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# dsbmv

```
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 vec-
tors 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 sup-
plied 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, sup-
plied 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, sup-
plied 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 ele-
ments 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 ele-
ments 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.
```