NAME

       STBMV - perform one of the matrix-vector operations   x := A*x, or x := A'*x,

SYNOPSIS

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

           INTEGER      INCX, K, LDA, N

           CHARACTER*1  DIAG, TRANS, UPLO

           REAL         A( LDA, * ), X( * )

PURPOSE

       STBMV  performs one of the matrix-vector operations

       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      - REAL             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      - REAL             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.

BLAS routine                                     16 October 1992                                        STBMV(3)