/*
 * RobustSingularValueDecomposition.java
 *
 * Copyright (c) 2002-2015 Alexei Drummond, Andrew Rambaut and Marc Suchard
 *
 * This file is part of BEAST.
 * See the NOTICE file distributed with this work for additional
 * information regarding copyright ownership and licensing.
 *
 * BEAST is free software; you can redistribute it and/or modify
 * it under the terms of the GNU Lesser General Public License as
 * published by the Free Software Foundation; either version 2
 * of the License, or (at your option) any later version.
 *
 *  BEAST is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU Lesser General Public License for more details.
 *
 * You should have received a copy of the GNU Lesser General Public
 * License along with BEAST; if not, write to the
 * Free Software Foundation, Inc., 51 Franklin St, Fifth Floor,
 * Boston, MA  02110-1301  USA
 */

package dr.math.matrixAlgebra;

import cern.colt.matrix.DoubleFactory2D;
import cern.colt.matrix.DoubleMatrix2D;
import cern.colt.matrix.linalg.Property;
import dr.math.MathUtils;


public class RobustSingularValueDecomposition implements java.io.Serializable {

            static final long serialVersionUID = 1020;
            /** 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;

//    private int maxIterations;

    static private int maxIterationsDefault = 100000;

    private static final String ERROR_STRING = "SVD is not converged.";

    /**
    Constructs and returns a new singular value decomposition object;
    The decomposed matrices can be retrieved via instance methods of the returned decomposition object.
    @param A    A rectangular matrix.
    @return     A decomposition object to access <tt>U</tt>, <tt>S</tt> and <tt>V</tt>.
    @throws IllegalArgumentException if <tt>A.rows() < A.columns()</tt>.
    */
    public RobustSingularValueDecomposition(DoubleMatrix2D Arg) throws ArithmeticException {
        this(Arg,maxIterationsDefault);
    }

    public RobustSingularValueDecomposition(DoubleMatrix2D Arg, int maxIterations) throws ArithmeticException {
            Property.DEFAULT.checkRectangular(Arg);

//            this.maxIterations = maxIterations;

              // Derived from LINPACK code.
              // Initialize.
              double[][] A = Arg.toArray();
              m = Arg.rows();
              n = Arg.columns();
              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] = MathUtils.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] = MathUtils.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);
              while (p > 0) {
                         int k,kase;

                         // Here is where a test for too many iterations would go.
                  if (iter > maxIterations)
                          throw new ArithmeticException(ERROR_STRING);

                         // 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.) +
                                                                  (ks != k+1 ? Math.abs(e[ks-1]) : 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 = MathUtils.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 = MathUtils.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 = MathUtils.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 = MathUtils.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;
                         }
              }
       }
    /**
    Returns the two norm condition number, which is <tt>max(S) / min(S)</tt>.
    */
    public double cond() {
            return s[0]/s[Math.min(m,n)-1];
    }
    /**
    Returns the diagonal matrix of singular values.
    @return     S
    */
    public DoubleMatrix2D getS() {
            double[][] S = new double[n][n];
            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 DoubleFactory2D.dense.make(S);
    }
    /**
    Returns the diagonal of <tt>S</tt>, which is a one-dimensional array of singular values
    @return     diagonal of <tt>S</tt>.
    */
    public double[] getSingularValues() {
            return s;
    }
    /**
    Returns the left singular vectors <tt>U</tt>.
    @return     <tt>U</tt>
    */
    public DoubleMatrix2D getU() {
            //return new DoubleMatrix2D(U,m,Math.min(m+1,n));
            return DoubleFactory2D.dense.make(U).viewPart(0,0,m,Math.min(m+1,n));
    }
    /**
    Returns the right singular vectors <tt>V</tt>.
    @return     <tt>V</tt>
    */
    public DoubleMatrix2D getV() {
            return DoubleFactory2D.dense.make(V);
    }
    /**
    Returns the two norm, which is <tt>max(S)</tt>.
    */
    public double norm2() {
            return s[0];
    }
    /**
    Returns the effective numerical matrix rank, which is the 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;
    }
    /**
    Returns a String with (propertyName, propertyValue) pairs.
    Useful for debugging or to quickly get the rough picture.
    For example,
    <pre>
    rank          : 3
    trace         : 0
    </pre>
    */
    public String toString() {
            StringBuffer buf = new StringBuffer();
            String unknown = "Illegal operation or error: ";

            buf.append("---------------------------------------------------------------------\n");
            buf.append("SingularValueDecomposition(A) --> cond(A), rank(A), norm2(A), U, S, V\n");
            buf.append("---------------------------------------------------------------------\n");

            buf.append("cond = ");
            try { buf.append(String.valueOf(this.cond()));}
            catch (IllegalArgumentException exc) { buf.append(unknown+exc.getMessage()); }

            buf.append("\nrank = ");
            try { buf.append(String.valueOf(this.rank()));}
            catch (IllegalArgumentException exc) { buf.append(unknown+exc.getMessage()); }

            buf.append("\nnorm2 = ");
            try { buf.append(String.valueOf(this.norm2()));}
            catch (IllegalArgumentException exc) { buf.append(unknown+exc.getMessage()); }

            buf.append("\n\nU = ");
            try { buf.append(String.valueOf(this.getU()));}
            catch (IllegalArgumentException exc) { buf.append(unknown+exc.getMessage()); }

            buf.append("\n\nS = ");
            try { buf.append(String.valueOf(this.getS()));}
            catch (IllegalArgumentException exc) { buf.append(unknown+exc.getMessage()); }

            buf.append("\n\nV = ");
            try { buf.append(String.valueOf(this.getV()));}
            catch (IllegalArgumentException exc) { buf.append(unknown+exc.getMessage()); }

            return buf.toString();
    }
    }



