# ?sbmv

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

## Syntax

FORTRAN 77:

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 )

Fortran 95:

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

## 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 `0` ≤ k.

alpha

  REAL for ssbmv

  DOUBLE PRECISION for dsbmv

  Specifies the scalar alpha.

a

  REAL for ssbmv

  DOUBLE PRECISION for dsbmv

  Array, 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 transfers 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 transfers 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`

lda

  INTEGER. Specifies the first 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, DIMENSION 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, DIMENSION 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.

## Fortran 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 Fortran 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 of length (n).

y

  Holds the vector of length (n).

uplo

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

alpha

  The default value is 1.

beta

  The default value is 1.

Parent topic: BLAS Level 2 Routines