package org.jmatrices.dbl.decomposition;

import org.jmatrices.dbl.Matrix;
import org.jmatrices.dbl.MatrixFactory;
import org.jmatrices.dbl.Matrix;
import org.jmatrices.dbl.MatrixFactory;

/**
 * SingularValueDecomposition
 * <P>
 * For an m-by-n matrix A with m >= n, the singular value decomposition is
 * an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and
 * an n-by-n orthogonal matrix V so that A = U*S*V'.
 * </p>
 * <P>
 * The singular values, sigma[k] = S[k][k], are ordered so that
 * sigma[0] >= sigma[1] >= ... >= sigma[n-1].
 * </p>
 * <P>
 * The singular value decompostion always exists, so the constructor will
 * never fail.  The matrix condition number and the effective numerical
 * rank can be computed from this decomposition.
 * </p>
 * <p>
 * http://kwon3d.com/theory/jkinem/svd.html
 * http://web.mit.edu/be.400/www/SVD/Singular_Value_Decomposition.htm
 * http://www.matheverywhere.com/mei/courseware/mgm/svdB/
 * http://www.netlib.org/lapack/lug/node53.html
 * http://www.mppmu.mpg.de/~schieck/svd.pdf
 * http://www.cs.ut.ee/~toomas_l/linalg/lin2/node14.html
 * </p>
 * <p>
 * A x X = B  <br/>
 * X = A<sup>-1</sup> x B <br/>
 * A = U x S x V<sup>T </sup> (Singular value decomposition of A)         <br/>
 * A-1 = V x S<sup>-1</sup> x U<sup>T</sup> (Generalized inverse or pseudoinverse of A)  <br/>
 * X = V x S<sup>-1</sup> x U<sup>T</sup> x B        <br/>
 * </p>
 * <p><font color="red">
 * The code is basically JAMA code with modifications made to fit in the scheme of things.
 * </font></p>
 * <p>
 * Author: purangp
 * </p>
 * Date: 12.03.2004
 * Time: 23:09:45
 */
public class SingularValueDecomposition {
    /* ------------------------
   Class variables
 * ------------------------ */

    /**
     * Arrays for internal storage of U and V.
     *
     * @serial internal storage of U.
     * @serial internal storage of V.
     */
    private double[][] U, V;

    /**
     * Array for internal storage of singular values.
     *
     * @serial internal storage of singular values.
     */
    private double[] s;

    /**
     * Row and column dimensions.
     *
     * @serial row dimension.
     * @serial column dimension.
     */
    private int m, n;
    private Matrix hint;
/* ------------------------
   Constructor
 * ------------------------ */

    /**
     * Construct the singular value decomposition
     *
     * @param matrix Rectangular matrix
     */

    public SingularValueDecomposition(Matrix matrix) {
         hint = matrix;
        // Derived from LINPACK code.
        // Initialize.
        double[][] A = matrix.get();   //changed
        m = matrix.rows();          //changed
        n = matrix.cols();          //changed
        int nu = Math.min(m, n);

        s = new double[Math.min(m + 1, n)];
        U = new double[m][nu];
        V = new double[n][n];

        double[] e = new double[n];
        double[] work = new double[m];

        boolean wantu = true;
        boolean wantv = true;

        // Reduce A to bidiagonal form, storing the diagonal elements
        // in s and the super-diagonal elements in e.

        int nct = Math.min(m - 1, n);
        int nrt = Math.max(0, Math.min(n - 2, m));
        for (int k = 0; k < Math.max(nct, nrt); k++) {
            if (k < nct) {

                // Compute the transformer for the k-th column and
                // place the k-th diagonal in s[k].
                // Compute 2-norm of k-th column without under/overflow.
                s[k] = 0;
                for (int i = k; i < m; i++) {
                    s[k] = Util.hypot(s[k], A[i][k]); //changed
                }
                if (s[k] != 0.0) {
                    if (A[k][k] < 0.0) {
                        s[k] = -s[k];
                    }
                    for (int i = k; i < m; i++) {
                        A[i][k] /= s[k];
                    }
                    A[k][k] += 1.0;
                }
                s[k] = -s[k];
            }
            for (int j = k + 1; j < n; j++) {
                if ((k < nct) & (s[k] != 0.0)) {

                    // Apply the transformer.

                    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 transformer.

                e[j] = A[k][j];
            }
            if (wantu & (k < nct)) {

                // Place the transformer 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 transformer 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] = Util.hypot(e[k], e[i]); //changed
                }
                if (e[k] != 0.0) {
                    if (e[k + 1] < 0.0) {
                        e[k] = -e[k];
                    }
                    for (int i = k + 1; i < n; i++) {
                        e[i] /= e[k];
                    }
                    e[k + 1] += 1.0;
                }
                e[k] = -e[k];
                if ((k + 1 < m) & (e[k] != 0.0)) {

                    // Apply the transformer.

                    for (int i = k + 1; i < m; i++) {
                        work[i] = 0.0;
                    }
                    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++) {
                        double t = -e[j] / e[k + 1];
                        for (int i = k + 1; i < m; i++) {
                            A[i][j] += t * work[i];
                        }
                    }
                }
                if (wantv) {

                    // Place the transformer 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 = Math.min(n, m + 1);
        if (nct < n) {
            s[nct] = A[nct][nct];
        }
        if (m < p) {
            s[p - 1] = 0.0;
        }
        if (nrt + 1 < p) {
            e[nrt] = A[nrt][p - 1];
        }
        e[p - 1] = 0.0;

        // If required, generate U.

        if (wantu) {
            for (int j = nct; j < nu; j++) {
                for (int i = 0; i < m; i++) {
                    U[i][j] = 0.0;
                }
                U[j][j] = 1.0;
            }
            for (int k = nct - 1; k >= 0; k--) {
                if (s[k] != 0.0) {
                    for (int j = k + 1; j < nu; 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] = 1.0 + U[k][k];
                    for (int i = 0; i < k - 1; i++) {
                        U[i][k] = 0.0;
                    }
                } else {
                    for (int i = 0; i < m; i++) {
                        U[i][k] = 0.0;
                    }
                    U[k][k] = 1.0;
                }
            }
        }

        // If required, generate V.

        if (wantv) {
            for (int k = n - 1; k >= 0; k--) {
                if ((k < nrt) & (e[k] != 0.0)) {
                    for (int j = k + 1; j < nu; 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] = 0.0;
                }
                V[k][k] = 1.0;
            }
        }

        // Main iteration loop for the singular values.
        int pp = p - 1;
        int iter = 0;
        double eps = Math.pow(2.0, -52.0);
        while (p > 0) {
            int k, 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 >= -1; k--) {
                if (k == -1) {
                    break;
                }
                if (Math.abs(e[k]) <= eps * (Math.abs(s[k]) + Math.abs(s[k + 1]))) {
                    e[k] = 0.0;
                    break;
                }
            }
            if (k == p - 2) {
                kase = 4;
            } else {
                int ks;
                for (ks = p - 1; ks >= k; ks--) {
                    if (ks == k) {
                        break;
                    }
                    double t = (ks != p ? Math.abs(e[ks]) : 0.) +
                            (ks != k + 1 ? Math.abs(e[ks - 1]) : 0.);
                    if (Math.abs(s[ks]) <= eps * t) {
                        s[ks] = 0.0;
                        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.

            switch (kase) {

                // Deflate negligible s(p).

                case 1:
                    {
                        double f = e[p - 2];
                        e[p - 2] = 0.0;
                        for (int j = p - 2; j >= k; j--) {
                            double t = Util.hypot(s[j], f);
                            double cs = s[j] / t;
                            double sn = f / t;
                            s[j] = t;
                            if (j != k) {
                                f = -sn * e[j - 1];
                                e[j - 1] = cs * e[j - 1];
                            }
                            if (wantv) {
                                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:
                    {
                        double f = e[k - 1];
                        e[k - 1] = 0.0;
                        for (int j = k; j < p; j++) {
                            double t = Util.hypot(s[j], f);
                            double cs = s[j] / t;
                            double sn = f / t;
                            s[j] = t;
                            f = -sn * e[j];
                            e[j] = cs * e[j];
                            if (wantu) {
                                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.
                        double scale = Math.max(Math.max(Math.max(Math.max(Math.abs(s[p - 1]), Math.abs(s[p - 2])), Math.abs(e[p - 2])),
                                Math.abs(s[k])), Math.abs(e[k]));
                        double sp = s[p - 1] / scale;
                        double spm1 = s[p - 2] / scale;
                        double epm1 = e[p - 2] / scale;
                        double sk = s[k] / scale;
                        double ek = e[k] / scale;
                        double b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2.0;
                        double c = (sp * epm1) * (sp * epm1);
                        double shift = 0.0;
                        if ((b != 0.0) | (c != 0.0)) {
                            shift = Math.sqrt(b * b + c);
                            if (b < 0.0) {
                                shift = -shift;
                            }
                            shift = c / (b + shift);
                        }
                        double f = (sk + sp) * (sk - sp) + shift;
                        double g = sk * ek;

                        // Chase zeros.

                        for (int j = k; j < p - 1; j++) {
                            double t = Util.hypot(f, g);   //changed
                            double cs = f / t;
                            double sn = g / t;
                            if (j != k) {
                                e[j - 1] = t;
                            }
                            f = cs * s[j] + sn * e[j];
                            e[j] = cs * e[j] - sn * s[j];
                            g = sn * s[j + 1];
                            s[j + 1] = cs * s[j + 1];
                            if (wantv) {
                                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 = Util.hypot(f, g);    //changed
                            cs = f / t;
                            sn = g / t;
                            s[j] = t;
                            f = cs * e[j] + sn * s[j + 1];
                            s[j + 1] = -sn * e[j] + cs * s[j + 1];
                            g = sn * e[j + 1];
                            e[j + 1] = cs * e[j + 1];
                            if (wantu && (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;
                        iter = iter + 1;
                    }
                    break;

                    // Convergence.

                case 4:
                    {

                        // Make the singular values positive.

                        if (s[k] <= 0.0) {
                            s[k] = (s[k] < 0.0 ? -s[k] : 0.0);
                            if (wantv) {
                                for (int i = 0; i <= pp; i++) {
                                    V[i][k] = -V[i][k];
                                }
                            }
                        }

                        // Order the singular values.

                        while (k < pp) {
                            if (s[k] >= s[k + 1]) {
                                break;
                            }
                            double t = s[k];
                            s[k] = s[k + 1];
                            s[k + 1] = t;
                            if (wantv && (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 (wantu && (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++;
                        }
                        iter = 0;
                        p--;
                    }
                    break;
            }
        }
    }



/* ------------------------
   Public Methods
 * ------------------------ */

    /**
     * Return the left singular vectors
     *
     * @return U
     */

    public Matrix getU() {
        //return new Matrix(U,m,Math.min(m+1,n));
        return MatrixFactory.getMatrix(m, Math.min(m + 1, n), hint, U);
    }

    /**
     * Return the right singular vectors
     *
     * @return V
     */

    public Matrix getV() {
        return MatrixFactory.getMatrix(n, n, hint, V);
    }

    /**
     * Return the one-dimensional array of singular values
     *
     * @return diagonal of S.
     */

    public double[] getSingularValues() {
        return s;
    }

    /**
     * Return the diagonal matrix of singular values
     *
     * @return S
     */

    public Matrix getS() {
        double[][] S = new double[m][n]; //changed n to m
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                S[i][j] = 0.0;
            }
            S[i][i] = this.s[i];
        }
        return MatrixFactory.getMatrix(m, n, hint, S);  //changed n to m
    }

    /**
     * Two norm
     *
     * @return max(S)
     */

    public double norm2() {
        return s[0];
    }

    /**
     * Two norm condition number
     *
     * @return max(S)/min(S)
     */

    public double cond() {
        return s[0] / s[Math.min(m, n) - 1];
    }

    /**
     * Effective numerical matrix rank
     *
     * @return Number of nonnegligible singular values.
     */

    public int rank() {
        double eps = Math.pow(2.0, -52.0);
        double tol = Math.max(m, n) * s[0] * eps;
        int r = 0;
        for (int i = 0; i < s.length; i++) {
            if (s[i] > tol) {
                r++;
            }
        }
        return r;
    }
}
