SUBROUTINE DSBMVF( UPLO, N, K, ALPHA, A, LDA, X, INCX,

     $                   BETA, Y, INCY )

*     .. Scalar Arguments ..

      DOUBLE PRECISION   ALPHA, BETA

      INTEGER            INCX, INCY, K, LDA, N

      CHARACTER*1        UPLO

*     .. Array Arguments ..

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

*     ..

*

*  Purpose

*  =======

*

*  DSBMV  performs the matrix-vector  operation

*

*     y := alpha*A*x + beta*y,

*

*  where alpha and beta are scalars, x and y are n element vectors and

*  A is an n by n symmetric band matrix, with k super-diagonals.

*

*  Parameters

*  ==========

*

*  UPLO   - CHARACTER*1.

*           On entry, UPLO specifies whether the upper or lower

*           triangular part of the band matrix A is being supplied as

*           follows:

*

*              UPLO = 'U' or 'u'   The upper triangular part of A is

*                                  being supplied.

*

*              UPLO = 'L' or 'l'   The lower triangular part of A is

*                                  being supplied.

*

*           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, K specifies the number of super-diagonals of the

*           matrix A. K must satisfy  0 .le. K.

*           Unchanged on exit.

*

*  ALPHA  - DOUBLE PRECISION.

*           On entry, ALPHA specifies the scalar alpha.

*           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 symmetric matrix, 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 the upper

*           triangular part of a symmetric 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 symmetric matrix, 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 the lower

*           triangular part of a symmetric 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

*

*           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

*           vector x.

*           Unchanged on exit.

*

*  INCX   - INTEGER.

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

*           X. INCX must not be zero.

*           Unchanged on exit.

*

*  BETA   - DOUBLE PRECISION.

*           On entry, BETA specifies the scalar beta.

*           Unchanged on exit.

*

*  Y      - DOUBLE PRECISION array of DIMENSION at least

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

*           Before entry, the incremented array Y must contain the

*           vector y. On exit, Y is overwritten by the updated vector y.

*

*  INCY   - INTEGER.

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

*           Y. INCY 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   ONE         , ZERO

      PARAMETER        ( ONE = 1.0D+0, ZERO = 0.0D+0 )

*     .. Local Scalars ..

      DOUBLE PRECISION   TEMP1, TEMP2

      INTEGER            I, INFO, IX, IY, J, JX, JY, KPLUS1, KX, KY, L

*     .. 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( N.LT.0 )THEN

         INFO = 2

      ELSE IF( K.LT.0 )THEN

         INFO = 3

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

         INFO = 6

      ELSE IF( INCX.EQ.0 )THEN

         INFO = 8

      ELSE IF( INCY.EQ.0 )THEN

         INFO = 11

      END IF

      IF( INFO.NE.0 )THEN

         CALL XERBLA( 'DSBMV ', INFO )

         RETURN

      END IF

*

*     Quick return if possible.

*

      IF( ( N.EQ.0 ).OR.( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) )

     $   RETURN

*

*     Set up the start points in  X  and  Y.

*

      IF( INCX.GT.0 )THEN

         KX = 1

      ELSE

         KX = 1 - ( N - 1 )*INCX

      END IF

      IF( INCY.GT.0 )THEN

         KY = 1

      ELSE

         KY = 1 - ( N - 1 )*INCY

      END IF

*

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

*     are accessed sequentially with one pass through A.

*

*     First form  y := beta*y.

*

      IF( BETA.NE.ONE )THEN

         IF( INCY.EQ.1 )THEN

            IF( BETA.EQ.ZERO )THEN

               DO 10, I = 1, N

                  Y( I ) = ZERO

   10          CONTINUE

            ELSE

               DO 20, I = 1, N

                  Y( I ) = BETA*Y( I )

   20          CONTINUE

            END IF

         ELSE

            IY = KY

            IF( BETA.EQ.ZERO )THEN

               DO 30, I = 1, N

                  Y( IY ) = ZERO

                  IY      = IY   + INCY

   30          CONTINUE

            ELSE

               DO 40, I = 1, N

                  Y( IY ) = BETA*Y( IY )

                  IY      = IY           + INCY

   40          CONTINUE

            END IF

         END IF

      END IF

      IF( ALPHA.EQ.ZERO )

     $   RETURN

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

*

*        Form  y  when upper triangle of A is stored.

*

         KPLUS1 = K + 1

         IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN

            DO 60, J = 1, N

               TEMP1 = ALPHA*X( J )

               TEMP2 = ZERO

               L     = KPLUS1 - J

               DO 50, I = MAX( 1, J - K ), J - 1

                  Y( I ) = Y( I ) + TEMP1*A( L + I, J )

                  TEMP2  = TEMP2  + A( L + I, J )*X( I )

   50          CONTINUE

               Y( J ) = Y( J ) + TEMP1*A( KPLUS1, J ) + ALPHA*TEMP2

   60       CONTINUE

         ELSE

            JX = KX

            JY = KY

            DO 80, J = 1, N

               TEMP1 = ALPHA*X( JX )

               TEMP2 = ZERO

               IX    = KX

               IY    = KY

               L     = KPLUS1 - J

               DO 70, I = MAX( 1, J - K ), J - 1

                  Y( IY ) = Y( IY ) + TEMP1*A( L + I, J )

                  TEMP2   = TEMP2   + A( L + I, J )*X( IX )

                  IX      = IX      + INCX

                  IY      = IY      + INCY

   70          CONTINUE

               Y( JY ) = Y( JY ) + TEMP1*A( KPLUS1, J ) + ALPHA*TEMP2

               JX      = JX      + INCX

               JY      = JY      + INCY

               IF( J.GT.K )THEN

                  KX = KX + INCX

                  KY = KY + INCY

               END IF

   80       CONTINUE

         END IF

      ELSE

*

*        Form  y  when lower triangle of A is stored.

*

         IF( ( INCX.EQ.1 ).AND.( INCY.EQ.1 ) )THEN

            DO 100, J = 1, N

               TEMP1  = ALPHA*X( J )

               TEMP2  = ZERO

               Y( J ) = Y( J )       + TEMP1*A( 1, J )

               L      = 1            - J

               DO 90, I = J + 1, MIN( N, J + K )

                  Y( I ) = Y( I ) + TEMP1*A( L + I, J )

                  TEMP2  = TEMP2  + A( L + I, J )*X( I )

   90          CONTINUE

               Y( J ) = Y( J ) + ALPHA*TEMP2

  100       CONTINUE

         ELSE

            JX = KX

            JY = KY

            DO 120, J = 1, N

               TEMP1   = ALPHA*X( JX )

               TEMP2   = ZERO

               Y( JY ) = Y( JY )       + TEMP1*A( 1, J )

               L       = 1             - J

               IX      = JX

               IY      = JY

               DO 110, I = J + 1, MIN( N, J + K )

                  IX      = IX      + INCX

                  IY      = IY      + INCY

                  Y( IY ) = Y( IY ) + TEMP1*A( L + I, J )

                  TEMP2   = TEMP2   + A( L + I, J )*X( IX )

  110          CONTINUE

               Y( JY ) = Y( JY ) + ALPHA*TEMP2

               JX      = JX      + INCX

               JY      = JY      + INCY

  120       CONTINUE

         END IF

      END IF

*

      RETURN

*

*     End of DSBMV .

*

      END