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    1:       SUBROUTINE ZHETRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
    2: *
    3: *  -- LAPACK routine (version 3.2) --
    4: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    5: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    6: *     November 2006
    7: *
    8: *     .. Scalar Arguments ..
    9:       CHARACTER          UPLO
   10:       INTEGER            INFO, LDA, LWORK, N
   11: *     ..
   12: *     .. Array Arguments ..
   13:       INTEGER            IPIV( * )
   14:       COMPLEX*16         A( LDA, * ), WORK( * )
   15: *     ..
   16: *
   17: *  Purpose
   18: *  =======
   19: *
   20: *  ZHETRF computes the factorization of a complex Hermitian matrix A
   21: *  using the Bunch-Kaufman diagonal pivoting method.  The form of the
   22: *  factorization is
   23: *
   24: *     A = U*D*U**H  or  A = L*D*L**H
   25: *
   26: *  where U (or L) is a product of permutation and unit upper (lower)
   27: *  triangular matrices, and D is Hermitian 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) COMPLEX*16 array, dimension (LDA,N)
   43: *          On entry, the Hermitian 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) COMPLEX*16 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: *  INFO    (output) INTEGER
   75: *          = 0:  successful exit
   76: *          < 0:  if INFO = -i, the i-th argument had an illegal value
   77: *          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
   78: *                has been completed, but the block diagonal matrix D is
   79: *                exactly singular, and division by zero will occur if it
   80: *                is used to solve a system of equations.
   81: *
   82: *  Further Details
   83: *  ===============
   84: *
   85: *  If UPLO = 'U', then A = U*D*U', where
   86: *     U = P(n)*U(n)* ... *P(k)U(k)* ...,
   87: *  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
   88: *  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
   89: *  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
   90: *  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
   91: *  that if the diagonal block D(k) is of order s (s = 1 or 2), then
   92: *
   93: *             (   I    v    0   )   k-s
   94: *     U(k) =  (   0    I    0   )   s
   95: *             (   0    0    I   )   n-k
   96: *                k-s   s   n-k
   97: *
   98: *  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
   99: *  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  100: *  and A(k,k), and v overwrites A(1:k-2,k-1:k).
  101: *
  102: *  If UPLO = 'L', then A = L*D*L', where
  103: *     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
  104: *  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
  105: *  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  106: *  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  107: *  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
  108: *  that if the diagonal block D(k) is of order s (s = 1 or 2), then
  109: *
  110: *             (   I    0     0   )  k-1
  111: *     L(k) =  (   0    I     0   )  s
  112: *             (   0    v     I   )  n-k-s+1
  113: *                k-1   s  n-k-s+1
  114: *
  115: *  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
  116: *  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  117: *  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
  118: *
  119: *  =====================================================================
  120: *
  121: *     .. Local Scalars ..
  122:       LOGICAL            LQUERY, UPPER
  123:       INTEGER            IINFO, IWS, J, K, KB, LDWORK, LWKOPT, NB, NBMIN
  124: *     ..
  125: *     .. External Functions ..
  126:       LOGICAL            LSAME
  127:       INTEGER            ILAENV
  128:       EXTERNAL           LSAME, ILAENV
  129: *     ..
  130: *     .. External Subroutines ..
  131:       EXTERNAL           XERBLA, ZHETF2, ZLAHEF
  132: *     ..
  133: *     .. Intrinsic Functions ..
  134:       INTRINSIC          MAX
  135: *     ..
  136: *     .. Executable Statements ..
  137: *
  138: *     Test the input parameters.
  139: *
  140:       INFO = 0
  141:       UPPER = LSAME( UPLO, 'U' )
  142:       LQUERY = ( LWORK.EQ.-1 )
  143:       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
  144:          INFO = -1
  145:       ELSE IF( N.LT.0 ) THEN
  146:          INFO = -2
  147:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
  148:          INFO = -4
  149:       ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
  150:          INFO = -7
  151:       END IF
  152: *
  153:       IF( INFO.EQ.0 ) THEN
  154: *
  155: *        Determine the block size
  156: *
  157:          NB = ILAENV( 1, 'ZHETRF', UPLO, N, -1, -1, -1 )
  158:          LWKOPT = N*NB
  159:          WORK( 1 ) = LWKOPT
  160:       END IF
  161: *
  162:       IF( INFO.NE.0 ) THEN
  163:          CALL XERBLA( 'ZHETRF', -INFO )
  164:          RETURN
  165:       ELSE IF( LQUERY ) THEN
  166:          RETURN
  167:       END IF
  168: *
  169:       NBMIN = 2
  170:       LDWORK = N
  171:       IF( NB.GT.1 .AND. NB.LT.N ) THEN
  172:          IWS = LDWORK*NB
  173:          IF( LWORK.LT.IWS ) THEN
  174:             NB = MAX( LWORK / LDWORK, 1 )
  175:             NBMIN = MAX( 2, ILAENV( 2, 'ZHETRF', UPLO, N, -1, -1, -1 ) )
  176:          END IF
  177:       ELSE
  178:          IWS = 1
  179:       END IF
  180:       IF( NB.LT.NBMIN )
  181:      $   NB = N
  182: *
  183:       IF( UPPER ) THEN
  184: *
  185: *        Factorize A as U*D*U' using the upper triangle of A
  186: *
  187: *        K is the main loop index, decreasing from N to 1 in steps of
  188: *        KB, where KB is the number of columns factorized by ZLAHEF;
  189: *        KB is either NB or NB-1, or K for the last block
  190: *
  191:          K = N
  192:    10    CONTINUE
  193: *
  194: *        If K < 1, exit from loop
  195: *
  196:          IF( K.LT.1 )
  197:      $      GO TO 40
  198: *
  199:          IF( K.GT.NB ) THEN
  200: *
  201: *           Factorize columns k-kb+1:k of A and use blocked code to
  202: *           update columns 1:k-kb
  203: *
  204:             CALL ZLAHEF( UPLO, K, NB, KB, A, LDA, IPIV, WORK, N, IINFO )
  205:          ELSE
  206: *
  207: *           Use unblocked code to factorize columns 1:k of A
  208: *
  209:             CALL ZHETF2( UPLO, K, A, LDA, IPIV, IINFO )
  210:             KB = K
  211:          END IF
  212: *
  213: *        Set INFO on the first occurrence of a zero pivot
  214: *
  215:          IF( INFO.EQ.0 .AND. IINFO.GT.0 )
  216:      $      INFO = IINFO
  217: *
  218: *        Decrease K and return to the start of the main loop
  219: *
  220:          K = K - KB
  221:          GO TO 10
  222: *
  223:       ELSE
  224: *
  225: *        Factorize A as L*D*L' using the lower triangle of A
  226: *
  227: *        K is the main loop index, increasing from 1 to N in steps of
  228: *        KB, where KB is the number of columns factorized by ZLAHEF;
  229: *        KB is either NB or NB-1, or N-K+1 for the last block
  230: *
  231:          K = 1
  232:    20    CONTINUE
  233: *
  234: *        If K > N, exit from loop
  235: *
  236:          IF( K.GT.N )
  237:      $      GO TO 40
  238: *
  239:          IF( K.LE.N-NB ) THEN
  240: *
  241: *           Factorize columns k:k+kb-1 of A and use blocked code to
  242: *           update columns k+kb:n
  243: *
  244:             CALL ZLAHEF( UPLO, N-K+1, NB, KB, A( K, K ), LDA, IPIV( K ),
  245:      $                   WORK, N, IINFO )
  246:          ELSE
  247: *
  248: *           Use unblocked code to factorize columns k:n of A
  249: *
  250:             CALL ZHETF2( UPLO, N-K+1, A( K, K ), LDA, IPIV( K ), IINFO )
  251:             KB = N - K + 1
  252:          END IF
  253: *
  254: *        Set INFO on the first occurrence of a zero pivot
  255: *
  256:          IF( INFO.EQ.0 .AND. IINFO.GT.0 )
  257:      $      INFO = IINFO + K - 1
  258: *
  259: *        Adjust IPIV
  260: *
  261:          DO 30 J = K, K + KB - 1
  262:             IF( IPIV( J ).GT.0 ) THEN
  263:                IPIV( J ) = IPIV( J ) + K - 1
  264:             ELSE
  265:                IPIV( J ) = IPIV( J ) - K + 1
  266:             END IF
  267:    30    CONTINUE
  268: *
  269: *        Increase K and return to the start of the main loop
  270: *
  271:          K = K + KB
  272:          GO TO 20
  273: *
  274:       END IF
  275: *
  276:    40 CONTINUE
  277:       WORK( 1 ) = LWKOPT
  278:       RETURN
  279: *
  280: *     End of ZHETRF
  281: *
  282:       END