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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.
```