/** 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;
   }