 0.0) {
		 return;
	  }

	  for (n = nn-1; n >= 0; n--) {
		 p = d[n];
		 q = e[n];

		 // Real vector

		 if (q == 0) {
			int l = n;
			H[n][n] = 1.0;
			for (int i = n-1; i >= 0; i--) {
			   w = H[i][i] - p;
			   r = 0.0;
			   for (int j = l; j <= n; j++) {
				  r = r + H[i][j] * H[j][n];
			   }
			   if (e[i] < 0.0) {
				  z = w;
				  s = r;
			   } else {
				  l = i;
				  if (e[i] == 0.0) {
					 if (w != 0.0) {
						H[i][n] = -r / w;
					 } else {
						H[i][n] = -r / (eps * norm);
					 }

				  // Solve real equations

				  } else {
					 x = H[i][i+1];
					 y = H[i+1][i];
					 q = (d[i] - p) * (d[i] - p) + e[i] * e[i];
					 t = (x * s - z * r) / q;
					 H[i][n] = t;
					 if (Math.abs(x) > Math.abs(z)) {
						H[i+1][n] = (-r - w * t) / x;
					 } else {
						H[i+1][n] = (-s - y * t) / z;
					 }
				  }

				  // Overflow control

				  t = Math.abs(H[i][n]);
				  if ((eps * t) * t > 1) {
					 for (int j = i; j <= n; j++) {
						H[j][n] = H[j][n] / t;
					 }
				  }
			   }
			}

		 // Complex vector

		 } else if (q < 0) {
			int l = n-1;

			// Last vector component imaginary so matrix is triangular

			if (Math.abs(H[n][n-1]) > Math.abs(H[n-1][n])) {
			   H[n-1][n-1] = q / H[n][n-1];
			   H[n-1][n] = -(H[n][n] - p) / H[n][n-1];
			} else {
			   cdiv(0.0,-H[n-1][n],H[n-1][n-1]-p,q);
			   H[n-1][n-1] = cdivr;
			   H[n-1][n] = cdivi;
			}
			H[n][n-1] = 0.0;
			H[n][n] = 1.0;
			for (int i = n-2; i >= 0; i--) {
			   double ra,sa,vr,vi;
			   ra = 0.0;
			   sa = 0.0;
			   for (int j = l; j <= n; j++) {
				  ra = ra + H[i][j] * H[j][n-1];
				  sa = sa + H[i][j] * H[j][n];
			   }
			   w = H[i][i] - p;

			   if (e[i] < 0.0) {
				  z = w;
				  r = ra;
				  s = sa;
			   } else {
				  l = i;
				  if (e[i] == 0) {
					 cdiv(-ra,-sa,w,q);
					 H[i][n-1] = cdivr;
					 H[i][n] = cdivi;
				  } else {

					 // Solve complex equations

					 x = H[i][i+1];
					 y = H[i+1][i];
					 vr = (d[i] - p) * (d[i] - p) + e[i] * e[i] - q * q;
					 vi = (d[i] - p) * 2.0 * q;
					 if (vr == 0.0 & vi == 0.0) {
						vr = eps * norm * (Math.abs(w) + Math.abs(q) +
						Math.abs(x) + Math.abs(y) + Math.abs(z));
					 }
					 cdiv(x*r-z*ra+q*sa,x*s-z*sa-q*ra,vr,vi);
					 H[i][n-1] = cdivr;
					 H[i][n] = cdivi;
					 if (Math.abs(x) > (Math.abs(z) + Math.abs(q))) {
						H[i+1][n-1] = (-ra - w * H[i][n-1] + q * H[i][n]) / x;
						H[i+1][n] = (-sa - w * H[i][n] - q * H[i][n-1]) / x;
					 } else {
						cdiv(-r-y*H[i][n-1],-s-y*H[i][n],z,q);
						H[i+1][n-1] = cdivr;
						H[i+1][n] = cdivi;
					 }
				  }

				  // Overflow control

				  t = Math.max(Math.abs(H[i][n-1]),Math.abs(H[i][n]));
				  if ((eps * t) * t > 1) {
					 for (int j = i; j <= n; j++) {
						H[j][n-1] = H[j][n-1] / t;
						H[j][n] = H[j][n] / t;
					 }
				  }
			   }
			}
		 }
	  }

	  // Vectors of isolated roots

	  for (int i = 0; i < nn; i++) {
		 if (i < low | i > high) {
			for (int j = i; j < nn; j++) {
			   V[i][j] = H[i][j];
			}
		 }
	  }

	  // Back transformation to get eigenvectors of original matrix

	  for (int j = nn-1; j >= low; j--) {
		 for (int i = low; i <= high; i++) {
			z = 0.0;
			for (int k = low; k <= Math.min(j,high); k++) {
			   z = z + V[i][k] * H[k][j];
			}
			V[i][j] = z;
		 }
	  }
   }
/**
Nonsymmetric reduction to Hessenberg form.
*/
private void orthes () {
	  //  This is derived from the Algol procedures orthes and ortran,
	  //  by Martin and Wilkinson, Handbook for Auto. Comp.,
	  //  Vol.ii-Linear Algebra, and the corresponding
	  //  Fortran subroutines in EISPACK.

	  int low = 0;
	  int high = n-1;

	  for (int m = low+1; m <= high-1; m++) {

		 // Scale column.

		 double scale = 0.0;
		 for (int i = m; i <= high; i++) {
			scale = scale + Math.abs(H[i][m-1]);
		 }
		 if (scale != 0.0) {

			// Compute Householder transformation.

			double h = 0.0;
			for (int i = high; i >= m; i--) {
			   ort[i] = H[i][m-1]/scale;
			   h += ort[i] * ort[i];
			}
			double g = Math.sqrt(h);
			if (ort[m] > 0) {
			   g = -g;
			}
			h = h - ort[m] * g;
			ort[m] = ort[m] - g;

			// Apply Householder similarity transformation
			// H = (I-u*u'/h)*H*(I-u*u')/h)

			for (int j = m; j < n; j++) {
			   double f = 0.0;
			   for (int i = high; i >= m; i--) {
				  f += ort[i]*H[i][j];
			   }
			   f = f/h;
			   for (int i = m; i <= high; i++) {
				  H[i][j] -= f*ort[i];
			   }
		   }

		   for (int i = 0; i <= high; i++) {
			   double f = 0.0;
			   for (int j = high; j >= m; j--) {
				  f += ort[j]*H[i][j];
			   }
			   f = f/h;
			   for (int j = m; j <= high; j++) {
				  H[i][j] -= f*ort[j];
			   }
			}
			ort[m] = scale*ort[m];
			H[m][m-1] = scale*g;
		 }
	  }

	  // Accumulate transformations (Algol's ortran).

	  for (int i = 0; i < n; i++) {
		 for (int j = 0; j < n; j++) {
			V[i][j] = (i == j ? 1.0 : 0.0);
		 }
	  }

	  for (int m = high-1; m >= low+1; m--) {
		 if (H[m][m-1] != 0.0) {
			for (int i = m+1; i <= high; i++) {
			   ort[i] = H[i][m-1];
			}
			for (int j = m; j <= high; j++) {
			   double g = 0.0;
			   for (int i = m; i <= high; i++) {
				  g += ort[i] * V[i][j];
			   }
			   // Double division avoids possible underflow
			   g = (g / ort[m]) / H[m][m-1];
			   for (int i = m; i <= high; i++) {
				  V[i][j] += g * ort[i];
			   }
			}
		 }
	  }
   }
/**
Returns a String with (propertyName, propertyValue) pairs.
Useful for debugging or to quickly get the rough picture.
For example,
<pre>
rank          : 3
trace         : 0
</pre>
*/
public String toString() {
	StringBuffer buf = new StringBuffer();
	String unknown = "Illegal operation or error: ";

	buf.append("---------------------------------------------------------------------\n");
	buf.append("EigenvalueDecomposition(A) --> D, V, realEigenvalues, imagEigenvalues\n");
	buf.append("---------------------------------------------------------------------\n");

	buf.append("realEigenvalues = ");
	try { buf.append(String.valueOf(this.getRealEigenvalues()));}
	catch (IllegalArgumentException exc) { buf.append(unknown+exc.getMessage()); }

	buf.append("\nimagEigenvalues = ");
	try { buf.append(String.valueOf(this.getImagEigenvalues()));}
	catch (IllegalArgumentException exc) { buf.append(unknown+exc.getMessage()); }

	buf.append("\n\nD = ");
	try { buf.append(String.valueOf(this.getD()));}
	catch (IllegalArgumentException exc) { buf.append(unknown+exc.getMessage()); }

	buf.append("\n\nV = ");
	try { buf.append(String.valueOf(this.getV()));}
	catch (IllegalArgumentException exc) { buf.append(unknown+exc.getMessage()); }

	return buf.toString();
}
/**
Symmetric tridiagonal QL algorithm.
*/
private void tql2 () throws ArithmeticException  {

	//  This is derived from the Algol procedures tql2, by
	//  Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
	//  Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
	//  Fortran subroutine in EISPACK.

	  for (int i = 1; i < n; i++) {
		 e[i-1] = e[i];
	  }
	  e[n-1] = 0.0;

	  double f = 0.0;
	  double tst1 = 0.0;
	  double eps = Math.pow(2.0,-52.0);
	  for (int l = 0; l < n; l++) {

		 // Find small subdiagonal element

		 tst1 = Math.max(tst1,Math.abs(d[l]) + Math.abs(e[l]));
		 int m = l;
		 while (m < n) {
			if (Math.abs(e[m]) <= eps*tst1) {
			   break;
			}
			m++;
		 }

		 // If m == l, d[l] is an eigenvalue,
		 // otherwise, iterate.

		 if (m > l) {
			int iter = 0;
			do {
			   iter = iter + 1;  // (Could check iteration count here.)
                                  if( iter > maxIterations)
                                           throw new ArithmeticException(ERROR_STRING);

			   // Compute implicit shift

			   double g = d[l];
			   double p = (d[l+1] - g) / (2.0 * e[l]);
			   double r = MathUtils.hypot(p,1.0);
			   if (p < 0) {
				  r = -r;
			   }
			   d[l] = e[l] / (p + r);
			   d[l+1] = e[l] * (p + r);
			   double dl1 = d[l+1];
			   double h = g - d[l];
			   for (int i = l+2; i < n; i++) {
				  d[i] -= h;
			   }
			   f = f + h;

			   // Implicit QL transformation.

			   p = d[m];
			   double c = 1.0;
			   double c2 = c;
			   double c3 = c;
			   double el1 = e[l+1];
			   double s = 0.0;
			   double s2 = 0.0;
			   for (int i = m-1; i >= l; i--) {
				  c3 = c2;
				  c2 = c;
				  s2 = s;
				  g = c * e[i];
				  h = c * p;
				  r = MathUtils.hypot(p,e[i]);
				  e[i+1] = s * r;
				  s = e[i] / r;
				  c = p / r;
				  p = c * d[i] - s * g;
				  d[i+1] = h + s * (c * g + s * d[i]);

				  // Accumulate transformation.

				  for (int k = 0; k < n; k++) {
					 h = V[k][i+1];
					 V[k][i+1] = s * V[k][i] + c * h;
					 V[k][i] = c * V[k][i] - s * h;
				  }
			   }
			   p = -s * s2 * c3 * el1 * e[l] / dl1;
			   e[l] = s * p;
			   d[l] = c * p;

			   // Check for convergence.

			} while (Math.abs(e[l]) > eps*tst1);
		 }
		 d[l] = d[l] + f;
		 e[l] = 0.0;
	  }

	  // Sort eigenvalues and corresponding vectors.

	  for (int i = 0; i < n-1; i++) {
		 int k = i;
		 double p = d[i];
		 for (int j = i+1; j < n; j++) {
			if (d[j] < p) {
			   k = j;
			   p = d[j];
			}
		 }
		 if (k != i) {
			d[k] = d[i];
			d[i] = p;
			for (int j = 0; j < n; j++) {
			   p = V[j][i];
			   V[j][i] = V[j][k];
			   V[j][k] = p;
			}
		 }
	  }
   }
/**
Symmetric Householder reduction to tridiagonal form.
*/
private void tred2 () {
   //  This is derived from the Algol procedures tred2 by
   //  Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
   //  Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
   //  Fortran subroutine in EISPACK.


	  for (int j = 0; j < n; j++) {
		 d[j] = V[n-1][j];
	  }


	  // Householder reduction to tridiagonal form.

	  for (int i = n-1; i > 0; i--) {

		 // Scale to avoid under/overflow.

		 double scale = 0.0;
		 double h = 0.0;
		 for (int k = 0; k < i; k++) {
			scale = scale + Math.abs(d[k]);
		 }
		 if (scale == 0.0) {
			e[i] = d[i-1];
			for (int j = 0; j < i; j++) {
			   d[j] = V[i-1][j];
			   V[i][j] = 0.0;
			   V[j][i] = 0.0;
			}
		 } else {

			// Generate Householder vector.

			for (int k = 0; k < i; k++) {
			   d[k] /= scale;
			   h += d[k] * d[k];
			}
			double f = d[i-1];
			double g = Math.sqrt(h);
			if (f > 0) {
			   g = -g;
			}
			e[i] = scale * g;
			h = h - f * g;
			d[i-1] = f - g;
			for (int j = 0; j < i; j++) {
			   e[j] = 0.0;
			}

			// Apply similarity transformation to remaining columns.

			for (int j = 0; j < i; j++) {
			   f = d[j];
			   V[j][i] = f;
			   g = e[j] + V[j][j] * f;
			   for (int k = j+1; k <= i-1; k++) {
				  g += V[k][j] * d[k];
				  e[k] += V[k][j] * f;
			   }
			   e[j] = g;
			}
			f = 0.0;
			for (int j = 0; j < i; j++) {
			   e[j] /= h;
			   f += e[j] * d[j];
			}
			double hh = f / (h + h);
			for (int j = 0; j < i; j++) {
			   e[j] -= hh * d[j];
			}
			for (int j = 0; j < i; j++) {
			   f = d[j];
			   g = e[j];
			   for (int k = j; k <= i-1; k++) {
				  V[k][j] -= (f * e[k] + g * d[k]);
			   }
			   d[j] = V[i-1][j];
			   V[i][j] = 0.0;
			}
		 }
		 d[i] = h;
	  }

	  // Accumulate transformations.

	  for (int i = 0; i < n-1; i++) {
		 V[n-1][i] = V[i][i];
		 V[i][i] = 1.0;
		 double h = d[i+1];
		 if (h != 0.0) {
			for (int k = 0; k <= i; k++) {
			   d[k] = V[k][i+1] / h;
			}
			for (int j = 0; j <= i; j++) {
			   double g = 0.0;
			   for (int k = 0; k <= i; k++) {
				  g += V[k][i+1] * V[k][j];
			   }
			   for (int k = 0; k <= i; k++) {
				  V[k][j] -= g * d[k];
			   }
			}
		 }
		 for (int k = 0; k <= i; k++) {
			V[k][i+1] = 0.0;
		 }
	  }
	  for (int j = 0; j < n; j++) {
		 d[j] = V[n-1][j];
		 V[n-1][j] = 0.0;
	  }
	  V[n-1][n-1] = 1.0;
	  e[0] = 0.0;
   }

}
