// ----------------------------------------------
// Lutz Roeder's Mapack for .NET, September 2000
// Adapted from Mapack for COM and Jama routines.
// http://www.aisto.com/roeder/dotnet
// ----------------------------------------------
namespace Mapack
{
	using System;

	/// <summary>
	/// Determines the eigenvalues and eigenvectors of a real square matrix.
	/// </summary>
	/// <remarks>
	/// If <c>A</c> is symmetric, then <c>A = V * D * V'</c> and <c>A = V * V'</c>
	/// where the eigenvalue matrix <c>D</c> is diagonal and the eigenvector matrix <c>V</c> is orthogonal.
	/// If <c>A</c> is not symmetric, the eigenvalue matrix <c>D</c> is block diagonal
	/// with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,
	/// <c>lambda+i*mu</c>, in 2-by-2 blocks, <c>[lambda, mu; -mu, lambda]</c>.
	/// The columns of <c>V</c> represent the eigenvectors in the sense that <c>A * V = V * D</c>.
	/// The matrix V may be badly conditioned, or even singular, so the validity of the equation
	/// <c>A=V*D*inverse(V)</c> depends upon the condition of <c>V</c>.
	/// </remarks>
	public class EigenvalueDecomposition
	{
		private int n;           	// matrix dimension
		private double[] d, e; 		// storage of eigenvalues.
		private Matrix V; 			// storage of eigenvectors.
		private Matrix H;  			// storage of nonsymmetric Hessenberg form.
		private double[] ort;    	// storage for nonsymmetric algorithm.
		private double cdivr, cdivi;
		private bool symmetric;

		/// <summary>Construct an eigenvalue decomposition.</summary>
		public EigenvalueDecomposition(Matrix value)
		{
			if (value == null)
			{
				throw new ArgumentNullException("value");				
			}

			if (value.Rows != value.Columns) 
			{
				throw new ArgumentException("Matrix is not a square matrix.", "value");
			}
			
			n = value.Columns;
			V = new Matrix(n,n);
			d = new double[n];
			e = new double[n];
	
			// Check for symmetry.
			this.symmetric = value.Symmetric;
	
			if (this.symmetric)
			{
				for (int i = 0; i < n; i++)
				{
					for (int j = 0; j < n; j++)
					{
						V[i,j] = value[i,j];
					}
				}
		 
				// Tridiagonalize.
				this.tred2();

				// Diagonalize.
				this.tql2();
			} 
			else 
			{
				H = new Matrix(n,n);
				ort = new double[n];
					 
				for (int j = 0; j < n; j++)
				{
					for (int i = 0; i < n; i++)
					{
						H[i,j] = value[i,j];
					}
				}
		 
				// Reduce to Hessenberg form.
				this.orthes();
		 
				// Reduce Hessenberg to real Schur form.
				this.hqr2();
			}
		}
		
		private void tred2() 
		{
			// Symmetric Householder reduction to tridiagonal form.
			// 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;
		} 
		 
		private void tql2() 
		{
			// Symmetric tridiagonal QL algorithm.
			// 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 = Hypotenuse(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 = Hypotenuse(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;
					}
				}
			}
		}
		 
		private void orthes() 
		{
			// Nonsymmetric reduction to Hessenberg form.
			// 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];
					}
				}
			}
		}
		 
		private void cdiv(double xr, double xi, double yr, double yi)
		{
			// Complex scalar division.
			double r;
			double 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;
			}
		}

		private void hqr2() 
		{
			// Nonsymmetric reduction from Hessenberg to real Schur form.   
			// 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.
			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;
			double q = 0;
			double r = 0;
			double s = 0;
			double z = 0;
			double t;
			double w;
			double x;
			double 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]);
			}