/** Singular Value Decomposition. 

                                U,S,V=svd(A);get...():      A=U*S*V^_T_ 

   <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).

   */

#ifdef JAMA_DYNAMIC

        int m, n;

        int nu,minm;

#else

                #define nu   MIN(m,n)

                #define minm MIN(m+1,n)

#endif

        JamaArray2(Real,m,nu) U;

        JamaArray2(Real,n,n)  V;

        JamaArray1(Real,minm) s;

PUBLIC

   member(SVD) (const JamaArray2(Real,m,n) &Arg) {

                 SVD0(Arg,1,1);

         }

   member(SVD) (const JamaArray2(Real,m,n) &Arg,int wantu,int wantv) {

                 SVD0(Arg,wantu,wantv);

         }

   void member(SVD0) (const JamaArray2(Real,m,n) &Arg,int wantu,int wantv) {

        //allocation

#ifdef JAMA_DYNAMIC

                m = Arg.NROWS();

                n = Arg.NCOLS();

                nu = MIN(m,n);//length of diagonal

                minm=MIN(m+1,n);//width of bidiagonal matrix.

      //s = JamaArray1(Real,minm)(minm); 

      //U = JamaArray2(Real,m,nu)(m, nu);//,   (0)

      //V = JamaArray2(Real,n,n)(n,n);

                RESIZE1(Real, s, minm);

                RESIZE2(Real, U, m ,nu);

                RESIZE2(Real, V, n, n);

                JamaArray1(Real,n) e(n); //PZ: e serves as storage for upper diagonal elements and also temporarily storage of last portion of rows. 

                                             //PZ: thus, minm-1 elements would be not enough. 

                                               //e[0]=D[0][1]

                JamaArray1(Real,m) work(m); 

                JamaArray2(Real,m,n) A(COPY(Arg));

        //end of allocation

                for (int i_=0;i_<m;i_++) {

                        for (int j_=0;j_<nu;j_++) {

                                $ U[i_][j_] = 0;

                        }

                }

#else

        //allocation

                memset(&U,0,sizeof(U));

      JamaArray1(Real,n) e;

      JamaArray1(Real,m) work;

                JamaArray2(Real,m,n) A;//(Arg.copy());

                memcpy(&A,&Arg,sizeof(A));

        //end of allocation

#endif

        if(m<n){

                throw("not safe operation!!!");

        }

//      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.

                  /* PZ:

                                        final result:

                                        Arg= M(U(:,m-1))*M(U(:,m-2)*...M(U(:,0))*D*( M(V(:,m-1))*M(V(:,m-2)*...M(V(:,0)) )^T

                                        where mirroring hauseholder transformation

                                            M(v)=I-2*v*v^T/||v||^2, but mirror plane vectors are constructed that

                                                                  U(k,k)=2*||U(:,k)||^2, thus M_k=I-U(:,k)*U(:,k)^T/U(k,k)

            and for all k, B[k][k]=s[k], B[k][k+1]=e[k], and B is 0 at all other locations. 

          Arrays U and V are not matrices, rather collection of vectors.

                                         during the operation, the first columns of U and V are filled, 

                                         while the bottom left portion of A remains inportant as other 

                                           parts would be zero-ed out. This is not administrated in matrix A. 

           For the kth iteration, the column transformation leaves the first k-1 rows the same, 

                                         and the kth row transformation leaves the first k column the same. 

                                         s[k] and e[k] is also filled successively, but unfilled portion of e[k]

                                         is used as temporary variable. 

                                */

      int nct = MIN(m-1,n);//PZ number of column transformations

      int nrt = MAX(0,MIN(n-2,m));//PZ: number of row transformations

      for (k = 0; k < MAX(nct,nrt); k++) {//PZ: for each element in the diagonal

         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.

                                         //PZ: s[k]=|A[k:end][k]|*sign(A[k,k])

                                         //PZ  if(s[k]!=0) {A[k:end,k]/=s[k]; A[k][k]+=1}

                                         //PZ  s[k]*=-1;

                                         //PZ azaz if(norm(A_:k)>0) {A_:k=A_:k/norm(A_:k)+I_:k}

                                         //         else 

            $ 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) {//PZ: two possible mirror vectors arose. one result -I_:k, other +I_:k after mirroring A_:k

                  $ s[k] = -$ s[k]; //PZ: this chooses the longer one, to avoid numeric instability if A_:k is very near to I_:k

               }

               for (i = k; i < m; i++) {

                  $ A[i][k] /= $ s[k];

               }

               $ A[k][k] += 1.0;//PZ: A_:k contains the vector perpendicular to the desired mirroring

            }

            $ s[k] = -$ s[k];//PZ: Mirror(A_:k) original=s_k*I_:k. The sign depends on the choosen vector

         }

         for (j = k+1; j < n; j++) {

            if ((k < nct) && ($ s[k] != 0.0))  {

            // Apply the transformation. //PZ: mirroring to the plane perp. to A_:k

               Real t(0.0);

               for (i = k; i < m; i++) {

                  t += $ A[i][k]*$ A[i][j];

               }//PZ: t=A[K:end][k]^_T_*A[k:end][j]

               t = -t/$ A[k][k];

                                                         //PZ: t=-A[K:end][k]^_T_*A[k:end][j]/A[k][k]

               for (i = k; i < m; i++) {

                  $ A[i][j] += t*$ A[i][k];

               }

                                                         //PZ: A[k:end][j]-=A[k:end][k]*(A[K:end][k]^_T_*A[k:end][j])/A[k][k]

                                                         //PZ: A[k:end][j]=A[k:end][j]-A[k:end][k]*(A[K:end][k]^_T_*A[k:end][j])/A[k][k]

                                                         //PZ: A_:j=(I -A_:k*A_:k^T/A_kk)*A_:j

                                                         //PZ: since A_kk = 2*||A_:k||^2, thus the transformation is I-2*A_:k*A_:k^T/||A_:k||^2 and is orthogonal. 

            }

            // 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];//PZ: the non-unit length normal vector of the mirroring plane, U_kk is twice the length squared

            }

         }

         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) {//PZ: Same formulation as for row transformations

                  $ 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++) {

                  Real 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];//PZ the non-uniform mirroring vector

               }

            }

         }

      }

      // 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) {//PZ: if(n>m && p==m+1)

         $ s[p-1] = 0.0; //PZ: s[m]=0.0

      }

      if (nrt+1 < p) {

         $ e[nrt] = $ A[nrt][p-1];

      }

      $ e[p-1] = 0.0;

      // If required, generate U.

      if (wantu) {

                                //PZ: transform the collection of mirror-plane normal vectors into a single orthogonal transformation

         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++) {

                  Real t(0.0);

                  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] = (Real) (1.0 + $ U[k][k] );

               for (i = 0; i < k-1; i++) {

                  $ U[i][k] = 0.0;

               }

            } else {

                                                        //PZ:  U_:k=I_:k

               for (i = 0; i < m; i++) {

                  $ U[i][k] = 0.0;

               }

               $ U[k][k] = 1.0;

            }

         }

      }

      // If required, generate V.

      if (wantv) {

                                //PZ: transform the collection of mirror-plane normal vectors into a single orthogonal transformation

         for (k = n-1; k >= 0; k--) {//PZ step back on each column

            if ((k < nrt) && ($ e[k] != 0.0)) {

               for (j = k+1; j < nu; j++) {//for all column to the right

                  Real t(0.0);

                  for (i = k+1; i < n; i++) {

                     t += $ V[i][k]*$ V[i][j];

                  }

                                                                        //substract the kth column from all righter...

                  t = t/$ V[k+1][k];

                  for (i = k+1; i < n; i++) {

                     $ V[i][j] -= t*$ V[i][k];

                  }

               }

            }

                                                //PZ write the kth. unit vector

            for (i = 0; i < n; i++) {

               $ V[i][k] = 0.0;

            }

            $ V[k][k] = 1.0;

         }

      }

      // Main iteration loop for the singular values.

    //PZ: at this point, if wantu=1 and wantv=1, then   Arg=U*D*V^T. 

                        //PZ: for all aplicable k, D[k][k]=s[k], D[k][k+1]=e[k]

      int pp = p-1;

      int iter = 0;

      Real eps = EPSILON(Real);

      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.

                     //                                                                action:

         // kase = 1     if s(p) and e[k-1] are negligible and k<p         Deflate negligible s(p).

         // kase = 2     if s(k) is negligible and k<p                     Split at negligible s(k).

         // kase = 3     if e[k-1] is negligible, k<p, and

         //              s(k), ..., s(p) are not negligible (qr step).     QR

         // kase = 4     if e(p-1) is negligible (convergence).            store singular value

         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;

               }

               Real t=(Real)( (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: {

               Real f($ e[p-2]);

               $ e[p-2] = 0.0;

               for (j = p-2; j >= k; j--) {

                  Real t( hypot_($ s[j],f));

                  Real cs($ s[j]/t);

                  Real 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: {

               Real f($ e[k-1]);

               $ e[k-1] = 0.0;

               for (j = k; j < p; j++) {

                  Real t(hypot_($ s[j],f));

                  Real cs( $ s[j]/t);

                  Real 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.

               Real scale = MAX(MAX(MAX(MAX(

                       ABS($ s[p-1]),ABS($ s[p-2])),ABS($ e[p-2])), 

                       ABS($ s[k])),ABS($ e[k]));

               Real sp = $ s[p-1]/scale;//PZ: calculating 1/scale would not include numeric overflow....?

               Real spm1 = $ s[p-2]/scale;

               Real epm1 = $ e[p-2]/scale;

               Real sk = $ s[k]/scale;

               Real ek = $ e[k]/scale;

               Real b = (Real)(((spm1 + sp)*(spm1 - sp) + epm1*epm1)/2.0);

               Real c = (sp*epm1)*(sp*epm1);

               Real 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);

               }

               Real f = (sk + sp)*(sk - sp) + shift;

               Real g = sk*ek;

               // Chase zeros.

               for (j = k; j < p-1; j++) {

                  Real t = hypot_(f,g);

                  Real cs = f/t;

                  Real 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] = (Real)($ 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 by bubble sort

               while (k < pp) {

                  if ($ s[k] >= $ s[k+1]) {

                     break;//nothing to do

                  }

                                                                        //PZ exchange s[k <> k+1], U[:,k<>k+1], V[:,k<>k+1], 

                  Real t = $ s[k];

                  $ s[k] = $ s[k+1];

                  $ s[k+1] = t;

                  if (wantv && (k < n-1)) {

                     for (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 (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;

         }

      }

   }

//   void member(getU) (JamaArray2(Real,m,minm) &A) 

   void member(getU) (JamaArray2(Real,m,nu) &A) //PZ

   {

//             int minm = MIN(m+1,n);

//                RESIZE2(Real,A,m,minm);

                RESIZE2(Real,A,m,nu);//PZ

//#ifdef __cplusplus

//          A = JamaArray2(Real,m,minm)(m, minm);

//#endif

          for (int i=0; i<m; i++)

                  for (int j=0; j<nu; j++)

                        $ A[i][j] = $ U[i][j];

   }

         //PZ: getLastColumnOfU

   void getLastColumnOfU(JamaArray1(Real,m) &u) {

           RESIZE1(Real,u,m);

                 if(m==n||($ s[nu-1])==0.0){

                   for(int i=0;i<m;i++) {

                                        $ u[i]= $ U[i][nu-1];

                         }                        

                 }else{

                         //m>n

                         //need to calculate one null vector of U

                         //return one normalized column of 1-U*U^T

                                JamaArray2(Real,m,m) A;

                                A=-U*(U^_T_);

                                for(int i=0;i<m;i++){

                                        $ A[i][i]+=1.0;

                                }

                                int M=m;

                                int N=n;

                                int NU=nu;

                                                //largest element in V_1

                                                Real max_=0.0;

                                                int maxi=-1;

                                                int maxj=-1;

                                                for(int i=0;i<M;i++){

                                                        for(int j=0;j<M;j++){

                                                                Real s=$ A[i][j];s*=s;

                                                                if(s>max_){

                                                                        max_=s;maxi=i;maxj=j;

                                                                }

                                                        }

                                                }

                                                //normalize its row

                                                Real l=0.0;

                                                        for(int j=0;j<M;j++){

                                                                Real &s=$ A[maxi][j];

                                                                l+=s*s;//hypot would be safe for extra large numbers near overflow limit...

                                                        }

                                                        l=1.0/sqrt(l);

                                                //put in V

                                                        for(int j=0;j<M;j++){

                                                                $ u[j]=l* $ A[maxi][j];

                                                        }

                 }

   }

   /* Return the right singular vectors */

   void member(getV) (JamaArray2(Real,n,n) &A) 

   {

           RESIZE2(Real,A,n,n);

             A = V;

   }

   void getLastColumnOfV(JamaArray1(Real,n) &v) {

           RESIZE1(Real,v,n);

           for(int i=0;i<n;i++) {

                $ v[i]= $ V[i][n-1];

           }                        

   }

   /** Return the one-dimensional array of singular values */

//   void member(getSingularValues) (JamaArray1(Real,minm) &x) 

   void member(getSingularValues) (JamaArray1(Real,nu) &x) //PZ

   {

                 RESIZE1(Real,x,nu);//PZ

//                  for (int j=0; j<minm; j++)

                  for (int j=0; j<nu; j++)//PZ

                           $ x[j] = $ s[j];

   }

   /** Return the diagonal matrix of singular values

   @return     S

   */

//   void member(getS) (JamaArray2(Real,n,n) &A) {

   void member(getS) (JamaArray2(Real,nu,n) &A) {//PZ

//                RESIZE2(Real,A,n,n);

                RESIZE2(Real,A,nu,n);//PZ

//#ifdef __cplusplus

//             A = JamaArray2(Real,n,n)(n,n);

//#endif

      for (int i = 0; i < nu; i++) {

         for (int j = 0; j < n; j++) {

            $ A[i][j] = 0.0;

         }

         $ A[i][i] = $ s[i];

      }

   }

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

   Real member(norm2) () {

      return $ s[0];

   }

   /** Two norm of condition number (max(S)/min(S)) */

   Real member(cond) () {

      return $ s[0]/$ s[MIN(m,n)-1];

   }

   /** Effective numerical matrix rank

   @return     Number of nonnegligible singular values.

   */

   int member(rank) () 

   {

      //Real eps = pow(2.0,-52.0); // Ez nagyon fugg a hasznalt tipustol!!! kell az epsilon konverzio

      Real eps = EPSILON(Real);

      Real tol = MAX(m,n)*$ s[0]*eps;

      int r = 0;

      for (int i = 0; i < s.DIM(); i++) {

         if ($ s[i] > tol) {

            r++;

         }

      }

      return r;

   }