Computes a matrix-vector product using a symmetric band matrix.

## Syntax

call ssbmv(uplo, n, k, alpha, a, lda, x, incx, beta, y, incy)

call dsbmv(uplo, n, k, alpha, a, lda, x, incx, beta, y, incy)

call sbmv(a, x, y [,uplo][,alpha] [,beta])

## Include Files

• mkl.fi, blas.f90

## Description

The ?sbmv routines perform a matrix-vector operation defined as

y := alpha*A*x + beta*y,

where:

alpha and beta are scalars,

x and y are n-element vectors,

A is an n-by-n symmetric band matrix, with k super-diagonals.

## Input Parameters

uplo

CHARACTER*1. Specifies whether the upper or lower triangular part of the band matrix A is used:

if uplo = 'U' or 'u' - upper triangular part;

if uplo = 'L' or 'l' - low triangular part.

n

INTEGER. Specifies the order of the matrix A. The value of n must be at least zero.

k

INTEGER. Specifies the number of super-diagonals of the matrix A.

The value of k must satisfy 0k.

alpha

REAL for ssbmv

DOUBLE PRECISION for dsbmv

Specifies the scalar alpha.

a

REAL for ssbmv

DOUBLE PRECISION for dsbmv

Array, size (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 transfers the upper triangular part of a symmetric band matrix from conventional full matrix storage (matrix) to band storage (a):

```       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 transfers the lower triangular part of a symmetric band matrix from conventional full matrix storage (matrix) to band storage (a):

```       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```
lda

INTEGER. Specifies the leading dimension of a as declared in the calling (sub)program. The value of lda must be at least (k + 1).

x

REAL for ssbmv

DOUBLE PRECISION for dsbmv

Array, size at least (1 + (n - 1)*abs(incx)). Before entry, the incremented array x must contain the vector x.

incx

INTEGER. Specifies the increment for the elements of x.

The value of incx must not be zero.

beta

REAL for ssbmv

DOUBLE PRECISION for dsbmv

Specifies the scalar beta.

y

REAL for ssbmv

DOUBLE PRECISION for dsbmv

Array, size at least (1 + (n - 1)*abs(incy)). Before entry, the incremented array y must contain the vector y.

incy

INTEGER. Specifies the increment for the elements of y.

The value of incy must not be zero.

## Output Parameters

y

Overwritten by the updated vector y.

## BLAS 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see BLAS 95 Interface Conventions.

Specific details for the routine sbmv interface are the following:

a

Holds the array a of size (k+1,n).

x

Holds the vector with the number of elements n.

y

Holds the vector with the number of elements n.

uplo

Must be 'U' or 'L'. The default value is 'U'.

alpha

The default value is 1.

beta

The default value is 0.

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