data/DWSIM.MathOps.Mapack/SingularValueDecomposition.cs

// ----------------------------------------------
// 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>
	/// 	Singular Value Decomposition for a rectangular matrix.
	/// </summary>
	/// <remarks>
	///	  For an m-by-n matrix <c>A</c> with <c>m >= n</c>, the singular value decomposition is
	///   an m-by-n orthogonal matrix <c>U</c>, an n-by-n diagonal matrix <c>S</c>, and
	///   an n-by-n orthogonal matrix <c>V</c> so that <c>A = U * S * V'</c>.
	///   The singular values, <c>sigma[k] = S[k,k]</c>, are ordered so that
	///   <c>sigma[0] >= sigma[1] >= ... >= sigma[n-1]</c>.
	///   The singular value decompostion always exists, so the constructor will
	///   never fail. The matrix condition number and the effective numerical
	///   rank can be computed from this decomposition.
	/// </remarks>
	public class SingularValueDecomposition
	{
		private Matrix U;
		private Matrix V;
		private double[] s; // singular values
		private int m;
		private int n;
	
		/// <summary>Construct singular value decomposition.</summary>
		public SingularValueDecomposition(Matrix value)
		{
			if (value == null)
			{
				throw new ArgumentNullException("value");
			}

			Matrix copy = (Matrix) value.Clone();
			double[][] a = copy.Array;
			m = value.Rows;
			n = value.Columns;
			int nu = Math.Min(m,n);
			s = new double [Math.Min(m+1,n)];
			U = new Matrix(m, nu);
			V = new Matrix(n, n);
			double[][] u = U.Array;
			double[][] v = V.Array;
			double[] e = new double [n];
			double[] work = new double [m];
			bool wantu = true;
			bool wantv = true;
	
			// Reduce A to bidiagonal form, storing the diagonal elements in s and the super-diagonal elements in e.
			int nct = Math.Min(m-1,n);
			int nrt = Math.Max(0,Math.Min(n-2,m));
			for (int k = 0; k < Math.Max(nct,nrt); k++) 
			{
				if (k < nct) 
				{
					// Compute the transformation for the k-th column and place the k-th diagonal in s[k].
					// Compute 2-norm of k-th column without under/overflow.
					s[k] = 0;
					for (int i = k; i < m; i++)
					{
						s[k] = Hypotenuse(s[k],a[i][k]);
					}
	
					if (s[k] != 0.0) 
					{
						if (a[k][k] < 0.0)
						{
							s[k] = -s[k];
						}
	
						for (int i = k; i < m; i++)
						{
							a[i][k] /= s[k];
						}
	
						a[k][k] += 1.0;
					}

					s[k] = -s[k];
				}
					
				for (int j = k+1; j < n; j++) 
				{
					if ((k < nct) & (s[k] != 0.0))  
					{
						// Apply the transformation.
						double t = 0;
						for (int i = k; i < m; i++)
							t += a[i][k]*a[i][j];
						t = -t/a[k][k];
						for (int i = k; i < m; i++)
							a[i][j] += t*a[i][k];
					}
	
					// Place the k-th row of A into e for the subsequent calculation of the row transformation.
					e[j] = a[k][j];
				}
					
				if (wantu & (k < nct)) 
				{
					// Place the transformation in U for subsequent back
					// multiplication.
					for (int i = k; i < m; i++)
						u[i][k] = a[i][k];
				}
	
				if (k < nrt) 
				{
					// Compute the k-th row transformation and place the k-th super-diagonal in e[k].
					// Compute 2-norm without under/overflow.
					e[k] = 0;
					for (int i = k+1; i < n; i++)
					{
						e[k] = Hypotenuse(e[k],e[i]);
					}
	
					if (e[k] != 0.0) 
					{
						if (e[k+1] < 0.0)
							e[k] = -e[k];
	
						for (int i = k+1; i < n; i++)
							e[i] /= e[k];
						
						e[k+1] += 1.0;
					}
						
					e[k] = -e[k];
					if ((k+1 < m) & (e[k] != 0.0)) 
					{
						// Apply the transformation.
						for (int i = k+1; i < m; i++)
							work[i] = 0.0;
	
						for (int j = k+1; j < n; j++)
							for (int i = k+1; i < m; i++)
								work[i] += e[j]*a[i][j];
	
						for (int j = k+1; j < n; j++) 
						{
							double t = -e[j]/e[k+1];
							for (int i = k+1; i < m; i++)
								a[i][j] += t*work[i];
						}
					}
							
					if (wantv) 
					{
						// Place the transformation in V for subsequent back multiplication.
						for (int i = k+1; i < n; i++)
							v[i][k] = e[i];
					}
				}
			}
	
			// Set up the final bidiagonal matrix or order p.
			int p = Math.Min(n,m+1);
			if (nct < n) s[nct] = a[nct][nct];
			if (m < p) s[p-1] = 0.0;
			if (nrt+1 < p) e[nrt] = a[nrt][p-1];
			e[p-1] = 0.0;
	
			// If required, generate U.
			if (wantu) 
			{
				for (int j = nct; j < nu; j++) 
				{
					for (int i = 0; i < m; i++) 
						u[i][j] = 0.0;
					u[j][j] = 1.0;
				}
					
				for (int k = nct-1; k >= 0; k--) 
				{
					if (s[k] != 0.0) 
					{
						for (int j = k+1; j < nu; j++) 
						{
							double t = 0;
							for (int i = k; i < m; i++) 
								t += u[i][k]*u[i][j];
	
							t = -t/u[k][k];
							for (int i = k; i < m; i++)
								u[i][j] += t*u[i][k];
						}
								
						for (int i = k; i < m; i++ )
							u[i][k] = -u[i][k];
	
						u[k][k] = 1.0 + u[k][k];
						for (int i = 0; i < k-1; i++) 
							u[i][k] = 0.0;
					} 
					else 
					{
						for (int i = 0; i < m; i++)
							u[i][k] = 0.0;
						u[k][k] = 1.0;
					}
				}
			}
	
			// If required, generate V.
			if (wantv) 
			{
				for (int k = n-1; k >= 0; k--) 
				{
					if ((k < nrt) & (e[k] != 0.0)) 
					{
						for (int j = k+1; j < nu; j++) 
						{
							double t = 0;
							for (int i = k+1; i < n; i++) 
								t += v[i][k]*v[i][j];
	
							t = -t/v[k+1][k];
							for (int i = k+1; i < n; i++)
								v[i][j] += t*v[i][k];
						}
					}
							
					for (int i = 0; i < n; i++) 
						v[i][k] = 0.0;
					v[k][k] = 1.0;
				}
			}
	
			// Main iteration loop for the singular values.
			int pp = p-1;
			int iter = 0;
			double eps = Math.Pow(2.0,-52.0);
			while (p > 0) 
			{
				int k,kase;
	
				// Here is where a test for too many iterations would go.
				// This section of the program inspects for
				// negligible elements in the s and e arrays.  On
				// completion the variables kase and k are set as follows.
				// kase = 1     if s(p) and e[k-1] are negligible and k<p
				// kase = 2     if s(k) is negligible and k<p
				// kase = 3     if e[k-1] is negligible, k<p, and s(k), ..., s(p) are not negligible (qr step).
				// kase = 4     if e(p-1) is negligible (convergence).
				for (k = p-2; k >= -1; k--) 
				{
					if (k == -1)
						break;
	
					if (Math.Abs(e[k]) <= eps*(Math.Abs(s[k]) + Math.Abs(s[k+1]))) 
					{
						e[k] = 0.0;
						break;
					}
				}
					 
				if (k == p-2) 
				{
					kase = 4;
				}
				else
				{
					int ks;
					for (ks = p-1; ks >= k; ks--) 
					{
						if (ks == k) 
							break;
								 
						double t = (ks != p ? Math.Abs(e[ks]) : 0.0) + (ks != k+1 ? Math.Abs(e[ks-1]) : 0.0);
						if (Math.Abs(s[ks]) <= eps*t)  
						{
							s[ks] = 0.0;
							break;
						}
					}
					
					if (ks == k) 
						kase = 3;
					else if (ks == p-1) 
						kase = 1;
					else
					{
						kase = 2;
						k = ks;
					}
				}
					
				k++;
	
				// Perform the task indicated by kase.
				switch (kase) 
				{
						// Deflate negligible s(p).
					case 1: 
					{
						double f = e[p-2];
						e[p-2] = 0.0;
						for (int j = p-2; j >= k; j--) 
						{
							double t = Hypotenuse(s[j],f);
							double cs = s[j]/t;
							double sn = f/t;
							s[j] = t;
							if (j != k) 
							{
								f = -sn*e[j-1];
								e[j-1] = cs*e[j-1];
							}
									
							if (wantv) 
							{
								for (int i = 0; i < n; i++) 
								{
									t = cs*v[i][j] + sn*v[i][p-1];
									v[i][p-1] = -sn*v[i][j] + cs*v[i][p-1];
									v[i][j] = t;
								}
							}
						}
					}
						break;
	
						// Split at negligible s(k).
					case 2: 
					{
						double f = e[k-1];
						e[k-1] = 0.0;
						for (int j = k; j < p; j++) 
						{
							double t = Hypotenuse(s[j],f);
							double cs = s[j]/t;
							double sn = f/t;
							s[j] = t;
							f = -sn*e[j];
							e[j] = cs*e[j];
							if (wantu) 
							{
								for (int i = 0; i < m; i++) 
								{
									t = cs*u[i][j] + sn*u[i][k-1];
									u[i][k-1] = -sn*u[i][j] + cs*u[i][k-1];
									u[i][j] = t;
								}
							}
						}
					}
						break;
	
						// Perform one qr step.
					case 3: 
					{
						// Calculate the shift.
						double scale = Math.Max(Math.Max(Math.Max(Math.Max(Math.Abs(s[p-1]),Math.Abs(s[p-2])),Math.Abs(e[p-2])), Math.Abs(s[k])),Math.Abs(e[k]));
						double sp = s[p-1]/scale;
						double spm1 = s[p-2]/scale;
						double epm1 = e[p-2]/scale;
						double sk = s[k]/scale;
						double ek = e[k]/scale;
						double b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/2.0;
						double c = (sp*epm1)*(sp*epm1);
						double shift = 0.0;
						if ((b != 0.0) | (c != 0.0)) 
						{
							shift = Math.Sqrt(b*b + c);
							if (b < 0.0)
								shift = -shift;
							shift = c/(b + shift);
						}
								
						double f = (sk + sp)*(sk - sp) + shift;
						double g = sk*ek;
		 
						// Chase zeros.
						for (int j = k; j < p-1; j++) 
						{
							double t = Hypotenuse(f,g);
							double cs = f/t;
							double sn = g/t;
							if (j != k)
								e[j-1] = t;
							f = cs*s[j] + sn*e[j];
							e[j] = cs*e[j] - sn*s[j];
							g = sn*s[j+1];
							s[j+1] = cs*s[j+1];
							if (wantv) 
							{
								for (int i = 0; i < n; i++) 
								{
									t = cs*v[i][j] + sn*v[i][j+1];
									v[i][j+1] = -sn*v[i][j] + cs*v[i][j+1];
									v[i][j] = t;
								}
							}
										
							t = Hypotenuse(f, g);
							cs = f/t;
							sn = g/t;
							s[j] = t;
							f = cs*e[j] + sn*s[j+1];
							s[j+1] = -sn*e[j] + cs*s[j+1];
							g = sn*e[j+1];
							e[j+1] = cs*e[j+1];
							if (wantu && (j < m-1)) 
							{
								for (int i = 0; i < m; i++) 
								{
									t = cs*u[i][j] + sn*u[i][j+1];
									u[i][j+1] = -sn*u[i][j] + cs*u[i][j+1];
									u[i][j] = t;
								}
							}
						}
								
						e[p-2] = f;
						iter = iter + 1;
					}
						break;
	
						// Convergence.
					case 4: 
					{
						// Make the singular values positive.
						if (s[k] <= 0.0) 
						{
							s[k] = (s[k] < 0.0 ? -s[k] : 0.0);
							if (wantv)
								for (int i = 0; i <= pp; i++)
									v[i][k] = -v[i][k];
						}
	 
						// Order the singular values.
						while (k < pp) 
						{
							if (s[k] >= s[k+1]) 
								break;
	
							double t = s[k];
							s[k] = s[k+1];
							s[k+1] = t;
							if (wantv && (k < n-1)) 
								for (int i = 0; i < n; i++) 
								{
									t = v[i][k+1]; 
									v[i][k+1] = v[i][k]; 
									v[i][k] = t;
								}
								
							if (wantu && (k < m-1)) 
								for (int i = 0; i < m; i++) 
								{
									t = u[i][k+1]; 
									u[i][k+1] = u[i][k]; 
									u[i][k] = t;
								}
									
							k++;
						}
							 
						iter = 0;
						p--;
					}
						break;
				}
			}
		}

		/// <summary>Returns the condition number <c>max(S) / min(S)</c>.</summary>
		public double Condition
		{
			get 
			{ 
				return s[0] / s[Math.Min(m, n) - 1]; 
			}
		}
	
		/// <summary>Returns the Two norm.</summary>
		public double Norm2
		{
			get 
			{ 
				return s[0]; 
			}
		}

		/// <summary>Returns the effective numerical matrix rank.</summary>
		/// <value>Number of non-negligible singular values.</value>
		public int Rank
		{
			get
			{
				double eps = Math.Pow(2.0,-52.0);
				double tol = Math.Max(m, n) * s[0] * eps;
				int r = 0;
				for (int i = 0; i < s.Length; i++)
				{
					if (s[i] > tol)
					{
						r++;
					}
				}

				return r;
			}
		}

		/// <summary>Return the one-dimensional array of singular values.</summary>		
		public double[] Diagonal
		{
			get 
			{ 
				return this.s; 
			}
		}

		private static double Hypotenuse(double a, double b) 
		{
			if (Math.Abs(a) > Math.Abs(b))
			{
				double r = b / a;
				return Math.Abs(a) * Math.Sqrt(1 + r * r);
			}

			if (b != 0)
			{
				double r = a / b;
				return Math.Abs(b) * Math.Sqrt(1 + r * r);
			}

			return 0.0;
		}

        /// <summary>Returns the U matrix.</summary>
        public Matrix UMatrix
        {
            get
            {
                return this.U;
            }
        }
        /// <summary>Returns the U matrix.</summary>
        public Matrix VMatrix
        {
            get
            {
                return this.V;
            }
        }
	}
}