/*
    * Licensed to the Apache Software Foundation (ASF) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * The ASF licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
    * the License.  You may obtain a copy of the License at
    *
    *      http://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   * See the License for the specific language governing permissions and
   * limitations under the License.
   */
  package org.apache.commons.math4.legacy.linear;
  
  import org.apache.commons.math4.legacy.exception.NumberIsTooLargeException;
  import org.apache.commons.math4.legacy.exception.util.LocalizedFormats;
  import org.apache.commons.math4.core.jdkmath.JdkMath;
  import org.apache.commons.numbers.core.Precision;
  
  /**
   * Calculates the compact Singular Value Decomposition of a matrix.
   * <p>
   * The Singular Value Decomposition of matrix A is a set of three matrices: U,
   * &Sigma; and V such that A = U &times; &Sigma; &times; V<sup>T</sup>. Let A be
   * a m &times; n matrix, then U is a m &times; p orthogonal matrix, &Sigma; is a
   * p &times; p diagonal matrix with positive or null elements, V is a p &times;
   * n orthogonal matrix (hence V<sup>T</sup> is also orthogonal) where
   * p=min(m,n).
   * </p>
   * <p>This class is similar to the class with similar name from the
   * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> library, with the
   * following changes:</p>
   * <ul>
   *   <li>the {@code norm2} method which has been renamed as {@link #getNorm()
   *   getNorm},</li>
   *   <li>the {@code cond} method which has been renamed as {@link
   *   #getConditionNumber() getConditionNumber},</li>
   *   <li>the {@code rank} method which has been renamed as {@link #getRank()
   *   getRank},</li>
   *   <li>a {@link #getUT() getUT} method has been added,</li>
   *   <li>a {@link #getVT() getVT} method has been added,</li>
   *   <li>a {@link #getSolver() getSolver} method has been added,</li>
   *   <li>a {@link #getCovariance(double) getCovariance} method has been added.</li>
   * </ul>
   * @see <a href="http://mathworld.wolfram.com/SingularValueDecomposition.html">MathWorld</a>
   * @see <a href="http://en.wikipedia.org/wiki/Singular_value_decomposition">Wikipedia</a>
   * @since 2.0 (changed to concrete class in 3.0)
   */
  public class SingularValueDecomposition {
      /** Relative threshold for small singular values. */
      private static final double EPS = 0x1.0p-52;
      /** Absolute threshold for small singular values. */
      private static final double TINY = 0x1.0p-966;
      /** Computed singular values. */
      private final double[] singularValues;
      /** max(row dimension, column dimension). */
      private final int m;
      /** min(row dimension, column dimension). */
      private final int n;
      /** Indicator for transposed matrix. */
      private final boolean transposed;
      /** Cached value of U matrix. */
      private final RealMatrix cachedU;
      /** Cached value of transposed U matrix. */
      private RealMatrix cachedUt;
      /** Cached value of S (diagonal) matrix. */
      private RealMatrix cachedS;
      /** Cached value of V matrix. */
      private final RealMatrix cachedV;
      /** Cached value of transposed V matrix. */
      private RealMatrix cachedVt;
      /**
       * Tolerance value for small singular values, calculated once we have
       * populated "singularValues".
       **/
      private final double tol;
  
      /**
       * Calculates the compact Singular Value Decomposition of the given matrix.
       *
       * @param matrix Matrix to decompose.
       */
      public SingularValueDecomposition(final RealMatrix matrix) {
          final double[][] A;
  
           // "m" is always the largest dimension.
          if (matrix.getRowDimension() < matrix.getColumnDimension()) {
              transposed = true;
              A = matrix.transpose().getData();
              m = matrix.getColumnDimension();
              n = matrix.getRowDimension();
          } else {
              transposed = false;
              A = matrix.getData();
              m = matrix.getRowDimension();
             n = matrix.getColumnDimension();
         }
 
         singularValues = new double[n];
         final double[][] U = new double[m][n];
         final double[][] V = new double[n][n];
         final double[] e = new double[n];
         final double[] work = new double[m];
         // Reduce A to bidiagonal form, storing the diagonal elements
         // in s and the super-diagonal elements in e.
         final int nct = JdkMath.min(m - 1, n);
         final int nrt = JdkMath.max(0, n - 2);
         for (int k = 0; k < JdkMath.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.
                 singularValues[k] = 0;
                 for (int i = k; i < m; i++) {
                     singularValues[k] = JdkMath.hypot(singularValues[k], A[i][k]);
                 }
                 if (singularValues[k] != 0) {
                     if (A[k][k] < 0) {
                         singularValues[k] = -singularValues[k];
                     }
                     for (int i = k; i < m; i++) {
                         A[i][k] /= singularValues[k];
                     }
                     A[k][k] += 1;
                 }
                 singularValues[k] = -singularValues[k];
             }
             for (int j = k + 1; j < n; j++) {
                 if (k < nct &&
                     singularValues[k] != 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 (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] = JdkMath.hypot(e[k], e[i]);
                 }
                 if (e[k] != 0) {
                     if (e[k + 1] < 0) {
                         e[k] = -e[k];
                     }
                     for (int i = k + 1; i < n; i++) {
                         e[i] /= e[k];
                     }
                     e[k + 1] += 1;
                 }
                 e[k] = -e[k];
                 if (k + 1 < m &&
                     e[k] != 0) {
                     // Apply the transformation.
                     for (int i = k + 1; i < m; i++) {
                         work[i] = 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++) {
                         final double t = -e[j] / e[k + 1];
                         for (int i = k + 1; i < m; i++) {
                             A[i][j] += t * work[i];
                         }
                     }
                 }
 
                 // 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 = n;
         if (nct < n) {
             singularValues[nct] = A[nct][nct];
         }
         if (m < p) {
             singularValues[p - 1] = 0;
         }
         if (nrt + 1 < p) {
             e[nrt] = A[nrt][p - 1];
         }
         e[p - 1] = 0;
 
         // Generate U.
         for (int j = nct; j < n; j++) {
             for (int i = 0; i < m; i++) {
                 U[i][j] = 0;
             }
             U[j][j] = 1;
         }
         for (int k = nct - 1; k >= 0; k--) {
             if (singularValues[k] != 0) {
                 for (int j = k + 1; j < n; 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 + U[k][k];
                 for (int i = 0; i < k - 1; i++) {
                     U[i][k] = 0;
                 }
             } else {
                 for (int i = 0; i < m; i++) {
                     U[i][k] = 0;
                 }
                 U[k][k] = 1;
             }
         }
 
         // Generate V.
         for (int k = n - 1; k >= 0; k--) {
             if (k < nrt &&
                 e[k] != 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;
             }
             V[k][k] = 1;
         }
 
         // Main iteration loop for the singular values.
         final int pp = p - 1;
         while (p > 0) {
             int k;
             int 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 >= 0; k--) {
                 final double threshold
                     = TINY + EPS * (JdkMath.abs(singularValues[k]) +
                                     JdkMath.abs(singularValues[k + 1]));
 
                 // the following condition is written this way in order
                 // to break out of the loop when NaN occurs, writing it
                 // as "if (JdkMath.abs(e[k]) <= threshold)" would loop
                 // indefinitely in case of NaNs because comparison on NaNs
                 // always return false, regardless of what is checked
                 // see issue MATH-947
                 if (!(JdkMath.abs(e[k]) > threshold)) {
                     e[k] = 0;
                     break;
                 }
             }
 
             if (k == p - 2) {
                 kase = 4;
             } else {
                 int ks;
                 for (ks = p - 1; ks >= k; ks--) {
                     if (ks == k) {
                         break;
                     }
                     final double t = (ks != p ? JdkMath.abs(e[ks]) : 0) +
                         (ks != k + 1 ? JdkMath.abs(e[ks - 1]) : 0);
                     if (JdkMath.abs(singularValues[ks]) <= TINY + EPS * t) {
                         singularValues[ks] = 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.
             double f;
             switch (kase) {
                 // Deflate negligible s(p).
                 case 1:
                     f = e[p - 2];
                     e[p - 2] = 0;
                     for (int j = p - 2; j >= k; j--) {
                         double t = JdkMath.hypot(singularValues[j], f);
                         final double cs = singularValues[j] / t;
                         final double sn = f / t;
                         singularValues[j] = t;
                         if (j != k) {
                             f = -sn * e[j - 1];
                             e[j - 1] = cs * e[j - 1];
                         }
 
                         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:
                     f = e[k - 1];
                     e[k - 1] = 0;
                     for (int j = k; j < p; j++) {
                         double t = JdkMath.hypot(singularValues[j], f);
                         final double cs = singularValues[j] / t;
                         final double sn = f / t;
                         singularValues[j] = t;
                         f = -sn * e[j];
                         e[j] = cs * e[j];
 
                         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.
                     final double maxPm1Pm2 = JdkMath.max(JdkMath.abs(singularValues[p - 1]),
                                                           JdkMath.abs(singularValues[p - 2]));
                     final double scale = JdkMath.max(JdkMath.max(JdkMath.max(maxPm1Pm2,
                                                                                 JdkMath.abs(e[p - 2])),
                                                                    JdkMath.abs(singularValues[k])),
                                                       JdkMath.abs(e[k]));
                     final double sp = singularValues[p - 1] / scale;
                     final double spm1 = singularValues[p - 2] / scale;
                     final double epm1 = e[p - 2] / scale;
                     final double sk = singularValues[k] / scale;
                     final double ek = e[k] / scale;
                     final double b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2.0;
                     final double c = (sp * epm1) * (sp * epm1);
                     double shift = 0;
                     if (b != 0 ||
                         c != 0) {
                         shift = JdkMath.sqrt(b * b + c);
                         if (b < 0) {
                             shift = -shift;
                         }
                         shift = c / (b + shift);
                     }
                     f = (sk + sp) * (sk - sp) + shift;
                     double g = sk * ek;
                     // Chase zeros.
                     for (int j = k; j < p - 1; j++) {
                         double t = JdkMath.hypot(f, g);
                         double cs = f / t;
                         double sn = g / t;
                         if (j != k) {
                             e[j - 1] = t;
                         }
                         f = cs * singularValues[j] + sn * e[j];
                         e[j] = cs * e[j] - sn * singularValues[j];
                         g = sn * singularValues[j + 1];
                         singularValues[j + 1] = cs * singularValues[j + 1];
 
                         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 = JdkMath.hypot(f, g);
                         cs = f / t;
                         sn = g / t;
                         singularValues[j] = t;
                         f = cs * e[j] + sn * singularValues[j + 1];
                         singularValues[j + 1] = -sn * e[j] + cs * singularValues[j + 1];
                         g = sn * e[j + 1];
                         e[j + 1] = cs * e[j + 1];
                         if (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;
                     break;
                 // Convergence.
                 default:
                     // Make the singular values positive.
                     if (singularValues[k] <= 0) {
                         singularValues[k] = singularValues[k] < 0 ? -singularValues[k] : 0;
 
                         for (int i = 0; i <= pp; i++) {
                             V[i][k] = -V[i][k];
                         }
                     }
                     // Order the singular values.
                     while (k < pp) {
                         if (singularValues[k] >= singularValues[k + 1]) {
                             break;
                         }
                         double t = singularValues[k];
                         singularValues[k] = singularValues[k + 1];
                         singularValues[k + 1] = t;
                         if (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 (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++;
                     }
                     p--;
                     break;
             }
         }
 
         // Set the small value tolerance used to calculate rank and pseudo-inverse
         tol = JdkMath.max(m * singularValues[0] * EPS,
                            JdkMath.sqrt(Precision.SAFE_MIN));
 
         if (!transposed) {
             cachedU = MatrixUtils.createRealMatrix(U);
             cachedV = MatrixUtils.createRealMatrix(V);
         } else {
             cachedU = MatrixUtils.createRealMatrix(V);
             cachedV = MatrixUtils.createRealMatrix(U);
         }
     }
 
     /**
      * Returns the matrix U of the decomposition.
      * <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
      * @return the U matrix
      * @see #getUT()
      */
     public RealMatrix getU() {
         // return the cached matrix
         return cachedU;
     }
 
     /**
      * Returns the transpose of the matrix U of the decomposition.
      * <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
      * @return the U matrix (or null if decomposed matrix is singular)
      * @see #getU()
      */
     public RealMatrix getUT() {
         if (cachedUt == null) {
             cachedUt = getU().transpose();
         }
         // return the cached matrix
         return cachedUt;
     }
 
     /**
      * Returns the diagonal matrix &Sigma; of the decomposition.
      * <p>&Sigma; is a diagonal matrix. The singular values are provided in
      * non-increasing order, for compatibility with Jama.</p>
      * @return the &Sigma; matrix
      */
     public RealMatrix getS() {
         if (cachedS == null) {
             // cache the matrix for subsequent calls
             cachedS = MatrixUtils.createRealDiagonalMatrix(singularValues);
         }
         return cachedS;
     }
 
     /**
      * Returns the diagonal elements of the matrix &Sigma; of the decomposition.
      * <p>The singular values are provided in non-increasing order, for
      * compatibility with Jama.</p>
      * @return the diagonal elements of the &Sigma; matrix
      */
     public double[] getSingularValues() {
         return singularValues.clone();
     }
 
     /**
      * Returns the matrix V of the decomposition.
      * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
      * @return the V matrix (or null if decomposed matrix is singular)
      * @see #getVT()
      */
     public RealMatrix getV() {
         // return the cached matrix
         return cachedV;
     }
 
     /**
      * Returns the transpose of the matrix V of the decomposition.
      * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
      * @return the V matrix (or null if decomposed matrix is singular)
      * @see #getV()
      */
     public RealMatrix getVT() {
         if (cachedVt == null) {
             cachedVt = getV().transpose();
         }
         // return the cached matrix
         return cachedVt;
     }
 
     /**
      * Returns the n &times; n covariance matrix.
      * <p>The covariance matrix is V &times; J &times; V<sup>T</sup>
      * where J is the diagonal matrix of the inverse of the squares of
      * the singular values.</p>
      * @param minSingularValue value below which singular values are ignored
      * (a 0 or negative value implies all singular value will be used)
      * @return covariance matrix
      * @exception IllegalArgumentException if minSingularValue is larger than
      * the largest singular value, meaning all singular values are ignored
      */
     public RealMatrix getCovariance(final double minSingularValue) {
         // get the number of singular values to consider
         final int p = singularValues.length;
         int dimension = 0;
         while (dimension < p &&
                singularValues[dimension] >= minSingularValue) {
             ++dimension;
         }
 
         if (dimension == 0) {
             throw new NumberIsTooLargeException(LocalizedFormats.TOO_LARGE_CUTOFF_SINGULAR_VALUE,
                                                 minSingularValue, singularValues[0], true);
         }
 
         final double[][] data = new double[dimension][p];
         getVT().walkInOptimizedOrder(new DefaultRealMatrixPreservingVisitor() {
             /** {@inheritDoc} */
             @Override
             public void visit(final int row, final int column,
                     final double value) {
                 data[row][column] = value / singularValues[row];
             }
         }, 0, dimension - 1, 0, p - 1);
 
         RealMatrix jv = new Array2DRowRealMatrix(data, false);
         return jv.transpose().multiply(jv);
     }
 
     /**
      * Returns the L<sub>2</sub> norm of the matrix.
      * <p>The L<sub>2</sub> norm is max(|A &times; u|<sub>2</sub> /
      * |u|<sub>2</sub>), where |.|<sub>2</sub> denotes the vectorial 2-norm
      * (i.e. the traditional euclidean norm).</p>
      * @return norm
      */
     public double getNorm() {
         return singularValues[0];
     }
 
     /**
      * Return the condition number of the matrix.
      * @return condition number of the matrix
      */
     public double getConditionNumber() {
         return singularValues[0] / singularValues[n - 1];
     }
 
     /**
      * Computes the inverse of the condition number.
      * In cases of rank deficiency, the {@link #getConditionNumber() condition
      * number} will become undefined.
      *
      * @return the inverse of the condition number.
      */
     public double getInverseConditionNumber() {
         return singularValues[n - 1] / singularValues[0];
     }
 
     /**
      * Return the effective numerical matrix rank.
      * <p>The effective numerical rank is the number of non-negligible
      * singular values. The threshold used to identify non-negligible
      * terms is max(m,n) &times; ulp(s<sub>1</sub>) where ulp(s<sub>1</sub>)
      * is the least significant bit of the largest singular value.</p>
      * @return effective numerical matrix rank
      */
     public int getRank() {
         int r = 0;
         for (int i = 0; i < singularValues.length; i++) {
             if (singularValues[i] > tol) {
                 r++;
             }
         }
         return r;
     }
 
     /**
      * Get a solver for finding the A &times; X = B solution in least square sense.
      * @return a solver
      */
     public DecompositionSolver getSolver() {
         return new Solver(singularValues, getUT(), getV(), getRank() == m, tol);
     }
 
     /** Specialized solver. */
     private static final class Solver implements DecompositionSolver {
         /** Pseudo-inverse of the initial matrix. */
         private final RealMatrix pseudoInverse;
         /** Singularity indicator. */
         private final boolean nonSingular;
 
         /**
          * Build a solver from decomposed matrix.
          *
          * @param singularValues Singular values.
          * @param uT U<sup>T</sup> matrix of the decomposition.
          * @param v V matrix of the decomposition.
          * @param nonSingular Singularity indicator.
          * @param tol tolerance for singular values
          */
         private Solver(final double[] singularValues, final RealMatrix uT,
                        final RealMatrix v, final boolean nonSingular, final double tol) {
             final double[][] suT = uT.getData();
             for (int i = 0; i < singularValues.length; ++i) {
                 final double a;
                 if (singularValues[i] > tol) {
                     a = 1 / singularValues[i];
                 } else {
                     a = 0;
                 }
                 final double[] suTi = suT[i];
                 for (int j = 0; j < suTi.length; ++j) {
                     suTi[j] *= a;
                 }
             }
             pseudoInverse = v.multiply(new Array2DRowRealMatrix(suT, false));
             this.nonSingular = nonSingular;
         }
 
         /**
          * Solve the linear equation A &times; X = B in least square sense.
          * <p>
          * The m&times;n matrix A may not be square, the solution X is such that
          * ||A &times; X - B|| is minimal.
          * </p>
          * @param b Right-hand side of the equation A &times; X = B
          * @return a vector X that minimizes the two norm of A &times; X - B
          * @throws org.apache.commons.math4.legacy.exception.DimensionMismatchException
          * if the matrices dimensions do not match.
          */
         @Override
         public RealVector solve(final RealVector b) {
             return pseudoInverse.operate(b);
         }
 
         /**
          * Solve the linear equation A &times; X = B in least square sense.
          * <p>
          * The m&times;n matrix A may not be square, the solution X is such that
          * ||A &times; X - B|| is minimal.
          * </p>
          *
          * @param b Right-hand side of the equation A &times; X = B
          * @return a matrix X that minimizes the two norm of A &times; X - B
          * @throws org.apache.commons.math4.legacy.exception.DimensionMismatchException
          * if the matrices dimensions do not match.
          */
         @Override
         public RealMatrix solve(final RealMatrix b) {
             return pseudoInverse.multiply(b);
         }
 
         /**
          * Check if the decomposed matrix is non-singular.
          *
          * @return {@code true} if the decomposed matrix is non-singular.
          */
         @Override
         public boolean isNonSingular() {
             return nonSingular;
         }
 
         /**
          * Get the pseudo-inverse of the decomposed matrix.
          *
          * @return the inverse matrix.
          */
         @Override
         public RealMatrix getInverse() {
             return pseudoInverse;
         }
     }
 }