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    1:       SUBROUTINE DGBTRS( TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB,
    2:      $                   INFO )
    3: *
    4: *  -- LAPACK routine (version 3.2) --
    5: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
    6: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
    7: *     November 2006
    8: *
    9: *     .. Scalar Arguments ..
   10:       CHARACTER          TRANS
   11:       INTEGER            INFO, KL, KU, LDAB, LDB, N, NRHS
   12: *     ..
   13: *     .. Array Arguments ..
   14:       INTEGER            IPIV( * )
   15:       DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * )
   16: *     ..
   17: *
   18: *  Purpose
   19: *  =======
   20: *
   21: *  DGBTRS solves a system of linear equations
   22: *     A * X = B  or  A**T * X = B
   23: *  with a general band matrix A using the LU factorization computed
   24: *  by DGBTRF.
   25: *
   26: *  Arguments
   27: *  =========
   28: *
   29: *  TRANS   (input) CHARACTER*1
   30: *          Specifies the form of the system of equations.
   31: *          = 'N':  A * X = B  (No transpose)
   32: *          = 'T':  A**T* X = B  (Transpose)
   33: *          = 'C':  A**T* X = B  (Conjugate transpose = Transpose)
   34: *
   35: *  N       (input) INTEGER
   36: *          The order of the matrix A.  N >= 0.
   37: *
   38: *  KL      (input) INTEGER
   39: *          The number of subdiagonals within the band of A.  KL >= 0.
   40: *
   41: *  KU      (input) INTEGER
   42: *          The number of superdiagonals within the band of A.  KU >= 0.
   43: *
   44: *  NRHS    (input) INTEGER
   45: *          The number of right hand sides, i.e., the number of columns
   46: *          of the matrix B.  NRHS >= 0.
   47: *
   48: *  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N)
   49: *          Details of the LU factorization of the band matrix A, as
   50: *          computed by DGBTRF.  U is stored as an upper triangular band
   51: *          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
   52: *          the multipliers used during the factorization are stored in
   53: *          rows KL+KU+2 to 2*KL+KU+1.
   54: *
   55: *  LDAB    (input) INTEGER
   56: *          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
   57: *
   58: *  IPIV    (input) INTEGER array, dimension (N)
   59: *          The pivot indices; for 1 <= i <= N, row i of the matrix was
   60: *          interchanged with row IPIV(i).
   61: *
   62: *  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
   63: *          On entry, the right hand side matrix B.
   64: *          On exit, the solution matrix X.
   65: *
   66: *  LDB     (input) INTEGER
   67: *          The leading dimension of the array B.  LDB >= max(1,N).
   68: *
   69: *  INFO    (output) INTEGER
   70: *          = 0:  successful exit
   71: *          < 0: if INFO = -i, the i-th argument had an illegal value
   72: *
   73: *  =====================================================================
   74: *
   75: *     .. Parameters ..
   76:       DOUBLE PRECISION   ONE
   77:       PARAMETER          ( ONE = 1.0D+0 )
   78: *     ..
   79: *     .. Local Scalars ..
   80:       LOGICAL            LNOTI, NOTRAN
   81:       INTEGER            I, J, KD, L, LM
   82: *     ..
   83: *     .. External Functions ..
   84:       LOGICAL            LSAME
   85:       EXTERNAL           LSAME
   86: *     ..
   87: *     .. External Subroutines ..
   88:       EXTERNAL           DGEMV, DGER, DSWAP, DTBSV, XERBLA
   89: *     ..
   90: *     .. Intrinsic Functions ..
   91:       INTRINSIC          MAX, MIN
   92: *     ..
   93: *     .. Executable Statements ..
   94: *
   95: *     Test the input parameters.
   96: *
   97:       INFO = 0
   98:       NOTRAN = LSAME( TRANS, 'N' )
   99:       IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
  100:      $    LSAME( TRANS, 'C' ) ) THEN
  101:          INFO = -1
  102:       ELSE IF( N.LT.0 ) THEN
  103:          INFO = -2
  104:       ELSE IF( KL.LT.0 ) THEN
  105:          INFO = -3
  106:       ELSE IF( KU.LT.0 ) THEN
  107:          INFO = -4
  108:       ELSE IF( NRHS.LT.0 ) THEN
  109:          INFO = -5
  110:       ELSE IF( LDAB.LT.( 2*KL+KU+1 ) ) THEN
  111:          INFO = -7
  112:       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
  113:          INFO = -10
  114:       END IF
  115:       IF( INFO.NE.0 ) THEN
  116:          CALL XERBLA( 'DGBTRS', -INFO )
  117:          RETURN
  118:       END IF
  119: *
  120: *     Quick return if possible
  121: *
  122:       IF( N.EQ.0 .OR. NRHS.EQ.0 )
  123:      $   RETURN
  124: *
  125:       KD = KU + KL + 1
  126:       LNOTI = KL.GT.0
  127: *
  128:       IF( NOTRAN ) THEN
  129: *
  130: *        Solve  A*X = B.
  131: *
  132: *        Solve L*X = B, overwriting B with X.
  133: *
  134: *        L is represented as a product of permutations and unit lower
  135: *        triangular matrices L = P(1) * L(1) * ... * P(n-1) * L(n-1),
  136: *        where each transformation L(i) is a rank-one modification of
  137: *        the identity matrix.
  138: *
  139:          IF( LNOTI ) THEN
  140:             DO 10 J = 1, N - 1
  141:                LM = MIN( KL, N-J )
  142:                L = IPIV( J )
  143:                IF( L.NE.J )
  144:      $            CALL DSWAP( NRHS, B( L, 1 ), LDB, B( J, 1 ), LDB )
  145:                CALL DGER( LM, NRHS, -ONE, AB( KD+1, J ), 1, B( J, 1 ),
  146:      $                    LDB, B( J+1, 1 ), LDB )
  147:    10       CONTINUE
  148:          END IF
  149: *
  150:          DO 20 I = 1, NRHS
  151: *
  152: *           Solve U*X = B, overwriting B with X.
  153: *
  154:             CALL DTBSV( 'Upper', 'No transpose', 'Non-unit', N, KL+KU,
  155:      $                  AB, LDAB, B( 1, I ), 1 )
  156:    20    CONTINUE
  157: *
  158:       ELSE
  159: *
  160: *        Solve A**T*X = B.
  161: *
  162:          DO 30 I = 1, NRHS
  163: *
  164: *           Solve U**T*X = B, overwriting B with X.
  165: *
  166:             CALL DTBSV( 'Upper', 'Transpose', 'Non-unit', N, KL+KU, AB,
  167:      $                  LDAB, B( 1, I ), 1 )
  168:    30    CONTINUE
  169: *
  170: *        Solve L**T*X = B, overwriting B with X.
  171: *
  172:          IF( LNOTI ) THEN
  173:             DO 40 J = N - 1, 1, -1
  174:                LM = MIN( KL, N-J )
  175:                CALL DGEMV( 'Transpose', LM, NRHS, -ONE, B( J+1, 1 ),
  176:      $                     LDB, AB( KD+1, J ), 1, ONE, B( J, 1 ), LDB )
  177:                L = IPIV( J )
  178:                IF( L.NE.J )
  179:      $            CALL DSWAP( NRHS, B( L, 1 ), LDB, B( J, 1 ), LDB )
  180:    40       CONTINUE
  181:          END IF
  182:       END IF
  183:       RETURN
  184: *
  185: *     End of DGBTRS
  186: *
  187:       END