# ?hbmv

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

## Syntax

FORTRAN 77:

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

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

Fortran 95:

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

## Description

The ?hbmv 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 Hermitian band matrix, with k super-diagonals.

## Input Parameters

uplo

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

  If `uplo = 'U'` or 'u', then the upper triangular part of the matrix A is is used.

  If `uplo = 'L'` or 'l', then the low triangular part of the matrix A is is used.

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

  COMPLEX for chbmv

  DOUBLE COMPLEX for zhbmv

  Specifies the scalar alpha.

a

  COMPLEX for chbmv

  DOUBLE COMPLEX for zhbmv

  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 Hermitian matrix. The matrix must be 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 Hermitian 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 Hermitian 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 Hermitian 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`

  The imaginary parts of the diagonal elements need not be set and are assumed to be zero.

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

  COMPLEX for chbmv

  DOUBLE COMPLEX for zhbmv

  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

  COMPLEX for chbmv

  DOUBLE COMPLEX for zhbmv

  Specifies the scalar beta.

y

  COMPLEX for chbmv

  DOUBLE COMPLEX for zhbmv

  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 hbmv 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