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## NAME

```       DSBMV - perform the matrix-vector operation   y := alpha*A*x + beta*y,

```

## SYNOPSIS

```       SUBROUTINE DSBMV ( UPLO, N, K, ALPHA, A, LDA, X, INCX, BETA, Y, INCY )

DOUBLE       PRECISION ALPHA, BETA

INTEGER      INCX, INCY, K, LDA, N

CHARACTER*1  UPLO

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

```

## PURPOSE

```       DSBMV  performs the matrix-vector  operation

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.
```