// Copyright (C) 2014-2015 Guibing Guo
//
// This file is part of LibRec.
//
// LibRec is free software: you can redistribute it and/or modify
// it under the terms of the GNU General Public License as published by
// the Free Software Foundation, either version 3 of the License, or
// (at your option) any later version.
//
// LibRec is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU General Public License
// along with LibRec. If not, see <http://www.gnu.org/licenses/>.
//

package librec.data;

import librec.util.Maths;

/**
 * <strong>Singular Value Decomposition: adapted from the JAMA implementations</strong><br>
 * 
 * <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'. Note that this implementation requires m>=n.
 * Otherwise, you'd better use the transpose of a matrix.
 * </p>
 * <p>
 * The singular values, sigma[k] = S[k][k], are ordered so that sigma[0] >= sigma[1] >= ... >= sigma[n-1].
 * </p>
 */

public class SVD {

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

	/**
	 * Array for internal storage of singular values.
	 */
	private double[] sigma;

	/**
	 * Row and column dimensions.
	 */
	private int m, n;

	/**
	 * Construct the singular value decomposition Structure to access U, S and V.
	 * 
	 * @param mat
	 *            Rectangular matrix
	 */

	public SVD(DenseMatrix mat) {

		// Derived from LINPACK code.

		// Initialize.
		DenseMatrix matClone = mat.clone();
		double[][] A = matClone.data;
		m = matClone.numRows;
		n = matClone.numColumns;

		/*
		 * Apparently the failing cases are only a proper subset of (m<n), so let's not throw error.
		 */
		int nu = Math.min(m, n);
		sigma = 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];

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

					// Apply the transformation.

					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 transformation.

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

				// Place the transformation 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 transformation 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] = Maths.hypot(e[k], e[i]);
				}
				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 transformation.

					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];
						}
					}
				}

				// Place the transformation 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) {
			sigma[nct] = A[nct][nct];
		}
		if (m < p) {
			sigma[p - 1] = 0.0;
		}
		if (nrt + 1 < p) {
			e[nrt] = A[nrt][p - 1];
		}
		e[p - 1] = 0.0;

		// Generate U
		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 (sigma[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;
			}
		}

		// Generate V

		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);
		double tiny = Math.pow(2.0, -966.0);
		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]) <= tiny + eps * (Math.abs(sigma[k]) + Math.abs(sigma[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(sigma[ks]) <= tiny + eps * t) {
						sigma[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 = Maths.hypot(sigma[j], f);
					double cs = sigma[j] / t;
					double sn = f / t;
					sigma[j] = t;
					if (j != k) {
						f = -sn * e[j - 1];
						e[j - 1] = cs * e[j - 1];
					}
					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 = Maths.hypot(sigma[j], f);
					double cs = sigma[j] / t;
					double sn = f / t;
					sigma[j] = t;
					f = -sn * e[j];
					e[j] = cs * e[j];
					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(sigma[p - 1]), Math.abs(sigma[p - 2])), Math.abs(e[p - 2])),
						Math.abs(sigma[k])), Math.abs(e[k]));
				double sp = sigma[p - 1] / scale;
				double spm1 = sigma[p - 2] / scale;
				double epm1 = e[p - 2] / scale;
				double sk = sigma[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 = Maths.hypot(f, g);
					double cs = f / t;
					double sn = g / t;
					if (j != k) {
						e[j - 1] = t;
					}
					f = cs * sigma[j] + sn * e[j];
					e[j] = cs * e[j] - sn * sigma[j];
					g = sn * sigma[j + 1];
					sigma[j + 1] = cs * sigma[j + 1];
					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 = Maths.hypot(f, g);
					cs = f / t;
					sn = g / t;
					sigma[j] = t;
					f = cs * e[j] + sn * sigma[j + 1];
					sigma[j + 1] = -sn * e[j] + cs * sigma[j + 1];
					g = sn * e[j + 1];
					e[j + 1] = cs * e[j + 1];
					if (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 (sigma[k] <= 0.0) {
					sigma[k] = (sigma[k] < 0.0 ? -sigma[k] : 0.0);
					for (int i = 0; i <= pp; i++) {
						V[i][k] = -V[i][k];
					}
				}

				// Order the singular values.

				while (k < pp) {
					if (sigma[k] >= sigma[k + 1]) {
						break;
					}
					double t = sigma[k];
					sigma[k] = sigma[k + 1];
					sigma[k + 1] = t;
					if (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 (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;
			}
		}
	}

	/**
	 * Return the left singular vectors
	 * 
	 * @return U
	 */
	public DenseMatrix getU() {
		return new DenseMatrix(U, m, Math.min(m + 1, n));
	}

	/**
	 * Return the right singular vectors
	 * 
	 * @return V
	 */
	public DenseMatrix getV() {
		return new DenseMatrix(V, n, n);
	}

	/**
	 * Return the one-dimensional array of singular values
	 * 
	 * @return diagonal of S.
	 */
	public double[] getSingularValues() {
		return sigma;
	}

	/**
	 * Return the diagonal matrix of singular values
	 * 
	 * @return S
	 */
	public DenseMatrix getS() {
		DenseMatrix res = new DenseMatrix(n, n);
		for (int i = 0; i < n; i++) {
			res.set(i, i, sigma[i]);
		}
		return res;
	}

	/**
	 * Two norm
	 * 
	 * @return max(S)
	 */
	public double norm2() {
		return sigma[0];
	}

	/**
	 * Two norm condition number
	 * 
	 * @return max(S)/min(S)
	 */
	public double cond() {
		return sigma[0] / sigma[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) * sigma[0] * eps;
		int r = 0;
		for (int i = 0; i < sigma.length; i++) {
			if (sigma[i] > tol) {
				r++;
			}
		}
		return r;
	}
}
