content: package edu.stanford.rsl.conrad.numerics;
  
  import Jama.Matrix;
  import Jama.util.*;
  
  /** Singular Value Decomposition.
     <P>
     For an m-by-n matrix A, the singular value decomposition is
     an m-by-(m or n) orthogonal matrix U, a (m or n)-by-n diagonal matrix S, and
     an n-by-n orthogonal matrix V so that A = U*S*V'.
     <P>
     The singular values, sigma[k] = S[k][k], are ordered so that
     sigma[0] >= sigma[1] >= ... >= sigma[n-1].
     <P>
     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.   
     <P>
     This class is mainly based on SingularValueDecomposition class in Jama in which 
     SVD sometimes fails on cases m < n. The bug has been fixed by Ron Boisvert <boisvert@nist.gov>. 
     Details of what were fixed can be found below:
     http://cio.nist.gov/esd/emaildir/lists/jama/msg01431.html
     http://cio.nist.gov/esd/emaildir/lists/jama/msg01430.html
     http://metamerist.blogspot.com/2008/04/svd-for-vertically-challenged.html
  
     Then, small changes were made to be compatible with CONRAD. 
     @author Jang-Hwan Choi
   */
  
  public class DecompositionSVD implements java.io.Serializable {
  
  	/**
  	 * 
  	 */
  	private static final long serialVersionUID = 201880439875521920L;
  
  	/* ------------------------
     Class variables
  	 * ------------------------ */
  
  	/** Arrays for internal storage of U and V.
     @serial internal storage of U.
     @serial internal storage of V.
  	 */
  	private double[][] U, V;
  
  	/** Array for internal storage of singular values.
     @serial internal storage of singular values.
  	 */
  	private double[] s;
  
  	/** Row and column dimensions.
     @serial row dimension.
     @serial column dimension.
     @serial U column dimension.
  	 */
  	private int m, n, ncu;
  
  	/** Column specification of matrix U
     @serial U column dimension toggle
  	 */
  
  	private boolean thin;
  
  
  	/** 
  	 * <b>Old Constructor</b><br>
  	 * Construct the singular value decomposition
  	 * Structure to access U, S and V.
  	 * @param Arg  Rectangular matrix
  	 * 
  	 */
  
  	public DecompositionSVD (SimpleMatrix Arg) {
  		this(Arg,true,true,true);
  	}
  
  	/* ------------------------
     Constructor
  	 * ------------------------ */
  
  	/** 
  	 * Construct the singular value decomposition, i.e. a
  	 * structure to access U, S and V.
  	 * @param Arg   Rectangular matrix
  	 * @param thin  If true U is economy sized
  	 * @param wantu If true generate the U matrix
  	 * @param wantv If true generate the V matrix
  	 * 
  	 */
  
  	public DecompositionSVD (SimpleMatrix Arg, boolean thin, boolean wantu,
  			boolean wantv) {
  
  		// Derived from LINPACK code.
  		// Initialize.
  		double[][] A = Arg.copyAsDoubleArray();
  		m = Arg.getRows();
  		n = Arg.getCols();
  		this.thin = thin;
  
  		ncu = thin?Math.min(m,n):m;
  		s = new double [Math.min(m+1,n)];
  		if (wantu) U = new double [m][ncu];
  		if (wantv) V = new double [n][n];
  		double[] e = new double [n];
  		double[] work = new double [m];
  
  		// 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));
  		int lu = Math.max(nct,nrt);
  		for (int k = 0; k < lu; 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] = Maths.hypot(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] = Maths.hypot(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 < ncu; 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 < ncu; 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;
  					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 < n; 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);
  		double tiny = Math.pow(2.0,-966.0);
  		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]) <=
  					tiny + 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.) + 
  					(ks != k+1 ? Math.abs(e[ks-1]) : 0.);
  					if (Math.abs(s[ks]) <= tiny + 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 = Maths.hypot(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 = Maths.hypot(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 = Maths.hypot(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 = Maths.hypot(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++;
  			}
  			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 < n; 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;
  			}
  		}
  		A = null;
  	}
  
  	/* ------------------------
     Public Methods
  	 * ------------------------ */
  
  	/** Return the left singular vectors
     @return     U
  	 */
  
  	public SimpleMatrix getU () {
  		return U==null?null:(new SimpleMatrix(new Matrix(U,m,m>=n?(thin?Math.min(m+1,n):ncu):ncu)));
  	}
  
  	/** Return the right singular vectors
     @return     V
  	 */
  
  	public SimpleMatrix getV () {
  		return V==null?null:new SimpleMatrix(new Matrix(V,n,n));
  	}
  
  	/** Return the one-dimensional array of singular values
     @return     diagonal of S.
  	 */
  
  	public double[] getSingularValues () {
  		return s;
  	}
  
  	/** Return the diagonal matrix of singular values
     @return     S
  	 */
  
  	public SimpleMatrix getS () {
  		SimpleMatrix X = new SimpleMatrix(new Matrix(m>=n?(thin?n:ncu):ncu,n));
  		for (int i = Math.min(m,n)-1; i>=0; i--)
  			X.setElementValue(i, i, s[i]);
  		return X;
  	}
  
  	/** Return the diagonal matrix of the reciprocals of the singular values
     @return     S+
  	 */
  
  	public SimpleMatrix getreciprocalS () {
  		SimpleMatrix X = new SimpleMatrix(new Matrix(n,m>=n?(thin?n:ncu):ncu));
  		for (int i = Math.min(m,n)-1; i>=0; i--)
  			X.setElementValue(i, i, s[i]==0.0?0.0:1.0/s[i]);
  		return X;
  	}
  
  	/** Return the Moore-Penrose (generalized) inverse
  	 *  Slightly modified version of Kim van der Linde's code
     @param omit if true tolerance based omitting of negligible singular values
     @return     A+
  	 */
  
  	public SimpleMatrix inverse(boolean omit) {
  		double[][] inverse = new double[n][m];
  		if(rank()> 0) {
  			double[] reciprocalS = new double[s.length];
  			if (omit) {
  				double tol = Math.max(m,n)*s[0]*Math.pow(2.0,-52.0);
  				for (int i = s.length-1;i>=0;i--)
  					reciprocalS[i] = Math.abs(s[i])<tol?0.0:1.0/s[i];
  			}
  			else
  				for (int i=s.length-1;i>=0;i--)
  					reciprocalS[i] = s[i]==0.0?0.0:1.0/s[i];
  			int min = Math.min(n, ncu);
  			for (int i = n-1; i >= 0; i--)
  				for (int j = m-1; j >= 0; j--)
  					for (int k = min-1; k >= 0; k--)
  						inverse[i][j] += V[i][k] * reciprocalS[k] * U[j][k];
  		} 
  		return new SimpleMatrix(new Matrix(inverse));
  	}
  	
  	/** Return the Moore-Penrose inverse including Tikhonov regularization.
  	 *  Reciprocal eigenvalues given as sigma / (sigma²+alpha), where alpha is the regularization parameter for A + alpha I.
  	 *  @param omit if true tolerance based omitting of negligible singular values
  	 *  @param alpha regularization parameter
  	 *  @return     A+
  	 *  @author Tobias Geimer
  	 */
  	public SimpleMatrix regularizedInverse(boolean omit, double alpha) {
  		// Regularization parameter < 0.0 is invalid. Call non-regularized pseudo-inverse instead.
  		if(alpha <= 0.0) return inverse(omit);
  		
  		double[][] inverse = new double[n][m];
  		if(rank()> 0) {
  			double[] reciprocalS = new double[s.length];
  			if (omit) {
  				double tol = Math.max(m,n)*s[0]*Math.pow(2.0,-52.0);
  				for (int i = s.length-1;i>=0;i--){
  					double filter = (s[i]) / (s[i]*s[i] + alpha);
  					reciprocalS[i] = Math.abs(s[i])<tol?0.0:filter;
  				}
  			} else {
  				for (int i=s.length-1;i>=0;i--) {
  					double filter = (s[i]) / (s[i]*s[i] + alpha);
  					reciprocalS[i] = s[i]==0.0?0.0:filter;
  				}
  			}
  			int min = Math.min(n, ncu);
  			for (int i = n-1; i >= 0; i--)
  				for (int j = m-1; j >= 0; j--)
  					for (int k = min-1; k >= 0; k--)
  						inverse[i][j] += V[i][k] * reciprocalS[k] * U[j][k];
  		} 
  		return new SimpleMatrix(new Matrix(inverse));
  	}
  
  	/** Two norm
     @return     max(S)
  	 */
  
  	public double norm2 () {
  		return s[0];
  	}
  
  	/** Two norm condition number
     @return     max(S)/min(S)
  	 */
  
  	public double cond () {
  		return s[0]/s[Math.min(m,n)-1];
  	}
  
  	/** Effective numerical matrix rank
     @return     Number of nonnegligible singular values.
  	 */
  
  	public int rank () {
  		double tol = Math.max(m,n)*s[0]*Math.pow(2.0,-52.0);
  		int r = 0;
  		for (int i = 0; i < s.length; i++) {
  			if (s[i] > tol) {
  				r++;
  			}
  		}
  		return r;
  	}
  }
  
  /*
   * Copyright (C) 2010-2014 Jang-Hwan Choi
   * CONRAD is developed as an Open Source project under the GNU General Public License (GPL).
  */