/* ========================================================================= */

 /* === AMD_aat ============================================================= */

 /* ========================================================================= */

 /* ------------------------------------------------------------------------- */

 /* AMD, Copyright (c) Timothy A. Davis,                                      */

 /* Patrick R. Amestoy, and Iain S. Duff.  See ../README.txt for License.     */

 /* email: davis at cise.ufl.edu    CISE Department, Univ. of Florida.        */

 /* web: http://www.cise.ufl.edu/research/sparse/amd                          */

 /* ------------------------------------------------------------------------- */

 /* AMD_aat:  compute the symmetry of the pattern of A, and count the number of

  * nonzeros each column of A+A' (excluding the diagonal).  Assumes the input

  * matrix has no errors, with sorted columns and no duplicates

  * (AMD_valid (n, n, Ap, Ai) must be AMD_OK, but this condition is not

  * checked).

  */

 #include "amd_internal.h"

 GLOBAL size_t AMD_aat   /* returns nz in A+A' */

 (

     Int n,

     const Int Ap [ ],

     const Int Ai [ ],

     Int Len [ ],        /* Len [j]: length of column j of A+A', excl diagonal*/

     Int Tp [ ],         /* workspace of size n */

     double Info [ ]

 )

 {

     Int p1, p2, p, i, j, pj, pj2, k, nzdiag, nzboth, nz ;

     double sym ;

     size_t nzaat ;

 #ifndef NDEBUG

     AMD_debug_init ("AMD AAT") ;

     for (k = 0 ; k < n ; k++) Tp [k] = EMPTY ;

     ASSERT (AMD_valid (n, n, Ap, Ai) == AMD_OK) ;

 #endif

     if (Info != (double *) NULL)

     {

         /* clear the Info array, if it exists */

         for (i = 0 ; i < AMD_INFO ; i++)

         {

             Info [i] = EMPTY ;

         }

         Info [AMD_STATUS] = AMD_OK ;

     }

     for (k = 0 ; k < n ; k++)

     {

         Len [k] = 0 ;

     }

     nzdiag = 0 ;

     nzboth = 0 ;

     nz = Ap [n] ;

     for (k = 0 ; k < n ; k++)

     {

         p1 = Ap [k] ;

         p2 = Ap [k+1] ;

         AMD_DEBUG2 (("\nAAT Column: "ID" p1: "ID" p2: "ID"\n", k, p1, p2)) ;

         /* construct A+A' */

         for (p = p1 ; p < p2 ; )

         {

             /* scan the upper triangular part of A */

             j = Ai [p] ;

             if (j < k)

             {

                 /* entry A (j,k) is in the strictly upper triangular part,

                  * add both A (j,k) and A (k,j) to the matrix A+A' */

                 Len [j]++ ;

                 Len [k]++ ;

                 AMD_DEBUG3 (("    upper ("ID","ID") ("ID","ID")\n", j,k, k,j));

                 p++ ;

             }

             else if (j == k)

             {

                 /* skip the diagonal */

                 p++ ;

                 nzdiag++ ;

                 break ;

             }

             else /* j > k */

             {

                 /* first entry below the diagonal */

                 break ;

             }

             /* scan lower triangular part of A, in column j until reaching

              * row k.  Start where last scan left off. */

             ASSERT (Tp [j] != EMPTY) ;

             ASSERT (Ap [j] <= Tp [j] && Tp [j] <= Ap [j+1]) ;

             pj2 = Ap [j+1] ;

             for (pj = Tp [j] ; pj < pj2 ; )

             {

                 i = Ai [pj] ;

                 if (i < k)

                 {

                     /* A (i,j) is only in the lower part, not in upper.

                      * add both A (i,j) and A (j,i) to the matrix A+A' */

                     Len [i]++ ;

                     Len [j]++ ;

                     AMD_DEBUG3 (("    lower ("ID","ID") ("ID","ID")\n",

                         i,j, j,i)) ;

                     pj++ ;

                 }

                 else if (i == k)

                 {

                     /* entry A (k,j) in lower part and A (j,k) in upper */

                     pj++ ;

                     nzboth++ ;

                     break ;

                 }

                 else /* i > k */

                 {

                     /* consider this entry later, when k advances to i */

                     break ;

                 }

             }

             Tp [j] = pj ;

         }

         /* Tp [k] points to the entry just below the diagonal in column k */

         Tp [k] = p ;

     }

     /* clean up, for remaining mismatched entries */

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

     {

         for (pj = Tp [j] ; pj < Ap [j+1] ; pj++)

         {

             i = Ai [pj] ;

             /* A (i,j) is only in the lower part, not in upper.

              * add both A (i,j) and A (j,i) to the matrix A+A' */

             Len [i]++ ;

             Len [j]++ ;

             AMD_DEBUG3 (("    lower cleanup ("ID","ID") ("ID","ID")\n",

                 i,j, j,i)) ;

         }

     }

     /* --------------------------------------------------------------------- */

     /* compute the symmetry of the nonzero pattern of A */

     /* --------------------------------------------------------------------- */

     /* Given a matrix A, the symmetry of A is:

      *  B = tril (spones (A), -1) + triu (spones (A), 1) ;

      *  sym = nnz (B & B') / nnz (B) ;

      *  or 1 if nnz (B) is zero.

      */

     if (nz == nzdiag)

     {

         sym = 1 ;

     }

     else

     {

         sym = (2 * (double) nzboth) / ((double) (nz - nzdiag)) ;

     }

     nzaat = 0 ;

     for (k = 0 ; k < n ; k++)

     {

         nzaat += Len [k] ;

     }

     AMD_DEBUG1 (("AMD nz in A+A', excluding diagonal (nzaat) = %g\n",

         (double) nzaat)) ;

     AMD_DEBUG1 (("   nzboth: "ID" nz: "ID" nzdiag: "ID" symmetry: %g\n",

                 nzboth, nz, nzdiag, sym)) ;

     if (Info != (double *) NULL)

     {

         Info [AMD_STATUS] = AMD_OK ;

         Info [AMD_N] = n ;

         Info [AMD_NZ] = nz ;

         Info [AMD_SYMMETRY] = sym ;         /* symmetry of pattern of A */

         Info [AMD_NZDIAG] = nzdiag ;        /* nonzeros on diagonal of A */

         Info [AMD_NZ_A_PLUS_AT] = nzaat ;   /* nonzeros in A+A' */

     }

     return (nzaat) ;

 }