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    1:       SUBROUTINE DSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
    2: *
    3: *  -- LAPACK routine (version 3.3.1) --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *  -- April 2011                                                      --
    7: *
    8: *     .. Scalar Arguments ..
    9:       CHARACTER          UPLO
   10:       INTEGER            INFO, LDA, LWORK, N
   11: *     ..
   12: *     .. Array Arguments ..
   13:       INTEGER            IPIV( * )
   14:       DOUBLE PRECISION   A( LDA, * ), WORK( * )
   15: *     ..
   16: *
   17: *  Purpose
   18: *  =======
   19: *
   20: *  DSYTRF computes the factorization of a real symmetric matrix A using
   21: *  the Bunch-Kaufman diagonal pivoting method.  The form of the
   22: *  factorization is
   23: *
   24: *     A = U*D*U**T  or  A = L*D*L**T
   25: *
   26: *  where U (or L) is a product of permutation and unit upper (lower)
   27: *  triangular matrices, and D is symmetric and block diagonal with
   28: *  1-by-1 and 2-by-2 diagonal blocks.
   29: *
   30: *  This is the blocked version of the algorithm, calling Level 3 BLAS.
   31: *
   32: *  Arguments
   33: *  =========
   34: *
   35: *  UPLO    (input) CHARACTER*1
   36: *          = 'U':  Upper triangle of A is stored;
   37: *          = 'L':  Lower triangle of A is stored.
   38: *
   39: *  N       (input) INTEGER
   40: *          The order of the matrix A.  N >= 0.
   41: *
   42: *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
   43: *          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
   44: *          N-by-N upper triangular part of A contains the upper
   45: *          triangular part of the matrix A, and the strictly lower
   46: *          triangular part of A is not referenced.  If UPLO = 'L', the
   47: *          leading N-by-N lower triangular part of A contains the lower
   48: *          triangular part of the matrix A, and the strictly upper
   49: *          triangular part of A is not referenced.
   50: *
   51: *          On exit, the block diagonal matrix D and the multipliers used
   52: *          to obtain the factor U or L (see below for further details).
   53: *
   54: *  LDA     (input) INTEGER
   55: *          The leading dimension of the array A.  LDA >= max(1,N).
   56: *
   57: *  IPIV    (output) INTEGER array, dimension (N)
   58: *          Details of the interchanges and the block structure of D.
   59: *          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
   60: *          interchanged and D(k,k) is a 1-by-1 diagonal block.
   61: *          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
   62: *          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
   63: *          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
   64: *          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
   65: *          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
   66: *
   67: *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
   68: *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   69: *
   70: *  LWORK   (input) INTEGER
   71: *          The length of WORK.  LWORK >=1.  For best performance
   72: *          LWORK >= N*NB, where NB is the block size returned by ILAENV.
   73: *
   74: *          If LWORK = -1, then a workspace query is assumed; the routine
   75: *          only calculates the optimal size of the WORK array, returns
   76: *          this value as the first entry of the WORK array, and no error
   77: *          message related to LWORK is issued by XERBLA.
   78: *
   79: *  INFO    (output) INTEGER
   80: *          = 0:  successful exit
   81: *          < 0:  if INFO = -i, the i-th argument had an illegal value
   82: *          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
   83: *                has been completed, but the block diagonal matrix D is
   84: *                exactly singular, and division by zero will occur if it
   85: *                is used to solve a system of equations.
   86: *
   87: *  Further Details
   88: *  ===============
   89: *
   90: *  If UPLO = 'U', then A = U*D*U**T, where
   91: *     U = P(n)*U(n)* ... *P(k)U(k)* ...,
   92: *  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
   93: *  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
   94: *  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
   95: *  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
   96: *  that if the diagonal block D(k) is of order s (s = 1 or 2), then
   97: *
   98: *             (   I    v    0   )   k-s
   99: *     U(k) =  (   0    I    0   )   s
  100: *             (   0    0    I   )   n-k
  101: *                k-s   s   n-k
  102: *
  103: *  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
  104: *  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  105: *  and A(k,k), and v overwrites A(1:k-2,k-1:k).
  106: *
  107: *  If UPLO = 'L', then A = L*D*L**T, where
  108: *     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
  109: *  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
  110: *  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  111: *  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  112: *  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
  113: *  that if the diagonal block D(k) is of order s (s = 1 or 2), then
  114: *
  115: *             (   I    0     0   )  k-1
  116: *     L(k) =  (   0    I     0   )  s
  117: *             (   0    v     I   )  n-k-s+1
  118: *                k-1   s  n-k-s+1
  119: *
  120: *  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
  121: *  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  122: *  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
  123: *
  124: *  =====================================================================
  125: *
  126: *     .. Local Scalars ..
  127:       LOGICAL            LQUERY, UPPER
  128:       INTEGER            IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
  129: *     ..
  130: *     .. External Functions ..
  131:       LOGICAL            LSAME
  132:       INTEGER            ILAENV
  133:       EXTERNAL           LSAME, ILAENV
  134: *     ..
  135: *     .. External Subroutines ..
  136:       EXTERNAL           DLASYF, DSYTF2, XERBLA
  137: *     ..
  138: *     .. Intrinsic Functions ..
  139:       INTRINSIC          MAX
  140: *     ..
  141: *     .. Executable Statements ..
  142: *
  143: *     Test the input parameters.
  144: *
  145:       INFO = 0
  146:       UPPER = LSAME( UPLO, 'U' )
  147:       LQUERY = ( LWORK.EQ.-1 )
  148:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
  149:          INFO = -1
  150:       ELSE IF( N.LT.0 ) THEN
  151:          INFO = -2
  152:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
  153:          INFO = -4
  154:       ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
  155:          INFO = -7
  156:       END IF
  157: *
  158:       IF( INFO.EQ.0 ) THEN
  159: *
  160: *        Determine the block size
  161: *
  162:          NB = ILAENV( 1, 'DSYTRF', UPLO, N, -1, -1, -1 )
  163:          LWKOPT = N*NB
  164:          WORK( 1 ) = LWKOPT
  165:       END IF
  166: *
  167:       IF( INFO.NE.0 ) THEN
  168:          CALL XERBLA( 'DSYTRF', -INFO )
  169:          RETURN
  170:       ELSE IF( LQUERY ) THEN
  171:          RETURN
  172:       END IF
  173: *
  174:       NBMIN = 2
  175:       LDWORK = N
  176:       IF( NB.GT.1 .AND. NB.LT.N ) THEN
  177:          IWS = LDWORK*NB
  178:          IF( LWORK.LT.IWS ) THEN
  179:             NB = MAX( LWORK / LDWORK, 1 )
  180:             NBMIN = MAX( 2, ILAENV( 2, 'DSYTRF', UPLO, N, -1, -1, -1 ) )
  181:          END IF
  182:       ELSE
  183:          IWS = 1
  184:       END IF
  185:       IF( NB.LT.NBMIN )
  186:      $   NB = N
  187: *
  188:       IF( UPPER ) THEN
  189: *
  190: *        Factorize A as U*D*U**T using the upper triangle of A
  191: *
  192: *        K is the main loop index, decreasing from N to 1 in steps of
  193: *        KB, where KB is the number of columns factorized by DLASYF;
  194: *        KB is either NB or NB-1, or K for the last block
  195: *
  196:          K = N
  197:    10    CONTINUE
  198: *
  199: *        If K < 1, exit from loop
  200: *
  201:          IF( K.LT.1 )
  202:      $      GO TO 40
  203: *
  204:          IF( K.GT.NB ) THEN
  205: *
  206: *           Factorize columns k-kb+1:k of A and use blocked code to
  207: *           update columns 1:k-kb
  208: *
  209:             CALL DLASYF( UPLO, K, NB, KB, A, LDA, IPIV, WORK, LDWORK,
  210:      $                   IINFO )
  211:          ELSE
  212: *
  213: *           Use unblocked code to factorize columns 1:k of A
  214: *
  215:             CALL DSYTF2( UPLO, K, A, LDA, IPIV, IINFO )
  216:             KB = K
  217:          END IF
  218: *
  219: *        Set INFO on the first occurrence of a zero pivot
  220: *
  221:          IF( INFO.EQ.0 .AND. IINFO.GT.0 )
  222:      $      INFO = IINFO
  223: *
  224: *        Decrease K and return to the start of the main loop
  225: *
  226:          K = K - KB
  227:          GO TO 10
  228: *
  229:       ELSE
  230: *
  231: *        Factorize A as L*D*L**T using the lower triangle of A
  232: *
  233: *        K is the main loop index, increasing from 1 to N in steps of
  234: *        KB, where KB is the number of columns factorized by DLASYF;
  235: *        KB is either NB or NB-1, or N-K+1 for the last block
  236: *
  237:          K = 1
  238:    20    CONTINUE
  239: *
  240: *        If K > N, exit from loop
  241: *
  242:          IF( K.GT.N )
  243:      $      GO TO 40
  244: *
  245:          IF( K.LE.N-NB ) THEN
  246: *
  247: *           Factorize columns k:k+kb-1 of A and use blocked code to
  248: *           update columns k+kb:n
  249: *
  250:             CALL DLASYF( UPLO, N-K+1, NB, KB, A( K, K ), LDA, IPIV( K ),
  251:      $                   WORK, LDWORK, IINFO )
  252:          ELSE
  253: *
  254: *           Use unblocked code to factorize columns k:n of A
  255: *
  256:             CALL DSYTF2( UPLO, N-K+1, A( K, K ), LDA, IPIV( K ), IINFO )
  257:             KB = N - K + 1
  258:          END IF
  259: *
  260: *        Set INFO on the first occurrence of a zero pivot
  261: *
  262:          IF( INFO.EQ.0 .AND. IINFO.GT.0 )
  263:      $      INFO = IINFO + K - 1
  264: *
  265: *        Adjust IPIV
  266: *
  267:          DO 30 J = K, K + KB - 1
  268:             IF( IPIV( J ).GT.0 ) THEN
  269:                IPIV( J ) = IPIV( J ) + K - 1
  270:             ELSE
  271:                IPIV( J ) = IPIV( J ) - K + 1
  272:             END IF
  273:    30    CONTINUE
  274: *
  275: *        Increase K and return to the start of the main loop
  276: *
  277:          K = K + KB
  278:          GO TO 20
  279: *
  280:       END IF
  281: *
  282:    40 CONTINUE
  283:       WORK( 1 ) = LWKOPT
  284:       RETURN
  285: *
  286: *     End of DSYTRF
  287: *
  288:       END