LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
cpstrf.f
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00001 *> \brief \b CPSTRF
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download CPSTRF + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE CPSTRF( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       REAL               TOL
00025 *       INTEGER            INFO, LDA, N, RANK
00026 *       CHARACTER          UPLO
00027 *       ..
00028 *       .. Array Arguments ..
00029 *       COMPLEX            A( LDA, * )
00030 *       REAL               WORK( 2*N )
00031 *       INTEGER            PIV( N )
00032 *       ..
00033 *  
00034 *
00035 *> \par Purpose:
00036 *  =============
00037 *>
00038 *> \verbatim
00039 *>
00040 *> CPSTRF computes the Cholesky factorization with complete
00041 *> pivoting of a complex Hermitian positive semidefinite matrix A.
00042 *>
00043 *> The factorization has the form
00044 *>    P**T * A * P = U**H * U ,  if UPLO = 'U',
00045 *>    P**T * A * P = L  * L**H,  if UPLO = 'L',
00046 *> where U is an upper triangular matrix and L is lower triangular, and
00047 *> P is stored as vector PIV.
00048 *>
00049 *> This algorithm does not attempt to check that A is positive
00050 *> semidefinite. This version of the algorithm calls level 3 BLAS.
00051 *> \endverbatim
00052 *
00053 *  Arguments:
00054 *  ==========
00055 *
00056 *> \param[in] UPLO
00057 *> \verbatim
00058 *>          UPLO is CHARACTER*1
00059 *>          Specifies whether the upper or lower triangular part of the
00060 *>          symmetric matrix A is stored.
00061 *>          = 'U':  Upper triangular
00062 *>          = 'L':  Lower triangular
00063 *> \endverbatim
00064 *>
00065 *> \param[in] N
00066 *> \verbatim
00067 *>          N is INTEGER
00068 *>          The order of the matrix A.  N >= 0.
00069 *> \endverbatim
00070 *>
00071 *> \param[in,out] A
00072 *> \verbatim
00073 *>          A is COMPLEX array, dimension (LDA,N)
00074 *>          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
00075 *>          n by n upper triangular part of A contains the upper
00076 *>          triangular part of the matrix A, and the strictly lower
00077 *>          triangular part of A is not referenced.  If UPLO = 'L', the
00078 *>          leading n by n lower triangular part of A contains the lower
00079 *>          triangular part of the matrix A, and the strictly upper
00080 *>          triangular part of A is not referenced.
00081 *>
00082 *>          On exit, if INFO = 0, the factor U or L from the Cholesky
00083 *>          factorization as above.
00084 *> \endverbatim
00085 *>
00086 *> \param[in] LDA
00087 *> \verbatim
00088 *>          LDA is INTEGER
00089 *>          The leading dimension of the array A.  LDA >= max(1,N).
00090 *> \endverbatim
00091 *>
00092 *> \param[out] PIV
00093 *> \verbatim
00094 *>          PIV is INTEGER array, dimension (N)
00095 *>          PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
00096 *> \endverbatim
00097 *>
00098 *> \param[out] RANK
00099 *> \verbatim
00100 *>          RANK is INTEGER
00101 *>          The rank of A given by the number of steps the algorithm
00102 *>          completed.
00103 *> \endverbatim
00104 *>
00105 *> \param[in] TOL
00106 *> \verbatim
00107 *>          TOL is REAL
00108 *>          User defined tolerance. If TOL < 0, then N*U*MAX( A(K,K) )
00109 *>          will be used. The algorithm terminates at the (K-1)st step
00110 *>          if the pivot <= TOL.
00111 *> \endverbatim
00112 *>
00113 *> \param[out] WORK
00114 *> \verbatim
00115 *>          WORK is REAL array, dimension (2*N)
00116 *>          Work space.
00117 *> \endverbatim
00118 *>
00119 *> \param[out] INFO
00120 *> \verbatim
00121 *>          INFO is INTEGER
00122 *>          < 0: If INFO = -K, the K-th argument had an illegal value,
00123 *>          = 0: algorithm completed successfully, and
00124 *>          > 0: the matrix A is either rank deficient with computed rank
00125 *>               as returned in RANK, or is indefinite.  See Section 7 of
00126 *>               LAPACK Working Note #161 for further information.
00127 *> \endverbatim
00128 *
00129 *  Authors:
00130 *  ========
00131 *
00132 *> \author Univ. of Tennessee 
00133 *> \author Univ. of California Berkeley 
00134 *> \author Univ. of Colorado Denver 
00135 *> \author NAG Ltd. 
00136 *
00137 *> \date November 2011
00138 *
00139 *> \ingroup complexOTHERcomputational
00140 *
00141 *  =====================================================================
00142       SUBROUTINE CPSTRF( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
00143 *
00144 *  -- LAPACK computational routine (version 3.4.0) --
00145 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00146 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00147 *     November 2011
00148 *
00149 *     .. Scalar Arguments ..
00150       REAL               TOL
00151       INTEGER            INFO, LDA, N, RANK
00152       CHARACTER          UPLO
00153 *     ..
00154 *     .. Array Arguments ..
00155       COMPLEX            A( LDA, * )
00156       REAL               WORK( 2*N )
00157       INTEGER            PIV( N )
00158 *     ..
00159 *
00160 *  =====================================================================
00161 *
00162 *     .. Parameters ..
00163       REAL               ONE, ZERO
00164       PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
00165       COMPLEX            CONE
00166       PARAMETER          ( CONE = ( 1.0E+0, 0.0E+0 ) )
00167 *     ..
00168 *     .. Local Scalars ..
00169       COMPLEX            CTEMP
00170       REAL               AJJ, SSTOP, STEMP
00171       INTEGER            I, ITEMP, J, JB, K, NB, PVT
00172       LOGICAL            UPPER
00173 *     ..
00174 *     .. External Functions ..
00175       REAL               SLAMCH
00176       INTEGER            ILAENV
00177       LOGICAL            LSAME, SISNAN
00178       EXTERNAL           SLAMCH, ILAENV, LSAME, SISNAN
00179 *     ..
00180 *     .. External Subroutines ..
00181       EXTERNAL           CGEMV, CHERK, CLACGV, CPSTF2, CSSCAL, CSWAP,
00182      $                   XERBLA
00183 *     ..
00184 *     .. Intrinsic Functions ..
00185       INTRINSIC          CONJG, MAX, MIN, REAL, SQRT, MAXLOC
00186 *     ..
00187 *     .. Executable Statements ..
00188 *
00189 *     Test the input parameters.
00190 *
00191       INFO = 0
00192       UPPER = LSAME( UPLO, 'U' )
00193       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00194          INFO = -1
00195       ELSE IF( N.LT.0 ) THEN
00196          INFO = -2
00197       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00198          INFO = -4
00199       END IF
00200       IF( INFO.NE.0 ) THEN
00201          CALL XERBLA( 'CPSTRF', -INFO )
00202          RETURN
00203       END IF
00204 *
00205 *     Quick return if possible
00206 *
00207       IF( N.EQ.0 )
00208      $   RETURN
00209 *
00210 *     Get block size
00211 *
00212       NB = ILAENV( 1, 'CPOTRF', UPLO, N, -1, -1, -1 )
00213       IF( NB.LE.1 .OR. NB.GE.N ) THEN
00214 *
00215 *        Use unblocked code
00216 *
00217          CALL CPSTF2( UPLO, N, A( 1, 1 ), LDA, PIV, RANK, TOL, WORK,
00218      $                INFO )
00219          GO TO 230
00220 *
00221       ELSE
00222 *
00223 *     Initialize PIV
00224 *
00225          DO 100 I = 1, N
00226             PIV( I ) = I
00227   100    CONTINUE
00228 *
00229 *     Compute stopping value
00230 *
00231          DO 110 I = 1, N
00232             WORK( I ) = REAL( A( I, I ) )
00233   110    CONTINUE
00234          PVT = MAXLOC( WORK( 1:N ), 1 )
00235          AJJ = REAL( A( PVT, PVT ) )
00236          IF( AJJ.EQ.ZERO.OR.SISNAN( AJJ ) ) THEN
00237             RANK = 0
00238             INFO = 1
00239             GO TO 230
00240          END IF
00241 *
00242 *     Compute stopping value if not supplied
00243 *
00244          IF( TOL.LT.ZERO ) THEN
00245             SSTOP = N * SLAMCH( 'Epsilon' ) * AJJ
00246          ELSE
00247             SSTOP = TOL
00248          END IF
00249 *
00250 *
00251          IF( UPPER ) THEN
00252 *
00253 *           Compute the Cholesky factorization P**T * A * P = U**H * U
00254 *
00255             DO 160 K = 1, N, NB
00256 *
00257 *              Account for last block not being NB wide
00258 *
00259                JB = MIN( NB, N-K+1 )
00260 *
00261 *              Set relevant part of first half of WORK to zero,
00262 *              holds dot products
00263 *
00264                DO 120 I = K, N
00265                   WORK( I ) = 0
00266   120          CONTINUE
00267 *
00268                DO 150 J = K, K + JB - 1
00269 *
00270 *              Find pivot, test for exit, else swap rows and columns
00271 *              Update dot products, compute possible pivots which are
00272 *              stored in the second half of WORK
00273 *
00274                   DO 130 I = J, N
00275 *
00276                      IF( J.GT.K ) THEN
00277                         WORK( I ) = WORK( I ) +
00278      $                              REAL( CONJG( A( J-1, I ) )*
00279      $                                    A( J-1, I ) )
00280                      END IF
00281                      WORK( N+I ) = REAL( A( I, I ) ) - WORK( I )
00282 *
00283   130             CONTINUE
00284 *
00285                   IF( J.GT.1 ) THEN
00286                      ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
00287                      PVT = ITEMP + J - 1
00288                      AJJ = WORK( N+PVT )
00289                      IF( AJJ.LE.SSTOP.OR.SISNAN( AJJ ) ) THEN
00290                         A( J, J ) = AJJ
00291                         GO TO 220
00292                      END IF
00293                   END IF
00294 *
00295                   IF( J.NE.PVT ) THEN
00296 *
00297 *                    Pivot OK, so can now swap pivot rows and columns
00298 *
00299                      A( PVT, PVT ) = A( J, J )
00300                      CALL CSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
00301                      IF( PVT.LT.N )
00302      $                  CALL CSWAP( N-PVT, A( J, PVT+1 ), LDA,
00303      $                              A( PVT, PVT+1 ), LDA )
00304                      DO 140 I = J + 1, PVT - 1
00305                         CTEMP = CONJG( A( J, I ) )
00306                         A( J, I ) = CONJG( A( I, PVT ) )
00307                         A( I, PVT ) = CTEMP
00308   140                CONTINUE
00309                      A( J, PVT ) = CONJG( A( J, PVT ) )
00310 *
00311 *                    Swap dot products and PIV
00312 *
00313                      STEMP = WORK( J )
00314                      WORK( J ) = WORK( PVT )
00315                      WORK( PVT ) = STEMP
00316                      ITEMP = PIV( PVT )
00317                      PIV( PVT ) = PIV( J )
00318                      PIV( J ) = ITEMP
00319                   END IF
00320 *
00321                   AJJ = SQRT( AJJ )
00322                   A( J, J ) = AJJ
00323 *
00324 *                 Compute elements J+1:N of row J.
00325 *
00326                   IF( J.LT.N ) THEN
00327                      CALL CLACGV( J-1, A( 1, J ), 1 )
00328                      CALL CGEMV( 'Trans', J-K, N-J, -CONE, A( K, J+1 ),
00329      $                           LDA, A( K, J ), 1, CONE, A( J, J+1 ),
00330      $                           LDA )
00331                      CALL CLACGV( J-1, A( 1, J ), 1 )
00332                      CALL CSSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
00333                   END IF
00334 *
00335   150          CONTINUE
00336 *
00337 *              Update trailing matrix, J already incremented
00338 *
00339                IF( K+JB.LE.N ) THEN
00340                   CALL CHERK( 'Upper', 'Conj Trans', N-J+1, JB, -ONE,
00341      $                        A( K, J ), LDA, ONE, A( J, J ), LDA )
00342                END IF
00343 *
00344   160       CONTINUE
00345 *
00346          ELSE
00347 *
00348 *        Compute the Cholesky factorization P**T * A * P = L * L**H
00349 *
00350             DO 210 K = 1, N, NB
00351 *
00352 *              Account for last block not being NB wide
00353 *
00354                JB = MIN( NB, N-K+1 )
00355 *
00356 *              Set relevant part of first half of WORK to zero,
00357 *              holds dot products
00358 *
00359                DO 170 I = K, N
00360                   WORK( I ) = 0
00361   170          CONTINUE
00362 *
00363                DO 200 J = K, K + JB - 1
00364 *
00365 *              Find pivot, test for exit, else swap rows and columns
00366 *              Update dot products, compute possible pivots which are
00367 *              stored in the second half of WORK
00368 *
00369                   DO 180 I = J, N
00370 *
00371                      IF( J.GT.K ) THEN
00372                         WORK( I ) = WORK( I ) +
00373      $                              REAL( CONJG( A( I, J-1 ) )*
00374      $                                    A( I, J-1 ) )
00375                      END IF
00376                      WORK( N+I ) = REAL( A( I, I ) ) - WORK( I )
00377 *
00378   180             CONTINUE
00379 *
00380                   IF( J.GT.1 ) THEN
00381                      ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
00382                      PVT = ITEMP + J - 1
00383                      AJJ = WORK( N+PVT )
00384                      IF( AJJ.LE.SSTOP.OR.SISNAN( AJJ ) ) THEN
00385                         A( J, J ) = AJJ
00386                         GO TO 220
00387                      END IF
00388                   END IF
00389 *
00390                   IF( J.NE.PVT ) THEN
00391 *
00392 *                    Pivot OK, so can now swap pivot rows and columns
00393 *
00394                      A( PVT, PVT ) = A( J, J )
00395                      CALL CSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ), LDA )
00396                      IF( PVT.LT.N )
00397      $                  CALL CSWAP( N-PVT, A( PVT+1, J ), 1,
00398      $                              A( PVT+1, PVT ), 1 )
00399                      DO 190 I = J + 1, PVT - 1
00400                         CTEMP = CONJG( A( I, J ) )
00401                         A( I, J ) = CONJG( A( PVT, I ) )
00402                         A( PVT, I ) = CTEMP
00403   190                CONTINUE
00404                      A( PVT, J ) = CONJG( A( PVT, J ) )
00405 *
00406 *                    Swap dot products and PIV
00407 *
00408                      STEMP = WORK( J )
00409                      WORK( J ) = WORK( PVT )
00410                      WORK( PVT ) = STEMP
00411                      ITEMP = PIV( PVT )
00412                      PIV( PVT ) = PIV( J )
00413                      PIV( J ) = ITEMP
00414                   END IF
00415 *
00416                   AJJ = SQRT( AJJ )
00417                   A( J, J ) = AJJ
00418 *
00419 *                 Compute elements J+1:N of column J.
00420 *
00421                   IF( J.LT.N ) THEN
00422                      CALL CLACGV( J-1, A( J, 1 ), LDA )
00423                      CALL CGEMV( 'No Trans', N-J, J-K, -CONE,
00424      $                           A( J+1, K ), LDA, A( J, K ), LDA, CONE,
00425      $                           A( J+1, J ), 1 )
00426                      CALL CLACGV( J-1, A( J, 1 ), LDA )
00427                      CALL CSSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
00428                   END IF
00429 *
00430   200          CONTINUE
00431 *
00432 *              Update trailing matrix, J already incremented
00433 *
00434                IF( K+JB.LE.N ) THEN
00435                   CALL CHERK( 'Lower', 'No Trans', N-J+1, JB, -ONE,
00436      $                        A( J, K ), LDA, ONE, A( J, J ), LDA )
00437                END IF
00438 *
00439   210       CONTINUE
00440 *
00441          END IF
00442       END IF
00443 *
00444 *     Ran to completion, A has full rank
00445 *
00446       RANK = N
00447 *
00448       GO TO 230
00449   220 CONTINUE
00450 *
00451 *     Rank is the number of steps completed.  Set INFO = 1 to signal
00452 *     that the factorization cannot be used to solve a system.
00453 *
00454       RANK = J - 1
00455       INFO = 1
00456 *
00457   230 CONTINUE
00458       RETURN
00459 *
00460 *     End of CPSTRF
00461 *
00462       END
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