/*
 *                    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;

	/** Singular Value Decomposition.
	<P>
	For an m-by-n matrix A with m >= n, the singular value decomposition is
	an m-by-n orthogonal matrix U, an 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.
	*/

public class SingularValueDecomposition implements java.io.Serializable {

	 static final long serialVersionUID = 640239472093534756l;

/* ------------------------
	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.
	*/
	private int m, n;

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

	/** Construct the singular value decomposition. Provides a data structure to access U, S and V.
	@param Arg    Rectangular matrix
	*/

	public SingularValueDecomposition (Matrix Arg) {

		// Derived from LINPACK code.
		// Initialize.
		double[][] A = Arg.getArrayCopy();
		m = Arg.getRowDimension();
		n = Arg.getColumnDimension();

		/* Apparently the failing cases are only a proper subset of (m<n),
	 so let's not throw error.  Correct fix to come later?
		if (m<n) {
	  throw new IllegalArgumentException("Jama SVD only works for m >= n"); }
		*/
		int nu = Math.min(m,n);
		s = new double [Math.min(m+1,n)];
		U = new double [m][nu];
		V = new double [n][n];
		double[] e = new double [n];
		double[] work = new double [m];
		boolean wantu = true;
		boolean 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] = 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 < 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);
		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 = 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;
			}
		}
	}

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

	/** Return the left singular vectors
	@return     U
	*/

	public Matrix getU () {
		return new Matrix(U,m,Math.min(m+1,n));
	}

	/** Return the right singular vectors
	@return     V
	*/

	public Matrix getV () {
		return 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 Matrix getS () {
		Matrix X = new Matrix(n,n);
		double[][] S = X.getArray();
		for (int i = 0; i < n; i++) {
			for (int j = 0; j < n; j++) {
				S[i][j] = 0.0;
			}
			S[i][i] = this.s[i];
		}
		return X;
	}

	/** 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 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;
	}
}
