sbmv(3F)

NAME
     SSBMV, DSBMV - Multiplies a real vector by a real symmetric band
     matrix

SYNOPSIS
     Real

        CALL SSBMV (uplo, n, k, alpha, a, lda, x, incx, beta, y, incy)

     Double precision

        CALL DSBMV (uplo, n, k, alpha, a, lda, x, incx, beta, y, incy)

DESCRIPTION
     These routines perform the following matrix-vector operation:

          y <- alpha Ax +
           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.

     This routine has the following arguments:

     uplo      Character*1.  (input)
               Specifies whether the upper or lower triangular part of band
               matrix A is 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.

     n         Integer.  (input)
               Specifies the order of matrix A.  n >= 0.

     k         Integer.  (input)
               Specifies the number of superdiagonals of matrix A.  k >= 0.

     alpha     Scalar alpha.  (input)
               SSBMV: Real.
               DSBMV: Double precision.

     a         Array of dimension (lda,n).  (input)
               SSBMV: Real array.
               DSBMV: Double precision array.

               Before entry with uplo = 'U' or 'u', the leading (k+1)-by-n
               part of 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 superdiagonal starting at position 2 in row k, and
               so on.  The top left k-by-k triangle of array a is not
               referenced.

               Before entry with uplo = 'L' or 'l', the leading (k+1)-by-n
               part of 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 subdiagonal starting at position 1 in row 2, and so
               on.  The bottom right k-by-k triangle of array a is not
               referenced.

               See the NOTES section for examples of Fortran code that
               transfer upper and lower parts of symmetric band matrices
               from conventional full matrix storage to band storage.

     lda       Integer.  (input)
               Specifies the first dimension of a as declared in the
               calling program.
               lda >= (k+1).

     x         Real array of dimension 1+(n-1) * |incx|.  (input)
               SSBMV: Real array.
               DSBMV: Double precision array.
               Contains vector x.

     incx      Integer.  (input)
               Specifies the increment for the elements of x.  incx must
               not be 0.

     beta      Scalar beta.  (input)
               SSBMV: Real.
               DSBMV: Double precision.

     y         Array of dimension 1+(n-1) * |incy|.  (input and output)
               SSBMV: Real array.
               DSBMV: Double precision array.
               Contains vector y.  On exit, the updated vector overwrites
               array y.

     incy      Integer.  (input)
               Specifies the increment for the elements of y.  incy must
               not be 0.

NOTES
     The following program segment transfers 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

     The following program segment transfers 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

     SSBMV/DSBMV is a Level 2 Basic Linear Algebra Subprogram (Level 2
     BLAS).

     When working backward (incx < 0 or incy < 0), this routine starts at
     the end of the vector and moves backward, as follows:

          x(1-incx * (n-1)), x(1-incx * (n-2)) , ..., x(1)

          y(1-incy * (n-1)), y(1-incy * (n-2)) , ..., y(1)

SEE ALSO
     HBMV(3F)