/*
 *                    BioJava development code
 *
 * This code may be freely distributed and modified under the
 * terms of the GNU Lesser General Public Licence.  This should
 * be distributed with the code.  If you do not have a copy,
 * see:
 *
 *      http://www.gnu.org/copyleft/lesser.html
 *
 * Copyright for this code is held jointly by the individual
 * authors.  These should be listed in @author doc comments.
 *
 * For more information on the BioJava project and its aims,
 * or to join the biojava-l mailing list, visit the home page
 * at:
 *
 *      http://www.biojava.org/
 *
 */
package org.biojava.nbio.structure.jama;

/** Eigenvalues and eigenvectors of a real matrix.
<P>
	If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is
	diagonal and the eigenvector matrix V is orthogonal.
	I.e. A = V.times(D.times(V.transpose())) and
	V.times(V.transpose()) equals the identity matrix.
<P>
	If A is not symmetric, then the eigenvalue matrix D is block diagonal
	with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,
	lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda].  The
	columns of V represent the eigenvectors in the sense that A*V = V*D,
	i.e. A.times(V) equals V.times(D).  The matrix V may be badly
	conditioned, or even singular, so the validity of the equation
	A = V*D*inverse(V) depends upon V.cond().
 */

public class EigenvalueDecomposition implements java.io.Serializable {

	/**
	 *
	 */
	private static final long serialVersionUID = 3557806515310435894L;


/* ------------------------
	Class variables
 * ------------------------ */

/** 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;

/* ------------------------
	Private Methods
 * ------------------------ */

	// 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;
	}

	// Symmetric tridiagonal QL algorithm.

	private void tql2 () {

	//  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.)

					// Compute implicit shift

					double g = d[l];
					double p = (d[l+1] - g) / (2.0 * e[l]);
					double r = Maths.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 = Maths.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;
				}
			}
		}
	}

	// 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];
					}
				}
			}
		}
	}


	// Complex scalar division.

	private transient double cdivr, cdivi;
	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;
		}
	}


	// Nonsymmetric reduction from Hessenberg to real Schur form.

	private void hqr2 () {

		//  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.)

				// 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) {
							 continue;
						}
						p = p / x;
						q = q / x;
						r = r / x;
					}

					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;
			}
		}
	}


/* ------------------------
	Constructor
 * ------------------------ */

	/** Check for symmetry, then construct the eigenvalue decomposition
		 Structure to access D and V.
	@param Arg    Square matrix
	*/

	public EigenvalueDecomposition (Matrix Arg) {
		double[][] A = Arg.getArray();
		n = Arg.getColumnDimension();
		V = new double[n][n];
		d = new double[n];
		e = new double[n];

		issymmetric = true;
		for (int j = 0; (j < n) & issymmetric; j++) {
			for (int i = 0; (i < n) & issymmetric; i++) {
				issymmetric = (A[i][j] == A[j][i]);
			}
		}

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

			// Reduce to Hessenberg form.
			orthes();

			// Reduce Hessenberg to real Schur form.
			hqr2();
		}
	}

/* ------------------------
	Public Methods
 * ------------------------ */

	/** Return the eigenvector matrix
	@return     V
	*/

	public Matrix getV () {
		return new Matrix(V,n,n);
	}

	/** Return the real parts of the eigenvalues
	@return     real(diag(D))
	*/

	public double[] getRealEigenvalues () {
		return d;
	}

	/** Return the imaginary parts of the eigenvalues
	@return     imag(diag(D))
	*/

	public double[] getImagEigenvalues () {
		return e;
	}

	/** Return the block diagonal eigenvalue matrix
	@return     D
	*/

	public Matrix getD () {
		Matrix X = new Matrix(n,n);
		double[][] D = X.getArray();
		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 X;
	}

}
