/*
    * 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 org.hipparchus.exception.LocalizedCoreFormats;
  import org.hipparchus.exception.MathIllegalArgumentException;
  import org.hipparchus.exception.MathIllegalStateException;
  import org.hipparchus.exception.MathRuntimeException;
  import org.hipparchus.util.FastMath;
  import org.hipparchus.util.Precision;
  
  /**
   * Calculates the eigen decomposition of a symmetric real matrix.
   * <p>
   * The eigen decomposition of matrix A is a set of two matrices:
   * \(V\) and \(D\) such that \(A V = V D\) where $\(A\),
   * \(V\) and \(D\) are all \(m \times m\) matrices.
   * <p>
   * This class is similar in spirit to the {@code EigenvalueDecomposition}
   * class from the <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a>
   * library, with the following changes:
   * </p>
   * <ul>
   *   <li>a {@link #getVT() getVt} method has been added,</li>
   *   <li>a {@link #getEigenvalue(int) getEigenvalue} method to pick up a
   *       single eigenvalue has been added,</li>
   *   <li>a {@link #getEigenvector(int) getEigenvector} method to pick up a
   *       single eigenvector has been added,</li>
   *   <li>a {@link #getDeterminant() getDeterminant} method has been added.</li>
   *   <li>a {@link #getSolver() getSolver} method has been added.</li>
   * </ul>
   * <p>
   * As \(A\) is symmetric, then \(A = V D V^T\) where the eigenvalue matrix \(D\)
   * is diagonal and the eigenvector matrix \(V\) is orthogonal, i.e.
   * {@code A = V.multiply(D.multiply(V.transpose()))} and
   * {@code V.multiply(V.transpose())} equals the identity matrix.
   * </p>
   * <p>
   * The columns of \(V\) represent the eigenvectors in the sense that \(A V = V D\),
   * i.e. {@code A.multiply(V)} equals {@code V.multiply(D)}.
   * The matrix \(V\) may be badly conditioned, or even singular, so the validity of the
   * equation \(A = V D V^{-1}\) depends upon the condition of \(V\).
   * </p>
   * This implementation is based on the paper by A. Drubrulle, R.S. Martin and
   * J.H. Wilkinson "The Implicit QL Algorithm" in Wilksinson and Reinsch (1971)
   * Handbook for automatic computation, vol. 2, Linear algebra, Springer-Verlag,
   * New-York.
   *
   * @see <a href="http://mathworld.wolfram.com/EigenDecomposition.html">MathWorld</a>
   * @see <a href="http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix">Wikipedia</a>
   */
  public class EigenDecompositionSymmetric {
  
      /** Default epsilon value to use for internal epsilon **/
      public static final double DEFAULT_EPSILON = 1e-12;
  
      /** Maximum number of iterations accepted in the implicit QL transformation */
      private static final byte MAX_ITER = 30;
  
      /** Internally used epsilon criteria. */
      private final double epsilon;
  
      /** Eigenvalues. */
      private double[] eigenvalues;
  
      /** Eigenvectors. */
      private ArrayRealVector[] eigenvectors;
  
      /** Cached value of V. */
      private RealMatrix cachedV;
  
      /** Cached value of D. */
      private DiagonalMatrix cachedD;
  
      /** Cached value of Vt. */
      private RealMatrix cachedVt;
  
      /**
       * Calculates the eigen decomposition of the given symmetric real matrix.
      * <p>
      * This constructor uses the {@link #DEFAULT_EPSILON default epsilon} and
      * decreasing order for eigenvalues.
      * </p>
      * @param matrix Matrix to decompose.
      * @throws MathIllegalStateException if the algorithm fails to converge.
      * @throws MathRuntimeException if the decomposition of a general matrix
      * results in a matrix with zero norm
      */
     public EigenDecompositionSymmetric(final RealMatrix matrix) {
         this(matrix, DEFAULT_EPSILON, true);
     }
 
     /**
      * Calculates the eigen decomposition of the given real matrix.
      * <p>
      * Supports decomposition of a general matrix since 3.1.
      *
      * @param matrix Matrix to decompose.
      * @param epsilon Epsilon used for internal tests (e.g. is singular, eigenvalue ratio, etc.)
      * @param decreasing if true, eigenvalues will be sorted in decreasing order
      * @throws MathIllegalStateException if the algorithm fails to converge.
      * @throws MathRuntimeException if the decomposition of a general matrix
      * results in a matrix with zero norm
      * @since 3.0
      */
     public EigenDecompositionSymmetric(final RealMatrix matrix,
                                        final double epsilon, final boolean decreasing)
         throws MathRuntimeException {
 
         this.epsilon = epsilon;
         MatrixUtils.checkSymmetric(matrix, epsilon);
 
         // transform the matrix to tridiagonal
         final TriDiagonalTransformer transformer = new TriDiagonalTransformer(matrix);
 
         findEigenVectors(transformer.getMainDiagonalRef(),
                          transformer.getSecondaryDiagonalRef(),
                          transformer.getQ().getData(),
                          decreasing);
 
     }
 
     /**
      * Calculates the eigen decomposition of the symmetric tridiagonal matrix.
      * <p>
      * The Householder matrix is assumed to be the identity matrix.
      * </p>
      * <p>
      * This constructor uses the {@link #DEFAULT_EPSILON default epsilon} and
      * decreasing order for eigenvalues.
      * </p>
      * @param main Main diagonal of the symmetric tridiagonal form.
      * @param secondary Secondary of the tridiagonal form.
      * @throws MathIllegalStateException if the algorithm fails to converge.
      */
     public EigenDecompositionSymmetric(final double[] main, final double[] secondary) {
         this(main, secondary, DEFAULT_EPSILON, true);
     }
 
     /**
      * Calculates the eigen decomposition of the symmetric tridiagonal
      * matrix.  The Householder matrix is assumed to be the identity matrix.
      *
      * @param main Main diagonal of the symmetric tridiagonal form.
      * @param secondary Secondary of the tridiagonal form.
      * @param epsilon Epsilon used for internal tests (e.g. is singular, eigenvalue ratio, etc.)
      * @param decreasing if true, eigenvalues will be sorted in decreasing order
      * @throws MathIllegalStateException if the algorithm fails to converge.
      * @since 3.0
      */
     public EigenDecompositionSymmetric(final double[] main, final double[] secondary,
                                        final double epsilon, final boolean decreasing) {
         this.epsilon = epsilon;
         final int size = main.length;
         final double[][] z = new double[size][size];
         for (int i = 0; i < size; i++) {
             z[i][i] = 1.0;
         }
         findEigenVectors(main.clone(), secondary.clone(), z, decreasing);
     }
 
     /**
      * Gets the matrix V of the decomposition.
      * V is an orthogonal matrix, i.e. its transpose is also its inverse.
      * The columns of V are the eigenvectors of the original matrix.
      * No assumption is made about the orientation of the system axes formed
      * by the columns of V (e.g. in a 3-dimension space, V can form a left-
      * or right-handed system).
      *
      * @return the V matrix.
      */
     public RealMatrix getV() {
 
         if (cachedV == null) {
             final int m = eigenvectors.length;
             cachedV = MatrixUtils.createRealMatrix(m, m);
             for (int k = 0; k < m; ++k) {
                 cachedV.setColumnVector(k, eigenvectors[k]);
             }
         }
         // return the cached matrix
         return cachedV;
     }
 
     /**
      * Gets the diagonal matrix D of the decomposition.
      * D is a diagonal matrix.
      * @return the D matrix.
      *
      * @see #getEigenvalues()
       */
     public DiagonalMatrix getD() {
 
         if (cachedD == null) {
             // cache the matrix for subsequent calls
             cachedD = new DiagonalMatrix(eigenvalues);
         }
 
         return cachedD;
 
     }
 
     /**
      * Get's the value for epsilon which is used for internal tests (e.g. is singular, eigenvalue ratio, etc.)
      *
      * @return the epsilon value.
      */
     public double getEpsilon() { return epsilon; }
 
     /**
      * Gets the transpose of the matrix V of the decomposition.
      * V is an orthogonal matrix, i.e. its transpose is also its inverse.
      * The columns of V are the eigenvectors of the original matrix.
      * No assumption is made about the orientation of the system axes formed
      * by the columns of V (e.g. in a 3-dimension space, V can form a left-
      * or right-handed system).
      *
      * @return the transpose of the V matrix.
      */
     public RealMatrix getVT() {
 
         if (cachedVt == null) {
             final int m = eigenvectors.length;
             cachedVt = MatrixUtils.createRealMatrix(m, m);
             for (int k = 0; k < m; ++k) {
                 cachedVt.setRowVector(k, eigenvectors[k]);
             }
         }
 
         // return the cached matrix
         return cachedVt;
     }
 
     /**
      * Gets a copy of the eigenvalues of the original matrix.
      *
      * @return a copy of the eigenvalues of the original matrix.
      *
      * @see #getD()
      * @see #getEigenvalue(int)
      */
     public double[] getEigenvalues() {
         return eigenvalues.clone();
     }
 
     /**
      * Returns the i<sup>th</sup> eigenvalue of the original matrix.
      *
      * @param i index of the eigenvalue (counting from 0)
      * @return real part of the i<sup>th</sup> eigenvalue of the original
      * matrix.
      *
      * @see #getD()
      * @see #getEigenvalues()
      */
     public double getEigenvalue(final int i) {
         return eigenvalues[i];
     }
 
     /**
      * Gets a copy of the i<sup>th</sup> eigenvector of the original matrix.
      * <p>
      * Note that if the the i<sup>th</sup> is complex this method will throw
      * an exception.
      * </p>
      * @param i Index of the eigenvector (counting from 0).
      * @return a copy of the i<sup>th</sup> eigenvector of the original matrix.
      * @see #getD()
      */
     public RealVector getEigenvector(final int i) {
         return eigenvectors[i].copy();
     }
 
     /**
      * Computes the determinant of the matrix.
      *
      * @return the determinant of the matrix.
      */
     public double getDeterminant() {
         double determinant = 1;
         for (int i = 0; i < eigenvalues.length; ++i) {
             determinant *= eigenvalues[i];
         }
         return determinant;
     }
 
     /**
      * Computes the square-root of the matrix.
      * This implementation assumes that the matrix is positive definite.
      *
      * @return the square-root of the matrix.
      * @throws MathRuntimeException if the matrix is not
      * symmetric or not positive definite.
      */
     public RealMatrix getSquareRoot() {
 
         final double[] sqrtEigenValues = new double[eigenvalues.length];
         for (int i = 0; i < eigenvalues.length; i++) {
             final double eigen = eigenvalues[i];
             if (eigen <= 0) {
                 throw new MathRuntimeException(LocalizedCoreFormats.UNSUPPORTED_OPERATION);
             }
             sqrtEigenValues[i] = FastMath.sqrt(eigen);
         }
         final RealMatrix sqrtEigen = MatrixUtils.createRealDiagonalMatrix(sqrtEigenValues);
         final RealMatrix v = getV();
         final RealMatrix vT = getVT();
 
         return v.multiply(sqrtEigen).multiply(vT);
 
     }
 
     /** Gets a solver for finding the \(A \times X = B\) solution in exact linear sense.
      * @return a solver
      */
     public DecompositionSolver getSolver() {
         return new Solver();
     }
 
     /** Specialized solver. */
     private class Solver implements DecompositionSolver {
 
         /**
          * Solves the linear equation \(A \times X = B\)for symmetric matrices A.
          * <p>
          * This method only finds exact linear solutions, i.e. solutions for
          * which ||A &times; X - B|| is exactly 0.
          * </p>
          *
          * @param b Right-hand side of the equation \(A \times X = B\).
          * @return a Vector X that minimizes the 2-norm of \(A \times X - B\).
          *
          * @throws MathIllegalArgumentException if the matrices dimensions do not match.
          * @throws MathIllegalArgumentException if the decomposed matrix is singular.
          */
         @Override
         public RealVector solve(final RealVector b) {
             if (!isNonSingular()) {
                 throw new MathIllegalArgumentException(LocalizedCoreFormats.SINGULAR_MATRIX);
             }
 
             final int m = eigenvalues.length;
             if (b.getDimension() != m) {
                 throw new MathIllegalArgumentException(LocalizedCoreFormats.DIMENSIONS_MISMATCH,
                                                        b.getDimension(), m);
             }
 
             final double[] bp = new double[m];
             for (int i = 0; i < m; ++i) {
                 final ArrayRealVector v = eigenvectors[i];
                 final double[] vData = v.getDataRef();
                 final double s = v.dotProduct(b) / eigenvalues[i];
                 for (int j = 0; j < m; ++j) {
                     bp[j] += s * vData[j];
                 }
             }
 
             return new ArrayRealVector(bp, false);
         }
 
         /** {@inheritDoc} */
         @Override
         public RealMatrix solve(RealMatrix b) {
 
             if (!isNonSingular()) {
                 throw new MathIllegalArgumentException(LocalizedCoreFormats.SINGULAR_MATRIX);
             }
 
             final int m = eigenvalues.length;
             if (b.getRowDimension() != m) {
                 throw new MathIllegalArgumentException(LocalizedCoreFormats.DIMENSIONS_MISMATCH,
                                                        b.getRowDimension(), m);
             }
 
             final int nColB = b.getColumnDimension();
             final double[][] bp = new double[m][nColB];
             final double[] tmpCol = new double[m];
             for (int k = 0; k < nColB; ++k) {
                 for (int i = 0; i < m; ++i) {
                     tmpCol[i] = b.getEntry(i, k);
                     bp[i][k]  = 0;
                 }
                 for (int i = 0; i < m; ++i) {
                     final ArrayRealVector v = eigenvectors[i];
                     final double[] vData = v.getDataRef();
                     double s = 0;
                     for (int j = 0; j < m; ++j) {
                         s += v.getEntry(j) * tmpCol[j];
                     }
                     s /= eigenvalues[i];
                     for (int j = 0; j < m; ++j) {
                         bp[j][k] += s * vData[j];
                     }
                 }
             }
 
             return new Array2DRowRealMatrix(bp, false);
 
         }
 
         /**
          * Checks whether the decomposed matrix is non-singular.
          *
          * @return true if the decomposed matrix is non-singular.
          */
         @Override
         public boolean isNonSingular() {
             double largestEigenvalueNorm = 0.0;
             // Looping over all values (in case they are not sorted in decreasing
             // order of their norm).
             for (int i = 0; i < eigenvalues.length; ++i) {
                 largestEigenvalueNorm = FastMath.max(largestEigenvalueNorm, FastMath.abs(eigenvalues[i]));
             }
             // Corner case: zero matrix, all exactly 0 eigenvalues
             if (largestEigenvalueNorm == 0.0) {
                 return false;
             }
             for (int i = 0; i < eigenvalues.length; ++i) {
                 // Looking for eigenvalues that are 0, where we consider anything much much smaller
                 // than the largest eigenvalue to be effectively 0.
                 if (Precision.equals(FastMath.abs(eigenvalues[i]) / largestEigenvalueNorm, 0, epsilon)) {
                     return false;
                 }
             }
             return true;
         }
 
         /**
          * Get the inverse of the decomposed matrix.
          *
          * @return the inverse matrix.
          * @throws MathIllegalArgumentException if the decomposed matrix is singular.
          */
         @Override
         public RealMatrix getInverse() {
             if (!isNonSingular()) {
                 throw new MathIllegalArgumentException(LocalizedCoreFormats.SINGULAR_MATRIX);
             }
 
             final int m = eigenvalues.length;
             final double[][] invData = new double[m][m];
 
             for (int i = 0; i < m; ++i) {
                 final double[] invI = invData[i];
                 for (int j = 0; j < m; ++j) {
                     double invIJ = 0;
                     for (int k = 0; k < m; ++k) {
                         final double[] vK = eigenvectors[k].getDataRef();
                         invIJ += vK[i] * vK[j] / eigenvalues[k];
                     }
                     invI[j] = invIJ;
                 }
             }
             return MatrixUtils.createRealMatrix(invData);
         }
 
         /** {@inheritDoc} */
         @Override
         public int getRowDimension() {
             return eigenvalues.length;
         }
 
         /** {@inheritDoc} */
         @Override
         public int getColumnDimension() {
             return eigenvalues.length;
         }
 
     }
 
     /**
      * Find eigenvalues and eigenvectors (Dubrulle et al., 1971)
      * @param main main diagonal of the tridiagonal matrix
      * @param secondary secondary diagonal of the tridiagonal matrix
      * @param householderMatrix Householder matrix of the transformation
      * @param decreasing if true, eigenvalues will be sorted in decreasing order
      * to tridiagonal form.
      */
     private void findEigenVectors(final double[] main, final double[] secondary,
                                   final double[][] householderMatrix, final boolean decreasing) {
         final double[][]z = householderMatrix.clone();
         final int n = main.length;
         eigenvalues = new double[n];
         final double[] e = new double[n];
         for (int i = 0; i < n - 1; i++) {
             eigenvalues[i] = main[i];
             e[i] = secondary[i];
         }
         eigenvalues[n - 1] = main[n - 1];
         e[n - 1] = 0;
 
         // Determine the largest main and secondary value in absolute term.
         double maxAbsoluteValue = 0;
         for (int i = 0; i < n; i++) {
             if (FastMath.abs(eigenvalues[i]) > maxAbsoluteValue) {
                 maxAbsoluteValue = FastMath.abs(eigenvalues[i]);
             }
             if (FastMath.abs(e[i]) > maxAbsoluteValue) {
                 maxAbsoluteValue = FastMath.abs(e[i]);
             }
         }
         // Make null any main and secondary value too small to be significant
         if (maxAbsoluteValue != 0) {
             for (int i=0; i < n; i++) {
                 if (FastMath.abs(eigenvalues[i]) <= Precision.EPSILON * maxAbsoluteValue) {
                     eigenvalues[i] = 0;
                 }
                 if (FastMath.abs(e[i]) <= Precision.EPSILON * maxAbsoluteValue) {
                     e[i]=0;
                 }
             }
         }
 
         for (int j = 0; j < n; j++) {
             int its = 0;
             int m;
             do {
                 for (m = j; m < n - 1; m++) {
                     double delta = FastMath.abs(eigenvalues[m]) +
                         FastMath.abs(eigenvalues[m + 1]);
                     if (FastMath.abs(e[m]) + delta == delta) {
                         break;
                     }
                 }
                 if (m != j) {
                     if (its == MAX_ITER) {
                         throw new MathIllegalStateException(LocalizedCoreFormats.CONVERGENCE_FAILED,
                                                             MAX_ITER);
                     }
                     its++;
                     double q = (eigenvalues[j + 1] - eigenvalues[j]) / (2 * e[j]);
                     double t = FastMath.sqrt(1 + q * q);
                     if (q < 0.0) {
                         q = eigenvalues[m] - eigenvalues[j] + e[j] / (q - t);
                     } else {
                         q = eigenvalues[m] - eigenvalues[j] + e[j] / (q + t);
                     }
                     double u = 0.0;
                     double s = 1.0;
                     double c = 1.0;
                     int i;
                     for (i = m - 1; i >= j; i--) {
                         double p = s * e[i];
                         double h = c * e[i];
                         if (FastMath.abs(p) >= FastMath.abs(q)) {
                             c = q / p;
                             t = FastMath.sqrt(c * c + 1.0);
                             e[i + 1] = p * t;
                             s = 1.0 / t;
                             c *= s;
                         } else {
                             s = p / q;
                             t = FastMath.sqrt(s * s + 1.0);
                             e[i + 1] = q * t;
                             c = 1.0 / t;
                             s *= c;
                         }
                         if (e[i + 1] == 0.0) {
                             eigenvalues[i + 1] -= u;
                             e[m] = 0.0;
                             break;
                         }
                         q = eigenvalues[i + 1] - u;
                         t = (eigenvalues[i] - q) * s + 2.0 * c * h;
                         u = s * t;
                         eigenvalues[i + 1] = q + u;
                         q = c * t - h;
                         for (int ia = 0; ia < n; ia++) {
                             p = z[ia][i + 1];
                             z[ia][i + 1] = s * z[ia][i] + c * p;
                             z[ia][i] = c * z[ia][i] - s * p;
                         }
                     }
                     if (t == 0.0 && i >= j) {
                         continue;
                     }
                     eigenvalues[j] -= u;
                     e[j] = q;
                     e[m] = 0.0;
                 }
             } while (m != j);
         }
 
         // Sort the eigen values (and vectors) in desired order
         for (int i = 0; i < n; i++) {
             int k = i;
             double p = eigenvalues[i];
             for (int j = i + 1; j < n; j++) {
                 if (eigenvalues[j] > p == decreasing) {
                     k = j;
                     p = eigenvalues[j];
                 }
             }
             if (k != i) {
                 eigenvalues[k] = eigenvalues[i];
                 eigenvalues[i] = p;
                 for (int j = 0; j < n; j++) {
                     p = z[j][i];
                     z[j][i] = z[j][k];
                     z[j][k] = p;
                 }
             }
         }
 
         // Determine the largest eigen value in absolute term.
         maxAbsoluteValue = 0;
         for (int i = 0; i < n; i++) {
             if (FastMath.abs(eigenvalues[i]) > maxAbsoluteValue) {
                 maxAbsoluteValue = FastMath.abs(eigenvalues[i]);
             }
         }
         // Make null any eigen value too small to be significant
         if (maxAbsoluteValue != 0.0) {
             for (int i=0; i < n; i++) {
                 if (FastMath.abs(eigenvalues[i]) < Precision.EPSILON * maxAbsoluteValue) {
                     eigenvalues[i] = 0;
                 }
             }
         }
         eigenvectors = new ArrayRealVector[n];
         for (int i = 0; i < n; i++) {
             eigenvectors[i] = new ArrayRealVector(n);
             for (int j = 0; j < n; j++) {
                 eigenvectors[i].setEntry(j, z[j][i]);
             }
         }
     }
 
 }