/**
  jama_svd.hpp

  From jama_svd.h with minor changes and adds by H.S.

  Changes:
  - Senseless inlining of the most of the member functions (e.g. of the
     480-line-constructor) removed.

  Adds:
  - A solve function

  @todo (HS):
  - Check the possibility of performing "in place" or "in situ" (or
     whatever it is called) on[in?] the input matrix \a Arg.
  - Make getS() safe for m .<. n.
  - Improve rank() for floats.

*/
#ifndef JAMA_SVD_H
#define JAMA_SVD_H

#include "tnt_array1d.hpp"
#include "tnt_array1d_utils.hpp"
#include "tnt_array2d.hpp"
#include "tnt_array2d_utils.hpp"
#include "tnt_math_utils.hpp"

#include <algorithm>		// HS: like TNT version 1.2.4
// for min(), max() below
#include <cmath>		// HS: like TNT version 1.2.4
// for abs() below

using namespace TNT;
using namespace std;

namespace JAMA
{
/** Singular Value Decomposition.
    <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>
    The singular values, sigma[k] = S[k][k], are ordered so that
    sigma[0] >= sigma[1] >= ... >= sigma[n-1].

    <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>
    (Adapted from JAMA, a Java Matrix Library, developed by jointly
    by the Mathworks and NIST; see  http://math.nist.gov/javanumerics/jama).

*/

template <class Real>
class SVD
{

  private:

    Array2D<Real> U, V;
    Array1D<Real> s;
    int m, n;                            // HS: Arg.dim1(), Arg.dim2()


  public:

    SVD (const Array2D<Real> &Arg);

    void getU (Array2D<Real> &A);

    /** Return the right singular vectors */
    void getV (Array2D<Real> &A)
    {
        A = V;
    }

    /** Return the one-dimensional array of singular values */
    void getSingularValues (Array1D<Real> &x)
    {
        x = s;
    }

    /** Return the diagonal matrix of singular values */
    void getS (Array2D<Real> &A);


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

    /** Two norm of condition number (max(S)/min(S)) */
    double cond ()
    {
        return s[0]/s[min(m,n)-1];
    }

    /**
      Effective numerical matrix rank
      @return    Number of nonnegligible singular values.
    */
    int rank ();                // HS: works satisfied only for double


    /** HS:
      Solve the (linear least square) problem \a Ax=b for given (m)-vector \a b.
      @return    normal solution (n)-vector \a x.
    */
    Array1D<Real> solve (const Array1D<Real> b);
    Array1D<Real> solve_1 (const Array1D<Real> b);
    Array1D<Real> solve_2 (const Array1D<Real> b);
};



template <class Real>
SVD <Real>::SVD (const Array2D<Real> &Arg)
{
      m = Arg.dim1();
      n = Arg.dim2();
      int nu = min(m,n);
      s = Array1D<Real>(min(m+1,n));
      U = Array2D<Real>(m, nu, Real(0));
      V = Array2D<Real>(n,n);
      Array1D<Real> e(n);
      Array1D<Real> work(m);
      Array2D<Real> A(Arg.copy());        // HS: copy() - muss das sein?
      int wantu = 1;                      /* boolean */
      int wantv = 1;                      /* boolean */
      int i=0, j=0, k=0;

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

      int nct = min(m-1,n);
      int nrt = max(0,min(n-2,m));
      for (k = 0; k < 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.
            s[k] = 0;
            for (i = k; i < m; i++) {
               s[k] = hypot(s[k],A[i][k]);
            }
            if (s[k] != 0.0) {
               if (A[k][k] < 0.0) {
                  s[k] = -s[k];
               }
               for (i = k; i < m; i++) {
                  A[i][k] /= s[k];
               }
               A[k][k] += 1.0;
            }
            s[k] = -s[k];
         }
         for (j = k+1; j < n; j++) {
            if ((k < nct) && (s[k] != 0.0))  {

               // Apply the transformation.

               double t = 0;
               for (i = k; i < m; i++) {
                  t += A[i][k]*A[i][j];
               }
               t = -t/A[k][k];
               for (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 (wantu & (k < nct)) {

            // Place the transformation in U for subsequent back
            // multiplication.

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

               // Place the transformation in V for subsequent
               // back multiplication.

               for (i = k+1; i < n; i++) {
                  V[i][k] = e[i];
               }
            }
         }
      }

      // Set up the final bidiagonal matrix or order p.

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

      // If required, generate V.

      if (wantv) {
         for (k = n-1; k >= 0; k--) {
            if ((k < nrt) & (e[k] != 0.0)) {
               for (j = k+1; j < nu; j++) {
                  double t = 0;
                  for (i = k+1; i < n; i++) {
                     t += V[i][k]*V[i][j];
                  }
                  t = -t/V[k+1][k];
                  for (i = k+1; i < n; i++) {
                     V[i][j] += t*V[i][k];
                  }
               }
            }
            for (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 = pow(2.0,-52.0);
      while (p > 0) {
         int k=0;
         int kase=0;

         // 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 (abs(e[k]) <= eps*(abs(s[k]) + 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 ? abs(e[ks]) : 0.) +
                          (ks != k+1 ? abs(e[ks-1]) : 0.);
               if (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 (j = p-2; j >= k; j--) {
                  double t = 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 (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 (j = k; j < p; j++) {
                  double t = 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 (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 = max(max(max(max(
                       abs(s[p-1]),abs(s[p-2])),abs(e[p-2])),
                       abs(s[k])),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 = 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 (j = k; j < p-1; j++) {
                  double t = hypot(f,g);
                  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 (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 = hypot(f,g);
                  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 (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 (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;
                  }
                  // swap (s[k],s[k+1]);
                  double t = s[k];
                  s[k] = s[k+1];
                  s[k+1] = t;
                  if (wantv && (k < n-1)) {
                     for (i = 0; i < n; i++) {
                        // swap (V[i][k], V[i][k+1]);
                        t = V[i][k+1]; V[i][k+1] = V[i][k]; V[i][k] = t;
                     }
                  }
                  if (wantu && (k < m-1)) {
                     for (i = 0; i < m; i++) {
                        // swap (U[i][k], U[i][k+1]);
                        t = U[i][k+1]; U[i][k+1] = U[i][k]; U[i][k] = t;
                     }
                  }
                  k++;
               }
               iter = 0;
               p--;
            }
            break;
         }
      }        // while(p>0)
}



/** HS: Why not simply return a ref to the intern U? -- Obviously the dimension
     may differ: intern-U[m,min(m,n)] vs. return-U[m,min(m+1,n)]. The intern
     dim is theoretical clear, but not to me the return dim.
*/
template <class Real>
void SVD<Real>::getU (Array2D<Real> &A)
{
    int minm = min(m+1,n);

    A = Array2D<Real>(m, minm);                 // HS: create new

    for (int i=0; i<m; i++)
        for (int j=0; j<minm; j++)
            A[i][j] = U[i][j];
}



/** Function fills in the argument Array2D \a A the diagonal matrix \a s of
     singular values.
     
    HS: 
    FRAGE: Warum \a A als Argument, warum keine Funktion, die \a A als 
     return-Wert liefert, allokiert wird \a A doch eh - ??
    
    @bug Der interne Vektor \a s wird kreiert mit der Dimension min(m+1,n).
     Falls in Ausgangs-(m,n)-Matrix <i>m+1 .<. n</i>, ist <i>dim(s) .<. n</i> 
     und es gibt eine  Bereichsueberschreitung bei <i> A[i][i] = s[i], </i> 
     da \a i bis inkl. \a n-1 laeuft.

    Kuerzere Formulierung der Funktion: <pre>
      A = Array2D<Real> (n,n, 0.);  // mit eventl. optimiertem Nullsetzen.
      for (int i=0; i < n; i++)  A[i][i] = s[i]; </pre>
*/
template <class Real>
void SVD<Real>::getS (Array2D<Real> &A)
{
    //std::cout << "getS(): (m,n)=" << m << ", " << n << "\n";
    //std::cout << s;
    A = Array2D<Real>(n,n);             // HS: why not: Array2D<Real>(n,n,0)?
    for (int i = 0; i < n; i++)         //  Because the i-loop then twice?
    {
        for (int j = 0; j < n; j++)
            A[i][j] = 0.0;

        A[i][i] = s[i];
    }
}


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

    TODO HS: works satisfied only for double, not for float.
*/
template <class Real>
int SVD<Real>::rank ()
{
    double eps = pow(2.0,-52.0);
    double tol = max(m,n)*s[0]*eps;
    //std::cout << "SVD::rank(): eps = " << eps << "  tol = " << tol << "\n";
    int r = 0;
    for (int i=0; i < s.dim(); i++)
    {
        if (s[i] > tol)
            r++;
    }
    return r;
}


/** Added by HS:
    Solve the (linear least square) problem \a Ax=b for a given (m)-vector \a b.
    @return: normal solution (n)-vector \a x.

    Algorithm: <i> x = V S^{-1} U^T b. </i>

    Regularisierte Loesung: Verwenden nur die numerisch relevanten Singulaerwerte
     as given by rank():

*/
template <class Real>
Array1D<Real> SVD<Real>::solve (const Array1D<Real> b)
{
    assert (m == b.dim());

    int rnk = rank();

    //std::cout << "SVD::solve()...:  ";
    //std::cout << "A[ " << m << ", " << n << " ],  rank = " << rnk << "\n";

    // Calculate (n)-vector beta = U^T * b:

    Array1D <Real> beta = matT_dot_vec (U,b);
    //std::cout << "beta = " << beta;

    // Calculate (n)-vector chi = S^+ beta = S^+ U^T b:

    Array1D <Real> chi (n);
    for (int i=0; i<rnk; i++)    chi[i] = beta[i] / s[i];
    for (int i=rnk; i<n; i++)    chi[i] = 0.0;
    //std::cout << "chi = " << chi;

    // Calculate (n)-vector x = V chi:

    Array1D <Real> x = mat_dot_vec (V,chi);
    //std::cout << "x = " << x;
    return x;
}

/** Added by HS:
    Alternative to solve(). 
    Differeneces: ...
*/
template <class Real>
Array1D<Real> SVD<Real>::solve_1 (const Array1D<Real> b)
{
    assert (m == b.dim());

    int rnk = rank();
    int d = min(m,n) - rnk;     // defect

    std::cout << "SVD::solve_1()...:  ";
    std::cout << "A[ " << m << ", " << n << " ],  rank = " << rnk << "\n";
    if (d)
    {   std::cout << "Rank defect: " << d << "-parametric solution "
                  << "resp. " << d << " irrelevant parameter(s).\n";
    }

    // Calculate (n)-vector y = U^T * b:
    // (For solution x we need only the first rnk elements of y.)

    Array1D <Real> y = matT_dot_vec (U,b);

    // Calculate (n)-vector chi = S^{-1} y = S^{-1} U^T b:

    for (int i=0; i<rnk; i++)    y[i] /=  s[i];
    for (int i=rnk; i<n; i++)    y[i] = 0.0;
    //std::cout << "chi = " << y;

    // Calculate (n)-vector x = V y = V S^{-1} U^T b:

    Array1D <Real> x(n);

    for (int i=0; i<n; i++)
        x[i] = dot (rnk, V[i], &y[0]);

    //std::cout << "x = " << x;
    return x;
}

/** Added by HS:
    Alternative to solve(). 
    Differeneces: ...
*/
template <class Real>
Array1D<Real> SVD<Real>::solve_2 (const Array1D<Real> b)
{
    assert (m == b.dim());

    int rnk = rank();
    int d = min(m,n) - rnk;     // rank defect

    std::cout << "SVD::solve_2()...:  ";
    std::cout << "A[ " << m << ", " << n << " ],  rank = " << rnk << "\n";
    if (d)
    {   std::cout << "Rank defect: " << d << "-parametric solution "
                  << "resp. " << d << " irrelevant parameter(s).\n";
    }

    // Calculate (n)-vector y = U^T * b:
    // (For solution x we use only the first rnk elements of y, so
    // we need a (rnk)-vector only.)

    Array1D <Real> y(rnk);

    for (int i=0; i<rnk; i++)
    {
        Real sum = 0;
        for (int j=0; j < b.dim(); j++)        // U^T * b
            sum += U[j][i] * b[j];

        y[i] = sum;
        //y[i] = dot (b.dim(), &U[0][i], U.stride(), &b[0]);
    }

    // Calculate the first rnk elements of  S^{-1} y

    for (int i=0; i<rnk; i++)    y[i] /= s[i];

    // Calculate (n)-vector x = V y = V S^{-1} U^T b:

    Array1D <Real> x(n);

    for (int i=0; i<n; i++)
        x[i] = dot (rnk, V[i], &y[0]);

    //std::cout << "x = " << x;
    return x;
}



}    // namespace JAMA

#endif
// JAMA_SVD_H
