/*
    * 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
    *
    *      https://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.
   */
  
  /*
   * This is not the original file distributed by the Apache Software Foundation
   * It has been modified by the Hipparchus project
   */
  
  package org.hipparchus.linear;
  
  import java.util.Arrays;
  
  import org.hipparchus.exception.LocalizedCoreFormats;
  import org.hipparchus.exception.MathIllegalArgumentException;
  import org.hipparchus.util.FastMath;
  
  
  /**
   * Class transforming a symmetrical matrix to tridiagonal shape.
   * <p>A symmetrical m &times; m matrix A can be written as the product of three matrices:
   * A = Q &times; T &times; Q<sup>T</sup> with Q an orthogonal matrix and T a symmetrical
   * tridiagonal matrix. Both Q and T are m &times; m matrices.</p>
   * <p>This implementation only uses the upper part of the matrix, the part below the
   * diagonal is not accessed at all.</p>
   * <p>Transformation to tridiagonal shape is often not a goal by itself, but it is
   * an intermediate step in more general decomposition algorithms like {@link
   * EigenDecompositionSymmetric 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>
   */
  class TriDiagonalTransformer {
      /** Householder vectors. */
      private final double[][] householderVectors;
      /** Main diagonal. */
      private final double[] main;
      /** Secondary diagonal. */
      private final double[] secondary;
      /** Cached value of Q. */
      private RealMatrix cachedQ;
      /** Cached value of Qt. */
      private RealMatrix cachedQt;
      /** Cached value of T. */
      private RealMatrix cachedT;
  
      /**
       * Build the transformation to tridiagonal shape of a symmetrical matrix.
       * <p>The specified matrix is assumed to be symmetrical without any check.
       * Only the upper triangular part of the matrix is used.</p>
       *
       * @param matrix Symmetrical matrix to transform.
       * @throws MathIllegalArgumentException if the matrix is not square.
       */
      TriDiagonalTransformer(RealMatrix matrix) {
          if (!matrix.isSquare()) {
              throw new MathIllegalArgumentException(LocalizedCoreFormats.NON_SQUARE_MATRIX,
                                                     matrix.getRowDimension(), matrix.getColumnDimension());
          }
  
          final int m = matrix.getRowDimension();
          householderVectors = matrix.getData();
          main      = new double[m];
          secondary = new double[m - 1];
          cachedQ   = null;
          cachedQt  = null;
          cachedT   = null;
  
          // transform matrix
          transform();
      }
  
      /**
       * Returns the matrix Q of the transform.
       * <p>Q is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
       * @return the Q matrix
       */
      public RealMatrix getQ() {
          if (cachedQ == null) {
              cachedQ = getQT().transpose();
          }
          return cachedQ;
      }
  
      /**
       * Returns the transpose of the matrix Q of the transform.
       * <p>Q is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
      * @return the Q matrix
      */
     public RealMatrix getQT() {
         if (cachedQt == null) {
             final int m = householderVectors.length;
             double[][] qta = new double[m][m];
 
             // build up first part of the matrix by applying Householder transforms
             for (int k = m - 1; k >= 1; --k) {
                 final double[] hK = householderVectors[k - 1];
                 qta[k][k] = 1;
                 if (hK[k] != 0.0) {
                     final double inv = 1.0 / (secondary[k - 1] * hK[k]);
                     double beta = 1.0 / secondary[k - 1];
                     qta[k][k] = 1 + beta * hK[k];
                     for (int i = k + 1; i < m; ++i) {
                         qta[k][i] = beta * hK[i];
                     }
                     for (int j = k + 1; j < m; ++j) {
                         beta = 0;
                         for (int i = k + 1; i < m; ++i) {
                             beta += qta[j][i] * hK[i];
                         }
                         beta *= inv;
                         qta[j][k] = beta * hK[k];
                         for (int i = k + 1; i < m; ++i) {
                             qta[j][i] += beta * hK[i];
                         }
                     }
                 }
             }
             qta[0][0] = 1;
             cachedQt = MatrixUtils.createRealMatrix(qta);
         }
 
         // return the cached matrix
         return cachedQt;
     }
 
     /**
      * Returns the tridiagonal matrix T of the transform.
      * @return the T matrix
      */
     public RealMatrix getT() {
         if (cachedT == null) {
             final int m = main.length;
             double[][] ta = new double[m][m];
             for (int i = 0; i < m; ++i) {
                 ta[i][i] = main[i];
                 if (i > 0) {
                     ta[i][i - 1] = secondary[i - 1];
                 }
                 if (i < main.length - 1) {
                     ta[i][i + 1] = secondary[i];
                 }
             }
             cachedT = MatrixUtils.createRealMatrix(ta);
         }
 
         // return the cached matrix
         return cachedT;
     }
 
     /**
      * Get the Householder vectors of the transform.
      * <p>Note that since this class is only intended for internal use,
      * it returns directly a reference to its internal arrays, not a copy.</p>
      * @return the main diagonal elements of the B matrix
      */
     double[][] getHouseholderVectorsRef() {
         return householderVectors; // NOPMD - returning an internal array is intentional and documented here
     }
 
     /**
      * Get the main diagonal elements of the matrix T of the transform.
      * <p>Note that since this class is only intended for internal use,
      * it returns directly a reference to its internal arrays, not a copy.</p>
      * @return the main diagonal elements of the T matrix
      */
     double[] getMainDiagonalRef() {
         return main; // NOPMD - returning an internal array is intentional and documented here
     }
 
     /**
      * Get the secondary diagonal elements of the matrix T of the transform.
      * <p>Note that since this class is only intended for internal use,
      * it returns directly a reference to its internal arrays, not a copy.</p>
      * @return the secondary diagonal elements of the T matrix
      */
     double[] getSecondaryDiagonalRef() {
         return secondary; // NOPMD - returning an internal array is intentional and documented here
     }
 
     /**
      * Transform original matrix to tridiagonal form.
      * <p>Transformation is done using Householder transforms.</p>
      */
     private void transform() {
         final int m = householderVectors.length;
         final double[] z = new double[m];
         for (int k = 0; k < m - 1; k++) {
 
             //zero-out a row and a column simultaneously
             final double[] hK = householderVectors[k];
             main[k] = hK[k];
             double xNormSqr = 0;
             for (int j = k + 1; j < m; ++j) {
                 final double c = hK[j];
                 xNormSqr += c * c;
             }
             final double a = (hK[k + 1] > 0) ? -FastMath.sqrt(xNormSqr) : FastMath.sqrt(xNormSqr);
             secondary[k] = a;
             if (a != 0.0) {
                 // apply Householder transform from left and right simultaneously
 
                 hK[k + 1] -= a;
                 final double beta = -1 / (a * hK[k + 1]);
 
                 // compute a = beta A v, where v is the Householder vector
                 // this loop is written in such a way
                 //   1) only the upper triangular part of the matrix is accessed
                 //   2) access is cache-friendly for a matrix stored in rows
                 Arrays.fill(z, k + 1, m, 0);
                 for (int i = k + 1; i < m; ++i) {
                     final double[] hI = householderVectors[i];
                     final double hKI = hK[i];
                     double zI = hI[i] * hKI;
                     for (int j = i + 1; j < m; ++j) {
                         final double hIJ = hI[j];
                         zI   += hIJ * hK[j];
                         z[j] += hIJ * hKI;
                     }
                     z[i] = beta * (z[i] + zI);
                 }
 
                 // compute gamma = beta vT z / 2
                 double gamma = 0;
                 for (int i = k + 1; i < m; ++i) {
                     gamma += z[i] * hK[i];
                 }
                 gamma *= beta / 2;
 
                 // compute z = z - gamma v
                 for (int i = k + 1; i < m; ++i) {
                     z[i] -= gamma * hK[i];
                 }
 
                 // update matrix: A = A - v zT - z vT
                 // only the upper triangular part of the matrix is updated
                 for (int i = k + 1; i < m; ++i) {
                     final double[] hI = householderVectors[i];
                     for (int j = i; j < m; ++j) {
                         hI[j] -= hK[i] * z[j] + z[i] * hK[j];
                     }
                 }
             }
         }
         main[m - 1] = householderVectors[m - 1][m - 1];
     }
 }