1
       2
       3
       4
       5
       6
       7
       8
       9
      10
      11
      12
      13
      14
      15
      16
      17
      18
      19
      20
      21
      22
      23
      24
      25
      26
      27
      28
      29
      30
      31
      32
      33
      34
      35
      36
      37
      38
      39
      40
      41
      42
      43
      44
      45
      46
      47
      48
      49
      50
      51
      52
      53
      54
      55
      56
      57
      58
      59
      60
      61
      62
      63
      64
      65
      66
      67
      68
      69
      70
      71
      72
      73
      74
      75
      76
      77
      78
      79
      80
      81
      82
      83
      84
      85
      86
      87
      88
      89
      90
      91
      92
      93
      94
      95
      96
      97
      98
      99
     100
     101
     102
     103
     104
     105
     106
     107
     108
     109
     110
     111
     112
     113
     114
     115
     116
     117
     118
     119
     120
     121
     122
     123
     124
     125
     126
     127
     128
     129
     130
     131
     132
     133
     134
     135
     136
     137
     138
     139
     140
     141
     142
     143
     144
     145
     146
     147
     148
     149
     150
     151
     152
     153
     154
     155
     156
     157
     158
     159
     160
     161
     162
     163
     164
     165
     166
     167
     168
     169
     170
     171
     172
     173
     174
     175
     176
     177
     178
     179
     180
     181
     182
     183
     184
     185
     186
     187
     188
     189
     190
     191
     192
     193
     194
     195
     196
     197
     198
     199
     200
     201
     202
     203
     204
     205
     206
     207
     208
     209
     210
     211
     212
     213
     214
     215
     216
     217
     218
     219
     220
     221
     222
     223
     224
     225
     226
     227
     228
     229
     230
     231
     232
     233
     234
     235
     236
     237
     238
     239
     240
     241
     242
     243
     244
     245
     246
     247
     248
     249
     250
     251
     252
     253
     254
     255
     256
     257
     258
     259
     260
     261
     262
     263
     264
     265
     266
     267
     268
     269
     270
     271
     272
     273
     274
     275
     276
     277
     278
     279
     280
     281
     282
     283
     284
     285
     286
     287
     288
     289
     290
     291
     292
     293
     294
     295
     296
     297
     298
     299
     300
     301
     302
     303
     304
     305
     306
     307
     308
      SUBROUTINE SPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
*
*  -- LAPACK PROTOTYPE routine (version 3.2.2) --
*     Craig Lucas, University of Manchester / NAG Ltd.
*     October, 2008
*
*     .. Scalar Arguments ..
      REAL               TOL
      INTEGER            INFO, LDA, N, RANK
      CHARACTER          UPLO
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), WORK( 2*N )
      INTEGER            PIV( N )
*     ..
*
*  Purpose
*  =======
*
*  SPSTF2 computes the Cholesky factorization with complete
*  pivoting of a real symmetric positive semidefinite matrix A.
*
*  The factorization has the form
*     P**T * A * P = U**T * U ,  if UPLO = 'U',
*     P**T * A * P = L  * L**T,  if UPLO = 'L',
*  where U is an upper triangular matrix and L is lower triangular, and
*  P is stored as vector PIV.
*
*  This algorithm does not attempt to check that A is positive
*  semidefinite. This version of the algorithm calls level 2 BLAS.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the upper or lower triangular part of the
*          symmetric matrix A is stored.
*          = 'U':  Upper triangular
*          = 'L':  Lower triangular
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) REAL array, dimension (LDA,N)
*          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*          n by n upper triangular part of A contains the upper
*          triangular part of the matrix A, and the strictly lower
*          triangular part of A is not referenced.  If UPLO = 'L', the
*          leading n by n lower triangular part of A contains the lower
*          triangular part of the matrix A, and the strictly upper
*          triangular part of A is not referenced.
*
*          On exit, if INFO = 0, the factor U or L from the Cholesky
*          factorization as above.
*
*  PIV     (output) INTEGER array, dimension (N)
*          PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
*
*  RANK    (output) INTEGER
*          The rank of A given by the number of steps the algorithm
*          completed.
*
*  TOL     (input) REAL
*          User defined tolerance. If TOL < 0, then N*U*MAX( A( K,K ) )
*          will be used. The algorithm terminates at the (K-1)st step
*          if the pivot <= TOL.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  WORK    (workspace) REAL array, dimension (2*N)
*          Work space.
*
*  INFO    (output) INTEGER
*          < 0: If INFO = -K, the K-th argument had an illegal value,
*          = 0: algorithm completed successfully, and
*          > 0: the matrix A is either rank deficient with computed rank
*               as returned in RANK, or is indefinite.  See Section 7 of
*               LAPACK Working Note #161 for further information.
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ONE, ZERO
      PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
*     ..
*     .. Local Scalars ..
      REAL               AJJ, SSTOP, STEMP
      INTEGER            I, ITEMP, J, PVT
      LOGICAL            UPPER
*     ..
*     .. External Functions ..
      REAL               SLAMCH
      LOGICAL            LSAME, SISNAN
      EXTERNAL           SLAMCH, LSAME, SISNAN
*     ..
*     .. External Subroutines ..
      EXTERNAL           SGEMV, SSCAL, SSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAXSQRTMAXLOC
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF.NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'SPSTF2'-INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Initialize PIV
*
      DO 100 I = 1, N
         PIV( I ) = I
  100 CONTINUE
*
*     Compute stopping value
*
      PVT = 1
      AJJ = A( PVT, PVT )
      DO I = 2, N
         IF( A( I, I ).GT.AJJ ) THEN
            PVT = I
            AJJ = A( PVT, PVT )
         END IF
      END DO
      IF( AJJ.EQ.ZERO.OR.SISNAN( AJJ ) ) THEN
         RANK = 0
         INFO = 1
         GO TO 170
      END IF
*
*     Compute stopping value if not supplied
*
      IF( TOL.LT.ZERO ) THEN
         SSTOP = N * SLAMCH( 'Epsilon' ) * AJJ
      ELSE
         SSTOP = TOL
      END IF
*
*     Set first half of WORK to zero, holds dot products
*
      DO 110 I = 1, N
         WORK( I ) = 0
  110 CONTINUE
*
      IF( UPPER ) THEN
*
*        Compute the Cholesky factorization P**T * A * P = U**T * U
*
         DO 130 J = 1, N
*
*        Find pivot, test for exit, else swap rows and columns
*        Update dot products, compute possible pivots which are
*        stored in the second half of WORK
*
            DO 120 I = J, N
*
               IF( J.GT.1 ) THEN
                  WORK( I ) = WORK( I ) + A( J-1, I )**2
               END IF
               WORK( N+I ) = A( I, I ) - WORK( I )
*
  120       CONTINUE
*
            IF( J.GT.1 ) THEN
               ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
               PVT = ITEMP + J - 1
               AJJ = WORK( N+PVT )
               IF( AJJ.LE.SSTOP.OR.SISNAN( AJJ ) ) THEN
                  A( J, J ) = AJJ
                  GO TO 160
               END IF
            END IF
*
            IF( J.NE.PVT ) THEN
*
*              Pivot OK, so can now swap pivot rows and columns
*
               A( PVT, PVT ) = A( J, J )
               CALL SSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
               IF( PVT.LT.N )
     $            CALL SSWAP( N-PVT, A( J, PVT+1 ), LDA,
     $                        A( PVT, PVT+1 ), LDA )
               CALL SSWAP( PVT-J-1, A( J, J+1 ), LDA, A( J+1, PVT ), 1 )
*
*              Swap dot products and PIV
*
               STEMP = WORK( J )
               WORK( J ) = WORK( PVT )
               WORK( PVT ) = STEMP
               ITEMP = PIV( PVT )
               PIV( PVT ) = PIV( J )
               PIV( J ) = ITEMP
            END IF
*
            AJJ = SQRT( AJJ )
            A( J, J ) = AJJ
*
*           Compute elements J+1:N of row J
*
            IF( J.LT.N ) THEN
               CALL SGEMV( 'Trans', J-1, N-J, -ONE, A( 1, J+1 ), LDA,
     $                     A( 1, J ), 1, ONE, A( J, J+1 ), LDA )
               CALL SSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
            END IF
*
  130    CONTINUE
*
      ELSE
*
*        Compute the Cholesky factorization P**T * A * P = L * L**T
*
         DO 150 J = 1, N
*
*        Find pivot, test for exit, else swap rows and columns
*        Update dot products, compute possible pivots which are
*        stored in the second half of WORK
*
            DO 140 I = J, N
*
               IF( J.GT.1 ) THEN
                  WORK( I ) = WORK( I ) + A( I, J-1 )**2
               END IF
               WORK( N+I ) = A( I, I ) - WORK( I )
*
  140       CONTINUE
*
            IF( J.GT.1 ) THEN
               ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
               PVT = ITEMP + J - 1
               AJJ = WORK( N+PVT )
               IF( AJJ.LE.SSTOP.OR.SISNAN( AJJ ) ) THEN
                  A( J, J ) = AJJ
                  GO TO 160
               END IF
            END IF
*
            IF( J.NE.PVT ) THEN
*
*              Pivot OK, so can now swap pivot rows and columns
*
               A( PVT, PVT ) = A( J, J )
               CALL SSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ), LDA )
               IF( PVT.LT.N )
     $            CALL SSWAP( N-PVT, A( PVT+1, J ), 1, A( PVT+1, PVT ),
     $                        1 )
               CALL SSWAP( PVT-J-1, A( J+1, J ), 1, A( PVT, J+1 ), LDA )
*
*              Swap dot products and PIV
*
               STEMP = WORK( J )
               WORK( J ) = WORK( PVT )
               WORK( PVT ) = STEMP
               ITEMP = PIV( PVT )
               PIV( PVT ) = PIV( J )
               PIV( J ) = ITEMP
            END IF
*
            AJJ = SQRT( AJJ )
            A( J, J ) = AJJ
*
*           Compute elements J+1:N of column J
*
            IF( J.LT.N ) THEN
               CALL SGEMV( 'No Trans', N-J, J-1-ONE, A( J+11 ), LDA,
     $                     A( J, 1 ), LDA, ONE, A( J+1, J ), 1 )
               CALL SSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
            END IF
*
  150    CONTINUE
*
      END IF
*
*     Ran to completion, A has full rank
*
      RANK = N
*
      GO TO 170
  160 CONTINUE
*
*     Rank is number of steps completed.  Set INFO = 1 to signal
*     that the factorization cannot be used to solve a system.
*
      RANK = J - 1
      INFO = 1
*
  170 CONTINUE
      RETURN
*
*     End of SPSTF2
*
      END