zpbtrf.f

          SUBROUTINE ZPBTRF( UPLO, N, KD, AB, LDAB, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          UPLO
          INTEGER            INFO, KD, LDAB, N
    *     ..
    *     .. Array Arguments ..
          COMPLEX*16         AB( LDAB, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  ZPBTRF computes the Cholesky factorization of a complex Hermitian
    *  positive definite band matrix A.
    *
    *  The factorization has the form
    *     A = U**H * U,  if UPLO = 'U', or
    *     A = L  * L**H,  if UPLO = 'L',
    *  where U is an upper triangular matrix and L is lower triangular.
    *
    *  Arguments
    *  =========
    *
    *  UPLO    (input) CHARACTER*1
    *          = 'U':  Upper triangle of A is stored;
    *          = 'L':  Lower triangle of A is stored.
    *
    *  N       (input) INTEGER
    *          The order of the matrix A.  N >= 0.
    *
    *  KD      (input) INTEGER
    *          The number of superdiagonals of the matrix A if UPLO = 'U',
    *          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    *
    *  AB      (input/output) COMPLEX*16 array, dimension (LDAB,N)
    *          On entry, the upper or lower triangle of the Hermitian band
    *          matrix A, stored in the first KD+1 rows of the array.  The
    *          j-th column of A is stored in the j-th column of the array AB
    *          as follows:
    *          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
    *
    *          On exit, if INFO = 0, the triangular factor U or L from the
    *          Cholesky factorization A = U**H*U or A = L*L**H of the band
    *          matrix A, in the same storage format as A.
    *
    *  LDAB    (input) INTEGER
    *          The leading dimension of the array AB.  LDAB >= KD+1.
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value
    *          > 0:  if INFO = i, the leading minor of order i is not
    *                positive definite, and the factorization could not be
    *                completed.
    *
    *  Further Details
    *  ===============
    *
    *  The band storage scheme is illustrated by the following example, when
    *  N = 6, KD = 2, and UPLO = 'U':
    *
    *  On entry:                       On exit:
    *
    *      *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
    *      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
    *     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
    *
    *  Similarly, if UPLO = 'L' the format of A is as follows:
    *
    *  On entry:                       On exit:
    *
    *     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
    *     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
    *     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *
    *
    *  Array elements marked * are not used by the routine.
    *
    *  Contributed by
    *  Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ONE, ZERO
          PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
          COMPLEX*16         CONE
          PARAMETER          ( CONE = ( 1.0D+0, 0.0D+0 ) )
          INTEGER            NBMAX, LDWORK
          PARAMETER          ( NBMAX = 32, LDWORK = NBMAX+1 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I, I2, I3, IB, II, J, JJ, NB
    *     ..
    *     .. Local Arrays ..
          COMPLEX*16         WORK( LDWORK, NBMAX )
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          INTEGER            ILAENV
          EXTERNAL           LSAME, ILAENV
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           XERBLA, ZGEMM, ZHERK, ZPBTF2, ZPOTF2, ZTRSM
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MIN
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
          IF( ( .NOT.LSAME( UPLO, 'U' ) ) .AND.
         $    ( .NOT.LSAME( UPLO, 'L' ) ) ) THEN
             INFO = -1
          ELSE IF( N.LT.0 ) THEN
             INFO = -2
          ELSE IF( KD.LT.0 ) THEN
             INFO = -3
          ELSE IF( LDAB.LT.KD+1 ) THEN
             INFO = -5
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'ZPBTRF', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 )
         $   RETURN
    *
    *     Determine the block size for this environment
    *
          NB = ILAENV( 1, 'ZPBTRF', UPLO, N, KD, -1, -1 )
    *
    *     The block size must not exceed the semi-bandwidth KD, and must not
    *     exceed the limit set by the size of the local array WORK.
    *
          NB = MIN( NB, NBMAX )
    *
          IF( NB.LE.1 .OR. NB.GT.KD ) THEN
    *
    *        Use unblocked code
    *
             CALL ZPBTF2( UPLO, N, KD, AB, LDAB, INFO )
          ELSE
    *
    *        Use blocked code
    *
             IF( LSAME( UPLO, 'U' ) ) THEN
    *
    *           Compute the Cholesky factorization of a Hermitian band
    *           matrix, given the upper triangle of the matrix in band
    *           storage.
    *
    *           Zero the upper triangle of the work array.
    *
                DO 20 J = 1, NB
                   DO 10 I = 1, J - 1
                      WORK( I, J ) = ZERO
       10          CONTINUE
       20       CONTINUE
    *
    *           Process the band matrix one diagonal block at a time.
    *
                DO 70 I = 1, N, NB
                   IB = MIN( NB, N-I+1 )
    *
    *              Factorize the diagonal block
    *
                   CALL ZPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II )
                   IF( II.NE.0 ) THEN
                      INFO = I + II - 1
                      GO TO 150
                   END IF
                   IF( I+IB.LE.N ) THEN
    *
    *                 Update the relevant part of the trailing submatrix.
    *                 If A11 denotes the diagonal block which has just been
    *                 factorized, then we need to update the remaining
    *                 blocks in the diagram:
    *
    *                    A11   A12   A13
    *                          A22   A23
    *                                A33
    *
    *                 The numbers of rows and columns in the partitioning
    *                 are IB, I2, I3 respectively. The blocks A12, A22 and
    *                 A23 are empty if IB = KD. The upper triangle of A13
    *                 lies outside the band.
    *
                      I2 = MIN( KD-IB, N-I-IB+1 )
                      I3 = MIN( IB, N-I-KD+1 )
    *
                      IF( I2.GT.0 ) THEN
    *
    *                    Update A12
    *
                         CALL ZTRSM( 'Left', 'Upper', 'Conjugate transpose',
         $                           'Non-unit', IB, I2, CONE,
         $                           AB( KD+1, I ), LDAB-1,
         $                           AB( KD+1-IB, I+IB ), LDAB-1 )
    *
    *                    Update A22
    *
                         CALL ZHERK( 'Upper', 'Conjugate transpose', I2, IB,
         $                           -ONE, AB( KD+1-IB, I+IB ), LDAB-1, ONE,
         $                           AB( KD+1, I+IB ), LDAB-1 )
                      END IF
    *
                      IF( I3.GT.0 ) THEN
    *
    *                    Copy the lower triangle of A13 into the work array.
    *
                         DO 40 JJ = 1, I3
                            DO 30 II = JJ, IB
                               WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 )
       30                   CONTINUE
       40                CONTINUE
    *
    *                    Update A13 (in the work array).
    *
                         CALL ZTRSM( 'Left', 'Upper', 'Conjugate transpose',
         $                           'Non-unit', IB, I3, CONE,
         $                           AB( KD+1, I ), LDAB-1, WORK, LDWORK )
    *
    *                    Update A23
    *
                         IF( I2.GT.0 )
         $                  CALL ZGEMM( 'Conjugate transpose',
         $                              'No transpose', I2, I3, IB, -CONE,
         $                              AB( KD+1-IB, I+IB ), LDAB-1, WORK,
         $                              LDWORK, CONE, AB( 1+IB, I+KD ),
         $                              LDAB-1 )
    *
    *                    Update A33
    *
                         CALL ZHERK( 'Upper', 'Conjugate transpose', I3, IB,
         $                           -ONE, WORK, LDWORK, ONE,
         $                           AB( KD+1, I+KD ), LDAB-1 )
    *
    *                    Copy the lower triangle of A13 back into place.
    *
                         DO 60 JJ = 1, I3
                            DO 50 II = JJ, IB
                               AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ )
       50                   CONTINUE
       60                CONTINUE
                      END IF
                   END IF
       70       CONTINUE
             ELSE
    *
    *           Compute the Cholesky factorization of a Hermitian band
    *           matrix, given the lower triangle of the matrix in band
    *           storage.
    *
    *           Zero the lower triangle of the work array.
    *
                DO 90 J = 1, NB
                   DO 80 I = J + 1, NB
                      WORK( I, J ) = ZERO
       80          CONTINUE
       90       CONTINUE
    *
    *           Process the band matrix one diagonal block at a time.
    *
                DO 140 I = 1, N, NB
                   IB = MIN( NB, N-I+1 )
    *
    *              Factorize the diagonal block
    *
                   CALL ZPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II )
                   IF( II.NE.0 ) THEN
                      INFO = I + II - 1
                      GO TO 150
                   END IF
                   IF( I+IB.LE.N ) THEN
    *
    *                 Update the relevant part of the trailing submatrix.
    *                 If A11 denotes the diagonal block which has just been
    *                 factorized, then we need to update the remaining
    *                 blocks in the diagram:
    *
    *                    A11
    *                    A21   A22
    *                    A31   A32   A33
    *
    *                 The numbers of rows and columns in the partitioning
    *                 are IB, I2, I3 respectively. The blocks A21, A22 and
    *                 A32 are empty if IB = KD. The lower triangle of A31
    *                 lies outside the band.
    *
                      I2 = MIN( KD-IB, N-I-IB+1 )
                      I3 = MIN( IB, N-I-KD+1 )
    *
                      IF( I2.GT.0 ) THEN
    *
    *                    Update A21
    *
                         CALL ZTRSM( 'Right', 'Lower',
         $                           'Conjugate transpose', 'Non-unit', I2,
         $                           IB, CONE, AB( 1, I ), LDAB-1,
         $                           AB( 1+IB, I ), LDAB-1 )
    *
    *                    Update A22
    *
                         CALL ZHERK( 'Lower', 'No transpose', I2, IB, -ONE,
         $                           AB( 1+IB, I ), LDAB-1, ONE,
         $                           AB( 1, I+IB ), LDAB-1 )
                      END IF
    *
                      IF( I3.GT.0 ) THEN
    *
    *                    Copy the upper triangle of A31 into the work array.
    *
                         DO 110 JJ = 1, IB
                            DO 100 II = 1, MIN( JJ, I3 )
                               WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 )
      100                   CONTINUE
      110                CONTINUE
    *
    *                    Update A31 (in the work array).
    *
                         CALL ZTRSM( 'Right', 'Lower',
         $                           'Conjugate transpose', 'Non-unit', I3,
         $                           IB, CONE, AB( 1, I ), LDAB-1, WORK,
         $                           LDWORK )
    *
    *                    Update A32
    *
                         IF( I2.GT.0 )
         $                  CALL ZGEMM( 'No transpose',
         $                              'Conjugate transpose', I3, I2, IB,
         $                              -CONE, WORK, LDWORK, AB( 1+IB, I ),
         $                              LDAB-1, CONE, AB( 1+KD-IB, I+IB ),
         $                              LDAB-1 )
    *
    *                    Update A33
    *
                         CALL ZHERK( 'Lower', 'No transpose', I3, IB, -ONE,
         $                           WORK, LDWORK, ONE, AB( 1, I+KD ),
         $                           LDAB-1 )
    *
    *                    Copy the upper triangle of A31 back into place.
    *
                         DO 130 JJ = 1, IB
                            DO 120 II = 1, MIN( JJ, I3 )
                               AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ )
      120                   CONTINUE
      130                CONTINUE
                      END IF
                   END IF
      140       CONTINUE
             END IF
          END IF
          RETURN
    *
      150 CONTINUE
          RETURN
    *
    *     End of ZPBTRF
    *
          END