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AIX Version 4.3 Base Operating System and Extensions Technical Reference, Volume 2

## SGBMV, DGBMV, CGBMV, or ZGBMV Subroutine

### Purpose

Performs matrix-vector operations with general banded matrices.

### Library

BLAS Library (libblas.a)

### FORTRAN Syntax

```SUBROUTINE SGBMV(TRANS, M, N, KL, KU, ALPHA, A, LDA,
X, INCX, BETA, Y, INCY)
REAL ALPHA,BETA
INTEGER INCX,INCY,KL,KU,LDA,M,N
CHARACTER*1 TRANS
REAL A(LDA,*), X(*), Y(*)```
```SUBROUTINE DGBMV(TRANS, M, N, KL, KU, ALPHA, A, LDA,
X, INCX, BETA, Y, INCY)
DOUBLE PRECISION ALPHA,BETA
INTEGER INCX,INCY,KL,KU,LDA,M,N
CHARACTER*1 TRANS
DOUBLE PRECISION A(LDA,*), X(*), Y(*)```
```SUBROUTINE CGBMV(TRANS, M, N, KL, KU, ALPHA, A, LDA,
X, INCX, BETA, Y, INCY)
COMPLEX ALPHA,BETA
INTEGER INCX,INCY,KL,KU,LDA,M,N
CHARACTER*1 TRANS
COMPLEX A(LDA,*), X(*), Y(*)```
```SUBROUTINE ZGBMV(TRANS, M, N, KL, KU, ALPHA, A, LDA,
X, INCX, BETA, Y, INCY)
COMPLEX*16 ALPHA,BETA
INTEGER INCX,INCY,KL,KU,LDA,M,N
CHARACTER*1 TRANS
COMPLEX*16 A(LDA,*), X(*), Y(*)```

### Description

The SGBMV, DGBMV, CGBMV, or ZGBMV subroutine performs one of the following matrix-vector operations:

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

OR

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

where alpha and beta are scalars, x and y are vectors and A is an M by N band matrix, with KL subdiagonals and KU superdiagonals.

### Parameters

TRANS On entry, TRANS specifies the operation to be performed as follows:
 TRANS = 'N' or 'n' y := alpha * A * x + beta * y TRANS = 'T' or 't' y := alpha * A' * x + beta * y TRANS = 'C' or 'c' y := alpha * A' * x + beta * y

Unchanged on exit.

M On entry, M specifies the number of rows of the matrix A; M must be at least 0; unchanged on exit.
N On entry, N specifies the number of columns of the matrix A; N must be at least 0; unchanged on exit.
KL On entry, KL specifies the number of subdiagonals of the matrix A; KL must satisfy 0 .le. KL; unchanged on exit.
KU On entry, KU specifies the number of superdiagonals of the matrix A; KU must satisfy 0 .le. KU; unchanged on exit.
ALPHA On entry, ALPHA specifies the scalar alpha; unchanged on exit.
A A vector of dimension ( LDA, N ); on entry, the leading ( KL + KU + 1 ) by N part of the array A must contain the matrix of coefficients, supplied column by column, with the leading diagonal of the matrix in row ( KU + 1 ) of the array, the first superdiagonal starting at position 2 in row KU, the first subdiagonal starting at position 1 in row ( KU + 2 ), and so on. Elements in the array A that do not correspond to elements in the band matrix (such as the top left KU by KU triangle) are not referenced. The following program segment transfers a band matrix from conventional full matrix storage to band storage:
```DO 20, J = 1, N
K = KU + 1 - J
DO 10, I = MAX( 1, J - KU ), MIN( M, J + KL )
A( K + I, J ) = matrix( I, J )
10 CONTINUE
20 CONTINUE```

Unchanged on exit.

LDA On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least ( KL + KU + 1 ); unchanged on exit.
X A vector of dimension at least (1 + (N-1) * abs( INCX ) ) when TRANS = 'N' or 'n', otherwise, at least (1 + (M-1) * abs( INCX ) ); on entry, the incremented array X must contain the vector x; unchanged on exit.
INCX On entry, INCX specifies the increment for the elements of X; INCX must not be 0; unchanged on exit.
BETA On entry, BETA specifies the scalar beta; when BETA is supplied as 0 then Y need not be set on input; unchanged on exit.
Y A vector of dimension at least (1 + (M-1) * abs( INCY ) ) when TRANS = 'N' or 'n' , otherwise, at least (1 + (N-1) * abs( INCY ) ); on entry, the incremented array Y must contain the vector y; on exit, Y is overwritten by the updated vector y.
INCY On entry, INCY specifies the increment for the elements of Y; INCY must not be 0; unchanged on exit.

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