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NAME

```       DGBMV  - perform one of the matrix-vector operations   y := alpha*A*x +
beta*y, or y := alpha*A’*x + beta*y,

```

SYNOPSIS

```       SUBROUTINE DGBMV ( TRANS, M, N, KL, KU, ALPHA, A, LDA, X,  INCX,  BETA,
Y, INCY )

DOUBLE       PRECISION ALPHA, BETA

INTEGER      INCX, INCY, KL, KU, LDA, M, N

CHARACTER*1  TRANS

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

```

PURPOSE

```       DGBMV  performs one of the matrix-vector operations

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.

```

PARAMETERS

```       TRANS  - 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’*x + beta*y.

TRANS = ’C’ or ’c’   y := alpha*A’*x + beta*y.

Unchanged on exit.

M      - INTEGER.
On  entry,  M  specifies  the number of rows of the matrix A.  M
must be at least zero.  Unchanged on exit.

N      - INTEGER.
On entry, N specifies the number of columns of the matrix A.   N
must be at least zero.  Unchanged on exit.

KL     - INTEGER.
On entry, KL specifies the number of sub-diagonals of the matrix
A. KL must satisfy  0 .le. KL.  Unchanged on exit.

KU     - INTEGER.
On entry, KU specifies the  number  of  super-diagonals  of  the
matrix A. KU must satisfy  0 .le. KU.  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, 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

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 ( kl + ku + 1 ).
Unchanged on exit.

X      - DOUBLE PRECISION array of 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.  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. When BETA is supplied
as zero then Y need not be set on input.  Unchanged on exit.

Y      - DOUBLE PRECISION array of 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.

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.
```