SGBMV  performs one of the matrix-vector operations


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


 where alpha and beta are scalars, x and y are vectors and A is an


 m by n band matrix, with kl sub-diagonals and ku super-diagonals.

TRANS

          TRANS is CHARACTER*1


           On entry, TRANS specifies the operation to be performed as


           follows:


              TRANS = 'N' or 'n'   y := alpha*A*x + beta*y.


              TRANS = 'T' or 't'   y := alpha*A**T*x + beta*y.


              TRANS = 'C' or 'c'   y := alpha*A**T*x + beta*y.

M

          M is INTEGER


           On entry, M specifies the number of rows of the matrix A.


           M must be at least zero.

N

          N is INTEGER


           On entry, N specifies the number of columns of the matrix A.


           N must be at least zero.

KL

          KL is INTEGER


           On entry, KL specifies the number of sub-diagonals of the


           matrix A. KL must satisfy  0 .le. KL.

KU

          KU is INTEGER


           On entry, KU specifies the number of super-diagonals of the


           matrix A. KU must satisfy  0 .le. KU.

ALPHA

          ALPHA is REAL


           On entry, ALPHA specifies the scalar alpha.

A

          A is REAL array, dimension ( LDA, N )


           Before entry, the leading ( kl + ku + 1 ) by n part of the


           array A must contain the matrix of coefficients, supplied


           column by column, with the leading diagonal of the matrix in


           row ( ku + 1 ) of the array, the first super-diagonal


           starting at position 2 in row ku, the first sub-diagonal


           starting at position 1 in row ( ku + 2 ), and so on.


           Elements in the array A that do not correspond to elements


           in the band matrix (such as the top left ku by ku triangle)


           are not referenced.


           The following program segment will transfer a band matrix


           from conventional full matrix storage to band storage:


                 DO 20, J = 1, N


                    K = KU + 1 - J


                    DO 10, I = MAX( 1, J - KU ), MIN( M, J + KL )


                       A( K + I, J ) = matrix( I, J )


              10    CONTINUE


              20 CONTINUE

LDA

          LDA is INTEGER


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


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


           ( kl + ku + 1 ).

X

          X is REAL array, dimension at least


           ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'


           and at least


           ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.


           Before entry, the incremented array X must contain the


           vector x.

INCX

          INCX is INTEGER


           On entry, INCX specifies the increment for the elements of


           X. INCX must not be zero.

BETA

          BETA is REAL


           On entry, BETA specifies the scalar beta. When BETA is


           supplied as zero then Y need not be set on input.

Y

          Y is REAL array, dimension at least


           ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'


           and at least


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


           Before entry, the incremented array Y must contain the


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


           If either m or n is zero, then Y not referenced and the function


           performs a quick return.

INCY

          INCY is INTEGER


           On entry, INCY specifies the increment for the elements of


           Y. INCY must not be zero.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  Level 2 Blas routine.


  The vector and matrix arguments are not referenced when N = 0, or M = 0


  -- 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.