cgbtrf.f

          SUBROUTINE CGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            INFO, KL, KU, LDAB, M, N
    *     ..
    *     .. Array Arguments ..
          INTEGER            IPIV( * )
          COMPLEX            AB( LDAB, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  CGBTRF computes an LU factorization of a complex m-by-n band matrix A
    *  using partial pivoting with row interchanges.
    *
    *  This is the blocked version of the algorithm, calling Level 3 BLAS.
    *
    *  Arguments
    *  =========
    *
    *  M       (input) INTEGER
    *          The number of rows of the matrix A.  M >= 0.
    *
    *  N       (input) INTEGER
    *          The number of columns of the matrix A.  N >= 0.
    *
    *  KL      (input) INTEGER
    *          The number of subdiagonals within the band of A.  KL >= 0.
    *
    *  KU      (input) INTEGER
    *          The number of superdiagonals within the band of A.  KU >= 0.
    *
    *  AB      (input/output) COMPLEX array, dimension (LDAB,N)
    *          On entry, the matrix A in band storage, in rows KL+1 to
    *          2*KL+KU+1; rows 1 to KL of the array need not be set.
    *          The j-th column of A is stored in the j-th column of the
    *          array AB as follows:
    *          AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
    *
    *          On exit, details of the factorization: U is stored as an
    *          upper triangular band matrix with KL+KU superdiagonals in
    *          rows 1 to KL+KU+1, and the multipliers used during the
    *          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
    *          See below for further details.
    *
    *  LDAB    (input) INTEGER
    *          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
    *
    *  IPIV    (output) INTEGER array, dimension (min(M,N))
    *          The pivot indices; for 1 <= i <= min(M,N), row i of the
    *          matrix was interchanged with row IPIV(i).
    *
    *  INFO    (output) INTEGER
    *          = 0: successful exit
    *          < 0: if INFO = -i, the i-th argument had an illegal value
    *          > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
    *               has been completed, but the factor U is exactly
    *               singular, and division by zero will occur if it is used
    *               to solve a system of equations.
    *
    *  Further Details
    *  ===============
    *
    *  The band storage scheme is illustrated by the following example, when
    *  M = N = 6, KL = 2, KU = 1:
    *
    *  On entry:                       On exit:
    *
    *      *    *    *    +    +    +       *    *    *   u14  u25  u36
    *      *    *    +    +    +    +       *    *   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
    *     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
    *     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
    *
    *  Array elements marked * are not used by the routine; elements marked
    *  + need not be set on entry, but are required by the routine to store
    *  elements of U because of fill-in resulting from the row interchanges.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          COMPLEX            ONE, ZERO
          PARAMETER          ( ONE = ( 1.0E+0, 0.0E+0 ),
         $                   ZERO = ( 0.0E+0, 0.0E+0 ) )
          INTEGER            NBMAX, LDWORK
          PARAMETER          ( NBMAX = 64, LDWORK = NBMAX+1 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            I, I2, I3, II, IP, J, J2, J3, JB, JJ, JM, JP,
         $                   JU, K2, KM, KV, NB, NW
          COMPLEX            TEMP
    *     ..
    *     .. Local Arrays ..
          COMPLEX            WORK13( LDWORK, NBMAX ),
         $                   WORK31( LDWORK, NBMAX )
    *     ..
    *     .. External Functions ..
          INTEGER            ICAMAX, ILAENV
          EXTERNAL           ICAMAX, ILAENV
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           CCOPY, CGBTF2, CGEMM, CGERU, CLASWP, CSCAL,
         $                   CSWAP, CTRSM, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, MIN
    *     ..
    *     .. Executable Statements ..
    *
    *     KV is the number of superdiagonals in the factor U, allowing for
    *     fill-in
    *
          KV = KU + KL
    *
    *     Test the input parameters.
    *
          INFO = 0
          IF( M.LT.0 ) THEN
             INFO = -1
          ELSE IF( N.LT.0 ) THEN
             INFO = -2
          ELSE IF( KL.LT.0 ) THEN
             INFO = -3
          ELSE IF( KU.LT.0 ) THEN
             INFO = -4
          ELSE IF( LDAB.LT.KL+KV+1 ) THEN
             INFO = -6
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'CGBTRF', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( M.EQ.0 .OR. N.EQ.0 )
         $   RETURN
    *
    *     Determine the block size for this environment
    *
          NB = ILAENV( 1, 'CGBTRF', ' ', M, N, KL, KU )
    *
    *     The block size must not exceed the limit set by the size of the
    *     local arrays WORK13 and WORK31.
    *
          NB = MIN( NB, NBMAX )
    *
          IF( NB.LE.1 .OR. NB.GT.KL ) THEN
    *
    *        Use unblocked code
    *
             CALL CGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
          ELSE
    *
    *        Use blocked code
    *
    *        Zero the superdiagonal elements of the work array WORK13
    *
             DO 20 J = 1, NB
                DO 10 I = 1, J - 1
                   WORK13( I, J ) = ZERO
       10       CONTINUE
       20    CONTINUE
    *
    *        Zero the subdiagonal elements of the work array WORK31
    *
             DO 40 J = 1, NB
                DO 30 I = J + 1, NB
                   WORK31( I, J ) = ZERO
       30       CONTINUE
       40    CONTINUE
    *
    *        Gaussian elimination with partial pivoting
    *
    *        Set fill-in elements in columns KU+2 to KV to zero
    *
             DO 60 J = KU + 2, MIN( KV, N )
                DO 50 I = KV - J + 2, KL
                   AB( I, J ) = ZERO
       50       CONTINUE
       60    CONTINUE
    *
    *        JU is the index of the last column affected by the current
    *        stage of the factorization
    *
             JU = 1
    *
             DO 180 J = 1, MIN( M, N ), NB
                JB = MIN( NB, MIN( M, N )-J+1 )
    *
    *           The active part of the matrix is partitioned
    *
    *              A11   A12   A13
    *              A21   A22   A23
    *              A31   A32   A33
    *
    *           Here A11, A21 and A31 denote the current block of JB columns
    *           which is about to be factorized. The number of rows in the
    *           partitioning are JB, I2, I3 respectively, and the numbers
    *           of columns are JB, J2, J3. The superdiagonal elements of A13
    *           and the subdiagonal elements of A31 lie outside the band.
    *
                I2 = MIN( KL-JB, M-J-JB+1 )
                I3 = MIN( JB, M-J-KL+1 )
    *
    *           J2 and J3 are computed after JU has been updated.
    *
    *           Factorize the current block of JB columns
    *
                DO 80 JJ = J, J + JB - 1
    *
    *              Set fill-in elements in column JJ+KV to zero
    *
                   IF( JJ+KV.LE.N ) THEN
                      DO 70 I = 1, KL
                         AB( I, JJ+KV ) = ZERO
       70             CONTINUE
                   END IF
    *
    *              Find pivot and test for singularity. KM is the number of
    *              subdiagonal elements in the current column.
    *
                   KM = MIN( KL, M-JJ )
                   JP = ICAMAX( KM+1, AB( KV+1, JJ ), 1 )
                   IPIV( JJ ) = JP + JJ - J
                   IF( AB( KV+JP, JJ ).NE.ZERO ) THEN
                      JU = MAX( JU, MIN( JJ+KU+JP-1, N ) )
                      IF( JP.NE.1 ) THEN
    *
    *                    Apply interchange to columns J to J+JB-1
    *
                         IF( JP+JJ-1.LT.J+KL ) THEN
    *
                            CALL CSWAP( JB, AB( KV+1+JJ-J, J ), LDAB-1,
         $                              AB( KV+JP+JJ-J, J ), LDAB-1 )
                         ELSE
    *
    *                       The interchange affects columns J to JJ-1 of A31
    *                       which are stored in the work array WORK31
    *
                            CALL CSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
         $                              WORK31( JP+JJ-J-KL, 1 ), LDWORK )
                            CALL CSWAP( J+JB-JJ, AB( KV+1, JJ ), LDAB-1,
         $                              AB( KV+JP, JJ ), LDAB-1 )
                         END IF
                      END IF
    *
    *                 Compute multipliers
    *
                      CALL CSCAL( KM, ONE / AB( KV+1, JJ ), AB( KV+2, JJ ),
         $                        1 )
    *
    *                 Update trailing submatrix within the band and within
    *                 the current block. JM is the index of the last column
    *                 which needs to be updated.
    *
                      JM = MIN( JU, J+JB-1 )
                      IF( JM.GT.JJ )
         $               CALL CGERU( KM, JM-JJ, -ONE, AB( KV+2, JJ ), 1,
         $                           AB( KV, JJ+1 ), LDAB-1,
         $                           AB( KV+1, JJ+1 ), LDAB-1 )
                   ELSE
    *
    *                 If pivot is zero, set INFO to the index of the pivot
    *                 unless a zero pivot has already been found.
    *
                      IF( INFO.EQ.0 )
         $               INFO = JJ
                   END IF
    *
    *              Copy current column of A31 into the work array WORK31
    *
                   NW = MIN( JJ-J+1, I3 )
                   IF( NW.GT.0 )
         $            CALL CCOPY( NW, AB( KV+KL+1-JJ+J, JJ ), 1,
         $                        WORK31( 1, JJ-J+1 ), 1 )
       80       CONTINUE
                IF( J+JB.LE.N ) THEN
    *
    *              Apply the row interchanges to the other blocks.
    *
                   J2 = MIN( JU-J+1, KV ) - JB
                   J3 = MAX( 0, JU-J-KV+1 )
    *
    *              Use CLASWP to apply the row interchanges to A12, A22, and
    *              A32.
    *
                   CALL CLASWP( J2, AB( KV+1-JB, J+JB ), LDAB-1, 1, JB,
         $                      IPIV( J ), 1 )
    *
    *              Adjust the pivot indices.
    *
                   DO 90 I = J, J + JB - 1
                      IPIV( I ) = IPIV( I ) + J - 1
       90          CONTINUE
    *
    *              Apply the row interchanges to A13, A23, and A33
    *              columnwise.
    *
                   K2 = J - 1 + JB + J2
                   DO 110 I = 1, J3
                      JJ = K2 + I
                      DO 100 II = J + I - 1, J + JB - 1
                         IP = IPIV( II )
                         IF( IP.NE.II ) THEN
                            TEMP = AB( KV+1+II-JJ, JJ )
                            AB( KV+1+II-JJ, JJ ) = AB( KV+1+IP-JJ, JJ )
                            AB( KV+1+IP-JJ, JJ ) = TEMP
                         END IF
      100             CONTINUE
      110          CONTINUE
    *
    *              Update the relevant part of the trailing submatrix
    *
                   IF( J2.GT.0 ) THEN
    *
    *                 Update A12
    *
                      CALL CTRSM( 'Left', 'Lower', 'No transpose', 'Unit',
         $                        JB, J2, ONE, AB( KV+1, J ), LDAB-1,
         $                        AB( KV+1-JB, J+JB ), LDAB-1 )
    *
                      IF( I2.GT.0 ) THEN
    *
    *                    Update A22
    *
                         CALL CGEMM( 'No transpose', 'No transpose', I2, J2,
         $                           JB, -ONE, AB( KV+1+JB, J ), LDAB-1,
         $                           AB( KV+1-JB, J+JB ), LDAB-1, ONE,
         $                           AB( KV+1, J+JB ), LDAB-1 )
                      END IF
    *
                      IF( I3.GT.0 ) THEN
    *
    *                    Update A32
    *
                         CALL CGEMM( 'No transpose', 'No transpose', I3, J2,
         $                           JB, -ONE, WORK31, LDWORK,
         $                           AB( KV+1-JB, J+JB ), LDAB-1, ONE,
         $                           AB( KV+KL+1-JB, J+JB ), LDAB-1 )
                      END IF
                   END IF
    *
                   IF( J3.GT.0 ) THEN
    *
    *                 Copy the lower triangle of A13 into the work array
    *                 WORK13
    *
                      DO 130 JJ = 1, J3
                         DO 120 II = JJ, JB
                            WORK13( II, JJ ) = AB( II-JJ+1, JJ+J+KV-1 )
      120                CONTINUE
      130             CONTINUE
    *
    *                 Update A13 in the work array
    *
                      CALL CTRSM( 'Left', 'Lower', 'No transpose', 'Unit',
         $                        JB, J3, ONE, AB( KV+1, J ), LDAB-1,
         $                        WORK13, LDWORK )
    *
                      IF( I2.GT.0 ) THEN
    *
    *                    Update A23
    *
                         CALL CGEMM( 'No transpose', 'No transpose', I2, J3,
         $                           JB, -ONE, AB( KV+1+JB, J ), LDAB-1,
         $                           WORK13, LDWORK, ONE, AB( 1+JB, J+KV ),
         $                           LDAB-1 )
                      END IF
    *
                      IF( I3.GT.0 ) THEN
    *
    *                    Update A33
    *
                         CALL CGEMM( 'No transpose', 'No transpose', I3, J3,
         $                           JB, -ONE, WORK31, LDWORK, WORK13,
         $                           LDWORK, ONE, AB( 1+KL, J+KV ), LDAB-1 )
                      END IF
    *
    *                 Copy the lower triangle of A13 back into place
    *
                      DO 150 JJ = 1, J3
                         DO 140 II = JJ, JB
                            AB( II-JJ+1, JJ+J+KV-1 ) = WORK13( II, JJ )
      140                CONTINUE
      150             CONTINUE
                   END IF
                ELSE
    *
    *              Adjust the pivot indices.
    *
                   DO 160 I = J, J + JB - 1
                      IPIV( I ) = IPIV( I ) + J - 1
      160          CONTINUE
                END IF
    *
    *           Partially undo the interchanges in the current block to
    *           restore the upper triangular form of A31 and copy the upper
    *           triangle of A31 back into place
    *
                DO 170 JJ = J + JB - 1, J, -1
                   JP = IPIV( JJ ) - JJ + 1
                   IF( JP.NE.1 ) THEN
    *
    *                 Apply interchange to columns J to JJ-1
    *
                      IF( JP+JJ-1.LT.J+KL ) THEN
    *
    *                    The interchange does not affect A31
    *
                         CALL CSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
         $                           AB( KV+JP+JJ-J, J ), LDAB-1 )
                      ELSE
    *
    *                    The interchange does affect A31
    *
                         CALL CSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
         $                           WORK31( JP+JJ-J-KL, 1 ), LDWORK )
                      END IF
                   END IF
    *
    *              Copy the current column of A31 back into place
    *
                   NW = MIN( I3, JJ-J+1 )
                   IF( NW.GT.0 )
         $            CALL CCOPY( NW, WORK31( 1, JJ-J+1 ), 1,
         $                        AB( KV+KL+1-JJ+J, JJ ), 1 )
      170       CONTINUE
      180    CONTINUE
          END IF
    *
          RETURN
    *
    *     End of CGBTRF
    *
          END