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    * 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
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    *
    *      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.MaxCountExceededException;
  import org.apache.commons.math4.legacy.exception.util.LocalizedFormats;
  import org.apache.commons.math4.core.jdkmath.JdkMath;
  import org.apache.commons.numbers.core.Precision;
  
  /**
   * Class transforming a general real matrix to Schur form.
   * <p>A m &times; m matrix A can be written as the product of three matrices: A = P
   * &times; T &times; P<sup>T</sup> with P an orthogonal matrix and T an quasi-triangular
   * matrix. Both P and T are m &times; m matrices.</p>
   * <p>Transformation to Schur form is often not a goal by itself, but it is an
   * intermediate step in more general decomposition algorithms like
   * {@link EigenDecomposition eigen decomposition}. This class is therefore
   * intended for internal use by the library and is not public. As a consequence
   * of this explicitly limited scope, many methods directly returns references to
   * internal arrays, not copies.</p>
   * <p>This class is based on the method hqr2 in class EigenvalueDecomposition
   * from the <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> library.</p>
   *
   * @see <a href="http://mathworld.wolfram.com/SchurDecomposition.html">Schur Decomposition - MathWorld</a>
   * @see <a href="http://en.wikipedia.org/wiki/Schur_decomposition">Schur Decomposition - Wikipedia</a>
   * @see <a href="http://en.wikipedia.org/wiki/Householder_transformation">Householder Transformations</a>
   * @since 3.1
   */
  class SchurTransformer {
      /** Maximum allowed iterations for convergence of the transformation. */
      private static final int MAX_ITERATIONS = 100;
  
      /** P matrix. */
      private final double[][] matrixP;
      /** T matrix. */
      private final double[][] matrixT;
      /** Cached value of P. */
      private RealMatrix cachedP;
      /** Cached value of T. */
      private RealMatrix cachedT;
      /** Cached value of PT. */
      private RealMatrix cachedPt;
  
      /** Epsilon criteria taken from JAMA code (originally was 2^-52). */
      private final double epsilon = Precision.EPSILON;
  
      /**
       * Build the transformation to Schur form of a general real matrix.
       *
       * @param matrix matrix to transform
       * @throws NonSquareMatrixException if the matrix is not square
       */
      SchurTransformer(final RealMatrix matrix) {
          if (!matrix.isSquare()) {
              throw new NonSquareMatrixException(matrix.getRowDimension(),
                                                 matrix.getColumnDimension());
          }
  
          HessenbergTransformer transformer = new HessenbergTransformer(matrix);
          matrixT = transformer.getH().getData();
          matrixP = transformer.getP().getData();
          cachedT = null;
          cachedP = null;
          cachedPt = null;
  
          // transform matrix
          transform();
      }
  
      /**
       * Returns the matrix P of the transform.
       * <p>P is an orthogonal matrix, i.e. its inverse is also its transpose.</p>
       *
       * @return the P matrix
       */
      public RealMatrix getP() {
          if (cachedP == null) {
              cachedP = MatrixUtils.createRealMatrix(matrixP);
          }
          return cachedP;
      }
  
      /**
       * Returns the transpose of the matrix P of the transform.
      * <p>P is an orthogonal matrix, i.e. its inverse is also its transpose.</p>
      *
      * @return the transpose of the P matrix
      */
     public RealMatrix getPT() {
         if (cachedPt == null) {
             cachedPt = getP().transpose();
         }
 
         // return the cached matrix
         return cachedPt;
     }
 
     /**
      * Returns the quasi-triangular Schur matrix T of the transform.
      *
      * @return the T matrix
      */
     public RealMatrix getT() {
         if (cachedT == null) {
             cachedT = MatrixUtils.createRealMatrix(matrixT);
         }
 
         // return the cached matrix
         return cachedT;
     }
 
     /**
      * Transform original matrix to Schur form.
      * @throws MaxCountExceededException if the transformation does not converge
      */
     private void transform() {
         final int n = matrixT.length;
 
         // compute matrix norm
         final double norm = getNorm();
 
         // shift information
         final ShiftInfo shift = new ShiftInfo();
 
         // Outer loop over eigenvalue index
         int iteration = 0;
         int iu = n - 1;
         while (iu >= 0) {
 
             // Look for single small sub-diagonal element
             final int il = findSmallSubDiagonalElement(iu, norm);
 
             // Check for convergence
             if (il == iu) {
                 // One root found
                 matrixT[iu][iu] += shift.exShift;
                 iu--;
                 iteration = 0;
             } else if (il == iu - 1) {
                 // Two roots found
                 double p = (matrixT[iu - 1][iu - 1] - matrixT[iu][iu]) / 2.0;
                 double q = p * p + matrixT[iu][iu - 1] * matrixT[iu - 1][iu];
                 matrixT[iu][iu] += shift.exShift;
                 matrixT[iu - 1][iu - 1] += shift.exShift;
 
                 if (q >= 0) {
                     double z = JdkMath.sqrt(JdkMath.abs(q));
                     if (p >= 0) {
                         z = p + z;
                     } else {
                         z = p - z;
                     }
                     final double x = matrixT[iu][iu - 1];
                     final double s = JdkMath.abs(x) + JdkMath.abs(z);
                     p = x / s;
                     q = z / s;
                     final double r = JdkMath.sqrt(p * p + q * q);
                     p /= r;
                     q /= r;
 
                     // Row modification
                     for (int j = iu - 1; j < n; j++) {
                         z = matrixT[iu - 1][j];
                         matrixT[iu - 1][j] = q * z + p * matrixT[iu][j];
                         matrixT[iu][j] = q * matrixT[iu][j] - p * z;
                     }
 
                     // Column modification
                     for (int i = 0; i <= iu; i++) {
                         z = matrixT[i][iu - 1];
                         matrixT[i][iu - 1] = q * z + p * matrixT[i][iu];
                         matrixT[i][iu] = q * matrixT[i][iu] - p * z;
                     }
 
                     // Accumulate transformations
                     for (int i = 0; i <= n - 1; i++) {
                         z = matrixP[i][iu - 1];
                         matrixP[i][iu - 1] = q * z + p * matrixP[i][iu];
                         matrixP[i][iu] = q * matrixP[i][iu] - p * z;
                     }
                 }
                 iu -= 2;
                 iteration = 0;
             } else {
                 // No convergence yet
                 computeShift(il, iu, iteration, shift);
 
                 // stop transformation after too many iterations
                 if (++iteration > MAX_ITERATIONS) {
                     throw new MaxCountExceededException(LocalizedFormats.CONVERGENCE_FAILED,
                                                         MAX_ITERATIONS);
                 }
 
                 // the initial houseHolder vector for the QR step
                 final double[] hVec = new double[3];
 
                 final int im = initQRStep(il, iu, shift, hVec);
                 performDoubleQRStep(il, im, iu, shift, hVec);
             }
         }
     }
 
     /**
      * Computes the L1 norm of the (quasi-)triangular matrix T.
      *
      * @return the L1 norm of matrix T
      */
     private double getNorm() {
         double norm = 0.0;
         for (int i = 0; i < matrixT.length; i++) {
             // as matrix T is (quasi-)triangular, also take the sub-diagonal element into account
             for (int j = JdkMath.max(i - 1, 0); j < matrixT.length; j++) {
                 norm += JdkMath.abs(matrixT[i][j]);
             }
         }
         return norm;
     }
 
     /**
      * Find the first small sub-diagonal element and returns its index.
      *
      * @param startIdx the starting index for the search
      * @param norm the L1 norm of the matrix
      * @return the index of the first small sub-diagonal element
      */
     private int findSmallSubDiagonalElement(final int startIdx, final double norm) {
         int l = startIdx;
         while (l > 0) {
             double s = JdkMath.abs(matrixT[l - 1][l - 1]) + JdkMath.abs(matrixT[l][l]);
             if (s == 0.0) {
                 s = norm;
             }
             if (JdkMath.abs(matrixT[l][l - 1]) < epsilon * s) {
                 break;
             }
             l--;
         }
         return l;
     }
 
     /**
      * Compute the shift for the current iteration.
      *
      * @param l the index of the small sub-diagonal element
      * @param idx the current eigenvalue index
      * @param iteration the current iteration
      * @param shift holder for shift information
      */
     private void computeShift(final int l, final int idx, final int iteration, final ShiftInfo shift) {
         // Form shift
         shift.x = matrixT[idx][idx];
         shift.y = shift.w = 0.0;
         if (l < idx) {
             shift.y = matrixT[idx - 1][idx - 1];
             shift.w = matrixT[idx][idx - 1] * matrixT[idx - 1][idx];
         }
 
         // Wilkinson's original ad hoc shift
         if (iteration == 10) {
             shift.exShift += shift.x;
             for (int i = 0; i <= idx; i++) {
                 matrixT[i][i] -= shift.x;
             }
             final double s = JdkMath.abs(matrixT[idx][idx - 1]) + JdkMath.abs(matrixT[idx - 1][idx - 2]);
             shift.x = 0.75 * s;
             shift.y = 0.75 * s;
             shift.w = -0.4375 * s * s;
         }
 
         // MATLAB's new ad hoc shift
         if (iteration == 30) {
             double s = (shift.y - shift.x) / 2.0;
             s = s * s + shift.w;
             if (s > 0.0) {
                 s = JdkMath.sqrt(s);
                 if (shift.y < shift.x) {
                     s = -s;
                 }
                 s = shift.x - shift.w / ((shift.y - shift.x) / 2.0 + s);
                 for (int i = 0; i <= idx; i++) {
                     matrixT[i][i] -= s;
                 }
                 shift.exShift += s;
                 shift.x = shift.y = shift.w = 0.964;
             }
         }
     }
 
     /**
      * Initialize the householder vectors for the QR step.
      *
      * @param il the index of the small sub-diagonal element
      * @param iu the current eigenvalue index
      * @param shift shift information holder
      * @param hVec the initial houseHolder vector
      * @return the start index for the QR step
      */
     private int initQRStep(int il, final int iu, final ShiftInfo shift, double[] hVec) {
         // Look for two consecutive small sub-diagonal elements
         int im = iu - 2;
         while (im >= il) {
             final double z = matrixT[im][im];
             final double r = shift.x - z;
             double s = shift.y - z;
             hVec[0] = (r * s - shift.w) / matrixT[im + 1][im] + matrixT[im][im + 1];
             hVec[1] = matrixT[im + 1][im + 1] - z - r - s;
             hVec[2] = matrixT[im + 2][im + 1];
 
             if (im == il) {
                 break;
             }
 
             final double lhs = JdkMath.abs(matrixT[im][im - 1]) * (JdkMath.abs(hVec[1]) + JdkMath.abs(hVec[2]));
             final double rhs = JdkMath.abs(hVec[0]) * (JdkMath.abs(matrixT[im - 1][im - 1]) +
                                                         JdkMath.abs(z) +
                                                         JdkMath.abs(matrixT[im + 1][im + 1]));
 
             if (lhs < epsilon * rhs) {
                 break;
             }
             im--;
         }
 
         return im;
     }
 
     /**
      * Perform a double QR step involving rows l:idx and columns m:n.
      *
      * @param il the index of the small sub-diagonal element
      * @param im the start index for the QR step
      * @param iu the current eigenvalue index
      * @param shift shift information holder
      * @param hVec the initial houseHolder vector
      */
     private void performDoubleQRStep(final int il, final int im, final int iu,
                                      final ShiftInfo shift, final double[] hVec) {
 
         final int n = matrixT.length;
         double p = hVec[0];
         double q = hVec[1];
         double r = hVec[2];
 
         for (int k = im; k <= iu - 1; k++) {
             boolean notlast = k != (iu - 1);
             if (k != im) {
                 p = matrixT[k][k - 1];
                 q = matrixT[k + 1][k - 1];
                 r = notlast ? matrixT[k + 2][k - 1] : 0.0;
                 shift.x = JdkMath.abs(p) + JdkMath.abs(q) + JdkMath.abs(r);
                 if (Precision.equals(shift.x, 0.0, epsilon)) {
                     continue;
                 }
                 p /= shift.x;
                 q /= shift.x;
                 r /= shift.x;
             }
             double s = JdkMath.sqrt(p * p + q * q + r * r);
             if (p < 0.0) {
                 s = -s;
             }
             if (s != 0.0) {
                 if (k != im) {
                     matrixT[k][k - 1] = -s * shift.x;
                 } else if (il != im) {
                     matrixT[k][k - 1] = -matrixT[k][k - 1];
                 }
                 p += s;
                 shift.x = p / s;
                 shift.y = q / s;
                 double z = r / s;
                 q /= p;
                 r /= p;
 
                 // Row modification
                 for (int j = k; j < n; j++) {
                     p = matrixT[k][j] + q * matrixT[k + 1][j];
                     if (notlast) {
                         p += r * matrixT[k + 2][j];
                         matrixT[k + 2][j] -= p * z;
                     }
                     matrixT[k][j] -= p * shift.x;
                     matrixT[k + 1][j] -= p * shift.y;
                 }
 
                 // Column modification
                 for (int i = 0; i <= JdkMath.min(iu, k + 3); i++) {
                     p = shift.x * matrixT[i][k] + shift.y * matrixT[i][k + 1];
                     if (notlast) {
                         p += z * matrixT[i][k + 2];
                         matrixT[i][k + 2] -= p * r;
                     }
                     matrixT[i][k] -= p;
                     matrixT[i][k + 1] -= p * q;
                 }
 
                 // Accumulate transformations
                 final int high = matrixT.length - 1;
                 for (int i = 0; i <= high; i++) {
                     p = shift.x * matrixP[i][k] + shift.y * matrixP[i][k + 1];
                     if (notlast) {
                         p += z * matrixP[i][k + 2];
                         matrixP[i][k + 2] -= p * r;
                     }
                     matrixP[i][k] -= p;
                     matrixP[i][k + 1] -= p * q;
                 }
             }  // (s != 0)
         }  // k loop
 
         // clean up pollution due to round-off errors
         for (int i = im + 2; i <= iu; i++) {
             matrixT[i][i-2] = 0.0;
             if (i > im + 2) {
                 matrixT[i][i-3] = 0.0;
             }
         }
     }
 
     /**
      * Internal data structure holding the current shift information.
      * Contains variable names as present in the original JAMA code.
      */
     private static final class ShiftInfo {
         // CHECKSTYLE: stop all
 
         /** x shift info. */
         double x;
         /** y shift info. */
         double y;
         /** w shift info. */
         double w;
         /** Indicates an exceptional shift. */
         double exShift;
 
         // CHECKSTYLE: resume all
     }
 }