/*
 * Copyright (c) 2009-2020, Peter Abeles. All Rights Reserved.
 *
 * This file is part of Efficient Java Matrix Library (EJML).
 *
 * Licensed 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.ejml.dense.row.decomposition.hessenberg;

import org.ejml.data.DMatrixRMaj;
import org.ejml.dense.row.decomposition.UtilDecompositons_DDRM;
import org.ejml.dense.row.decomposition.qr.QrHelperFunctions_DDRM;
import org.ejml.interfaces.decomposition.DecompositionInterface;
import org.jetbrains.annotations.Nullable;

/**
 * <p>
 * Finds the decomposition of a matrix in the form of:<br>
 * <br>
 * A = OHO<sup>T</sup><br>
 * <br>
 * where A is an m by m matrix, O is an orthogonal matrix, and H is an upper Hessenberg matrix.
 * </p>
 *
 * <p>
 * A matrix is upper Hessenberg if a<sup>ij</sup> = 0 for all i &gt; j+1. For example, the following matrix
 * is upper Hessenberg.<br>
 * <br>
 * WRITE IT OUT USING A TABLE
 * </p>
 *
 * <p>
 * This decomposition is primarily used as a step for computing the eigenvalue decomposition of a matrix.
 * The basic algorithm comes from David S. Watkins, "Fundamentals of MatrixComputations" Second Edition.
 * </p>
 */
// TODO create a column based one similar to what was done for QR decomposition?
@SuppressWarnings("NullAway.Init")
public class HessenbergSimilarDecomposition_DDRM
        implements DecompositionInterface<DMatrixRMaj> {
    // A combined matrix that stores te upper Hessenberg matrix and the orthogonal matrix.
    private DMatrixRMaj QH;
    // number of rows and columns of the matrix being decompose
    private int N;

    // the first element in the orthogonal vectors
    private double[] gammas;
    // temporary storage
    protected double[] b;
    protected double[] u;

    /**
     * Creates a decomposition that won't need to allocate new memory if it is passed matrices up to
     * the specified size.
     *
     * @param initialSize Expected size of the matrices it will decompose.
     */
    public HessenbergSimilarDecomposition_DDRM( int initialSize ) {
        gammas = new double[initialSize];
        b = new double[initialSize];
        u = new double[initialSize];
    }

    public HessenbergSimilarDecomposition_DDRM() {
        this(5);
    }

    /**
     * Computes the decomposition of the provided matrix.  If no errors are detected then true is returned,
     * false otherwise.
     *
     * @param A The matrix that is being decomposed.  Not modified.
     * @return If it detects any errors or not.
     */
    @Override
    public boolean decompose( DMatrixRMaj A ) {
        if (A.numRows != A.numCols)
            throw new IllegalArgumentException("A must be square.");
        if (A.numRows <= 0)
            return false;

        QH = A;

        N = A.numCols;

        if (b.length < N) {
            b = new double[N];
            gammas = new double[N];
            u = new double[N];
        }
        return _decompose();
    }

    @Override
    public boolean inputModified() {
        return true;
    }

    /**
     * The raw QH matrix that is stored internally.
     *
     * @return QH matrix.
     */
    public DMatrixRMaj getQH() {
        return QH;
    }

    /**
     * An upper Hessenberg matrix from the decomposition.
     *
     * @param H If not null then the results will be stored here.  Otherwise a new matrix will be created.
     * @return The extracted H matrix.
     */
    public DMatrixRMaj getH( @Nullable DMatrixRMaj H ) {
        H = UtilDecompositons_DDRM.ensureZeros(H, N, N);

        // copy the first row
        System.arraycopy(QH.data, 0, H.data, 0, N);

        for (int i = 1; i < N; i++) {
            for (int j = i - 1; j < N; j++) {
                H.set(i, j, QH.get(i, j));
            }
        }

        return H;
    }

    /**
     * An orthogonal matrix that has the following property: H = Q<sup>T</sup>AQ
     *
     * @param Q If not null then the results will be stored here.  Otherwise a new matrix will be created.
     * @return The extracted Q matrix.
     */
    public DMatrixRMaj getQ( @Nullable DMatrixRMaj Q ) {
        Q = UtilDecompositons_DDRM.ensureIdentity(Q, N, N);

        for (int j = N - 2; j >= 0; j--) {
            u[j + 1] = 1;
            for (int i = j + 2; i < N; i++) {
                u[i] = QH.get(i, j);
            }
            rank1UpdateMultR(Q, gammas[j], j + 1, j + 1, N);
        }

        return Q;
    }

    /**
     * Internal function for computing the decomposition.
     */
    private boolean _decompose() {
        double[] h = QH.data;

        for (int k = 0; k < N - 2; k++) {
            // find the largest value in this column
            // this is used to normalize the column and mitigate overflow/underflow
            double max = 0;

            for (int i = k + 1; i < N; i++) {
                // copy the householder vector to vector outside of the matrix to reduce caching issues
                // big improvement on larger matrices and a relatively small performance hit on small matrices.
                double val = u[i] = h[i*N + k];
                val = Math.abs(val);
                if (val > max)
                    max = val;
            }

            if (max > 0) {
                // -------- set up the reflector Q_k

                double tau = 0;
                // normalize to reduce overflow/underflow
                // and compute tau for the reflector
                for (int i = k + 1; i < N; i++) {
                    double val = u[i] /= max;
                    tau += val*val;
                }

                tau = Math.sqrt(tau);

                if (u[k + 1] < 0)
                    tau = -tau;

                // write the reflector into the lower left column of the matrix
                double nu = u[k + 1] + tau;
                u[k + 1] = 1.0;

                for (int i = k + 2; i < N; i++) {
                    h[i*N + k] = u[i] /= nu;
                }

                double gamma = nu/tau;
                gammas[k] = gamma;

                // ---------- multiply on the left by Q_k
                rank1UpdateMultR(QH, gamma, k + 1, k + 1, N);

                // ---------- multiply on the right by Q_k
                rank1UpdateMultL(QH, gamma, 0, k + 1, N);

                // since the first element in the householder vector is known to be 1
                // store the full upper hessenberg
                h[(k + 1)*N + k] = -tau*max;
            } else {
                gammas[k] = 0;
            }
        }

        return true;
    }

    protected void rank1UpdateMultL( DMatrixRMaj A, double gamma, int colA0, int w0, int w1 ) {
        QrHelperFunctions_DDRM.rank1UpdateMultL(A, u, gamma, colA0, w0, w1);
    }

    protected void rank1UpdateMultR( DMatrixRMaj A, double gamma, int colA0, int w0, int w1 ) {
        QrHelperFunctions_DDRM.rank1UpdateMultR(A, u, gamma, colA0, w0, w1, this.b);
    }

    public double[] getGammas() {
        return gammas;
    }
}
