SUBROUTINE DTBMVF( UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX )

*     .. Scalar Arguments ..

      INTEGER            INCX, K, LDA, N

      CHARACTER*1        DIAG, TRANS, UPLO

*     .. Array Arguments ..

      DOUBLE PRECISION   A( LDA, * ), X( * )

*     ..

*

*  Purpose

*  =======

*

*  DTBMV  performs one of the matrix-vector operations

*

*     x := A*x,   or   x := A'*x,

*

*  where x is an n element vector and  A is an n by n unit, or non-unit,

*  upper or lower triangular band matrix, with ( k + 1 ) diagonals.

*

*  Parameters

*  ==========

*

*  UPLO   - CHARACTER*1.

*           On entry, UPLO specifies whether the matrix is an upper or

*           lower triangular matrix as follows:

*

*              UPLO = 'U' or 'u'   A is an upper triangular matrix.

*

*              UPLO = 'L' or 'l'   A is a lower triangular matrix.

*

*           Unchanged on exit.

*

*  TRANS  - CHARACTER*1.

*           On entry, TRANS specifies the operation to be performed as

*           follows:

*

*              TRANS = 'N' or 'n'   x := A*x.

*

*              TRANS = 'T' or 't'   x := A'*x.

*

*              TRANS = 'C' or 'c'   x := A'*x.

*

*           Unchanged on exit.

*

*  DIAG   - CHARACTER*1.

*           On entry, DIAG specifies whether or not A is unit

*           triangular as follows:

*

*              DIAG = 'U' or 'u'   A is assumed to be unit triangular.

*

*              DIAG = 'N' or 'n'   A is not assumed to be unit

*                                  triangular.

*

*           Unchanged on exit.

*

*  N      - INTEGER.

*           On entry, N specifies the order of the matrix A.

*           N must be at least zero.

*           Unchanged on exit.

*

*  K      - INTEGER.

*           On entry with UPLO = 'U' or 'u', K specifies the number of

*           super-diagonals of the matrix A.

*           On entry with UPLO = 'L' or 'l', K specifies the number of

*           sub-diagonals of the matrix A.

*           K must satisfy  0 .le. K.

*           Unchanged on exit.

*

*  A      - DOUBLE PRECISION array of DIMENSION ( LDA, n ).

*           Before entry with UPLO = 'U' or 'u', the leading ( k + 1 )

*           by n part of the array A must contain the upper triangular

*           band part of the matrix of coefficients, supplied column by

*           column, with the leading diagonal of the matrix in row

*           ( k + 1 ) of the array, the first super-diagonal starting at

*           position 2 in row k, and so on. The top left k by k triangle

*           of the array A is not referenced.

*           The following program segment will transfer an upper

*           triangular band matrix from conventional full matrix storage

*           to band storage:

*

*                 DO 20, J = 1, N

*                    M = K + 1 - J

*                    DO 10, I = MAX( 1, J - K ), J

*                       A( M + I, J ) = matrix( I, J )

*              10    CONTINUE

*              20 CONTINUE

*

*           Before entry with UPLO = 'L' or 'l', the leading ( k + 1 )

*           by n part of the array A must contain the lower triangular

*           band part of the matrix of coefficients, supplied column by

*           column, with the leading diagonal of the matrix in row 1 of

*           the array, the first sub-diagonal starting at position 1 in

*           row 2, and so on. The bottom right k by k triangle of the

*           array A is not referenced.

*           The following program segment will transfer a lower

*           triangular band matrix from conventional full matrix storage

*           to band storage:

*

*                 DO 20, J = 1, N

*                    M = 1 - J

*                    DO 10, I = J, MIN( N, J + K )

*                       A( M + I, J ) = matrix( I, J )

*              10    CONTINUE

*              20 CONTINUE

*

*           Note that when DIAG = 'U' or 'u' the elements of the array A

*           corresponding to the diagonal elements of the matrix are not

*           referenced, but are assumed to be unity.

*           Unchanged on exit.

*

*  LDA    - INTEGER.

*           On entry, LDA specifies the first dimension of A as declared

*           in the calling (sub) program. LDA must be at least

*           ( k + 1 ).

*           Unchanged on exit.

*

*  X      - DOUBLE PRECISION array of dimension at least

*           ( 1 + ( n - 1 )*abs( INCX ) ).

*           Before entry, the incremented array X must contain the n

*           element vector x. On exit, X is overwritten with the

*           tranformed vector x.

*

*  INCX   - INTEGER.

*           On entry, INCX specifies the increment for the elements of

*           X. INCX must not be zero.

*           Unchanged on exit.

*

*

*  Level 2 Blas routine.

*

*  -- Written on 22-October-1986.

*     Jack Dongarra, Argonne National Lab.

*     Jeremy Du Croz, Nag Central Office.

*     Sven Hammarling, Nag Central Office.

*     Richard Hanson, Sandia National Labs.

*

*

*     .. Parameters ..

      DOUBLE PRECISION   ZERO

      PARAMETER        ( ZERO = 0.0D+0 )

*     .. Local Scalars ..

      DOUBLE PRECISION   TEMP

      INTEGER            I, INFO, IX, J, JX, KPLUS1, KX, L

      LOGICAL            NOUNIT

*     .. External Functions ..

      LOGICAL            LSAME

      EXTERNAL           LSAME

*     .. External Subroutines ..

      EXTERNAL           XERBLA

*     .. Intrinsic Functions ..

      INTRINSIC          MAX, MIN

*     ..

*     .. Executable Statements ..

*

*     Test the input parameters.

*

      INFO = 0

      IF     ( .NOT.LSAME( UPLO , 'U' ).AND.

     $         .NOT.LSAME( UPLO , 'L' )      )THEN

         INFO = 1

      ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND.

     $         .NOT.LSAME( TRANS, 'T' ).AND.

     $         .NOT.LSAME( TRANS, 'C' )      )THEN

         INFO = 2

      ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND.

     $         .NOT.LSAME( DIAG , 'N' )      )THEN

         INFO = 3

      ELSE IF( N.LT.0 )THEN

         INFO = 4

      ELSE IF( K.LT.0 )THEN

         INFO = 5

      ELSE IF( LDA.LT.( K + 1 ) )THEN

         INFO = 7

      ELSE IF( INCX.EQ.0 )THEN

         INFO = 9

      END IF

      IF( INFO.NE.0 )THEN

         CALL XERBLA( 'DTBMV ', INFO )

         RETURN

      END IF

*

*     Quick return if possible.

*

      IF( N.EQ.0 )

     $   RETURN

*

      NOUNIT = LSAME( DIAG, 'N' )

*

*     Set up the start point in X if the increment is not unity. This

*     will be  ( N - 1 )*INCX   too small for descending loops.

*

      IF( INCX.LE.0 )THEN

         KX = 1 - ( N - 1 )*INCX

      ELSE IF( INCX.NE.1 )THEN

         KX = 1

      END IF

*

*     Start the operations. In this version the elements of A are

*     accessed sequentially with one pass through A.

*

      IF( LSAME( TRANS, 'N' ) )THEN

*

*         Form  x := A*x.

*

         IF( LSAME( UPLO, 'U' ) )THEN

            KPLUS1 = K + 1

            IF( INCX.EQ.1 )THEN

               DO 20, J = 1, N

                  IF( X( J ).NE.ZERO )THEN

                     TEMP = X( J )

                     L    = KPLUS1 - J

                     DO 10, I = MAX( 1, J - K ), J - 1

                        X( I ) = X( I ) + TEMP*A( L + I, J )

   10                CONTINUE

                     IF( NOUNIT )

     $                  X( J ) = X( J )*A( KPLUS1, J )

                  END IF

   20          CONTINUE

            ELSE

               JX = KX

               DO 40, J = 1, N

                  IF( X( JX ).NE.ZERO )THEN

                     TEMP = X( JX )

                     IX   = KX

                     L    = KPLUS1  - J

                     DO 30, I = MAX( 1, J - K ), J - 1

                        X( IX ) = X( IX ) + TEMP*A( L + I, J )

                        IX      = IX      + INCX

   30                CONTINUE

                     IF( NOUNIT )

     $                  X( JX ) = X( JX )*A( KPLUS1, J )

                  END IF

                  JX = JX + INCX

                  IF( J.GT.K )

     $               KX = KX + INCX

   40          CONTINUE

            END IF

         ELSE

            IF( INCX.EQ.1 )THEN

               DO 60, J = N, 1, -1

                  IF( X( J ).NE.ZERO )THEN

                     TEMP = X( J )

                     L    = 1      - J

                     DO 50, I = MIN( N, J + K ), J + 1, -1

                        X( I ) = X( I ) + TEMP*A( L + I, J )

   50                CONTINUE

                     IF( NOUNIT )

     $                  X( J ) = X( J )*A( 1, J )

                  END IF

   60          CONTINUE

            ELSE

               KX = KX + ( N - 1 )*INCX

               JX = KX

               DO 80, J = N, 1, -1

                  IF( X( JX ).NE.ZERO )THEN

                     TEMP = X( JX )

                     IX   = KX

                     L    = 1       - J

                     DO 70, I = MIN( N, J + K ), J + 1, -1

                        X( IX ) = X( IX ) + TEMP*A( L + I, J )

                        IX      = IX      - INCX

   70                CONTINUE

                     IF( NOUNIT )

     $                  X( JX ) = X( JX )*A( 1, J )

                  END IF

                  JX = JX - INCX

                  IF( ( N - J ).GE.K )

     $               KX = KX - INCX

   80          CONTINUE

            END IF

         END IF

      ELSE

*

*        Form  x := A'*x.

*

         IF( LSAME( UPLO, 'U' ) )THEN

            KPLUS1 = K + 1

            IF( INCX.EQ.1 )THEN

               DO 100, J = N, 1, -1

                  TEMP = X( J )

                  L    = KPLUS1 - J

                  IF( NOUNIT )

     $               TEMP = TEMP*A( KPLUS1, J )

                  DO 90, I = J - 1, MAX( 1, J - K ), -1

                     TEMP = TEMP + A( L + I, J )*X( I )

   90             CONTINUE

                  X( J ) = TEMP

  100          CONTINUE

            ELSE

               KX = KX + ( N - 1 )*INCX

               JX = KX

               DO 120, J = N, 1, -1

                  TEMP = X( JX )

                  KX   = KX      - INCX

                  IX   = KX

                  L    = KPLUS1  - J

                  IF( NOUNIT )

     $               TEMP = TEMP*A( KPLUS1, J )

                  DO 110, I = J - 1, MAX( 1, J - K ), -1

                     TEMP = TEMP + A( L + I, J )*X( IX )

                     IX   = IX   - INCX

  110             CONTINUE

                  X( JX ) = TEMP

                  JX      = JX   - INCX

  120          CONTINUE

            END IF

         ELSE

            IF( INCX.EQ.1 )THEN

               DO 140, J = 1, N

                  TEMP = X( J )

                  L    = 1      - J

                  IF( NOUNIT )

     $               TEMP = TEMP*A( 1, J )

                  DO 130, I = J + 1, MIN( N, J + K )

                     TEMP = TEMP + A( L + I, J )*X( I )

  130             CONTINUE

                  X( J ) = TEMP

  140          CONTINUE

            ELSE

               JX = KX

               DO 160, J = 1, N

                  TEMP = X( JX )

                  KX   = KX      + INCX

                  IX   = KX

                  L    = 1       - J

                  IF( NOUNIT )

     $               TEMP = TEMP*A( 1, J )

                  DO 150, I = J + 1, MIN( N, J + K )

                     TEMP = TEMP + A( L + I, J )*X( IX )

                     IX   = IX   + INCX

  150             CONTINUE

                  X( JX ) = TEMP

                  JX      = JX   + INCX

  160          CONTINUE

            END IF

         END IF

      END IF

*

      RETURN

*

*     End of DTBMV .

*

      END