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

SUBROUTINE SSBMV

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

REAL ALPHA, BETA

INTEGER INCX, INCY, K, LDA, N

CHARACTER*1 UPLO

REAL A( LDA, * ), X( * ), Y( * )

SSBMV 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.

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 - REAL .

On entry, ALPHA specifies the scalar alpha. Unchanged on exit.

A - REAL 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 - REAL 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 - REAL .

On entry, BETA specifies the scalar beta. Unchanged on exit.

Y - REAL 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.