# gbmv - Man Page

gbmv: general matrix-vector multiply

## Synopsis

### Functions

subroutine cgbmv (trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y, incy)
CGBMV
subroutine dgbmv (trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y, incy)
DGBMV
subroutine sgbmv (trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y, incy)
SGBMV
subroutine zgbmv (trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y, incy)
ZGBMV

## Function Documentation

### subroutine cgbmv (character trans, integer m, integer n, integer kl, integer ku, complex alpha, complex, dimension(lda,*) a, integer lda, complex, dimension(*) x, integer incx, complex beta, complex, dimension(*) y, integer incy)

CGBMV

Purpose:

``` CGBMV  performs one of the matrix-vector operations

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

y := alpha*A**H*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 sub-diagonals and ku super-diagonals.```
Parameters

TRANS

```          TRANS is CHARACTER*1
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**T*x + beta*y.

TRANS = 'C' or 'c'   y := alpha*A**H*x + beta*y.```

M

```          M is INTEGER
On entry, M specifies the number of rows of the matrix A.
M must be at least zero.```

N

```          N is INTEGER
On entry, N specifies the number of columns of the matrix A.
N must be at least zero.```

KL

```          KL is INTEGER
On entry, KL specifies the number of sub-diagonals of the
matrix A. KL must satisfy  0 .le. KL.```

KU

```          KU is INTEGER
On entry, KU specifies the number of super-diagonals of the
matrix A. KU must satisfy  0 .le. KU.```

ALPHA

```          ALPHA is COMPLEX
On entry, ALPHA specifies the scalar alpha.```

A

```          A is COMPLEX array, dimension ( LDA, N )
Before 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 super-diagonal
starting at position 2 in row ku, the first sub-diagonal
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 will transfer 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```

LDA

```          LDA is INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
( kl + ku + 1 ).```

X

```          X is COMPLEX array, dimension at least
( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
and at least
( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
Before entry, the incremented array X must contain the
vector x.```

INCX

```          INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.```

BETA

```          BETA is COMPLEX
On entry, BETA specifies the scalar beta. When BETA is
supplied as zero then Y need not be set on input.```

Y

```          Y is COMPLEX array, dimension at least
( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
and at least
( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
Before entry, the incremented array Y must contain the
vector y. On exit, Y is overwritten by the updated vector y.
If either m or n is zero, then Y not referenced and the function
performs a quick return.```

INCY

```          INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y. INCY must not be zero.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

```  Level 2 Blas routine.
The vector and matrix arguments are not referenced when N = 0, or M = 0

-- 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.```

Definition at line 188 of file cgbmv.f.

### subroutine dgbmv (character trans, integer m, integer n, integer kl, integer ku, double precision alpha, double precision, dimension(lda,*) a, integer lda, double precision, dimension(*) x, integer incx, double precision beta, double precision, dimension(*) y, integer incy)

DGBMV

Purpose:

``` DGBMV  performs one of the matrix-vector operations

y := alpha*A*x + beta*y,   or   y := alpha*A**T*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 sub-diagonals and ku super-diagonals.```
Parameters

TRANS

```          TRANS is CHARACTER*1
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**T*x + beta*y.

TRANS = 'C' or 'c'   y := alpha*A**T*x + beta*y.```

M

```          M is INTEGER
On entry, M specifies the number of rows of the matrix A.
M must be at least zero.```

N

```          N is INTEGER
On entry, N specifies the number of columns of the matrix A.
N must be at least zero.```

KL

```          KL is INTEGER
On entry, KL specifies the number of sub-diagonals of the
matrix A. KL must satisfy  0 .le. KL.```

KU

```          KU is INTEGER
On entry, KU specifies the number of super-diagonals of the
matrix A. KU must satisfy  0 .le. KU.```

ALPHA

```          ALPHA is DOUBLE PRECISION.
On entry, ALPHA specifies the scalar alpha.```

A

```          A is DOUBLE PRECISION array, dimension ( LDA, N )
Before 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 super-diagonal
starting at position 2 in row ku, the first sub-diagonal
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 will transfer 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```

LDA

```          LDA is INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
( kl + ku + 1 ).```

X

```          X is DOUBLE PRECISION array, dimension at least
( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
and at least
( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
Before entry, the incremented array X must contain the
vector x.```

INCX

```          INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.```

BETA

```          BETA is DOUBLE PRECISION.
On entry, BETA specifies the scalar beta. When BETA is
supplied as zero then Y need not be set on input.```

Y

```          Y is DOUBLE PRECISION array, dimension at least
( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
and at least
( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
Before entry, the incremented array Y must contain the
vector y. On exit, Y is overwritten by the updated vector y.
If either m or n is zero, then Y not referenced and the function
performs a quick return.```

INCY

```          INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y. INCY must not be zero.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

```  Level 2 Blas routine.
The vector and matrix arguments are not referenced when N = 0, or M = 0

-- 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.```

Definition at line 186 of file dgbmv.f.

### subroutine sgbmv (character trans, integer m, integer n, integer kl, integer ku, real alpha, real, dimension(lda,*) a, integer lda, real, dimension(*) x, integer incx, real beta, real, dimension(*) y, integer incy)

SGBMV

Purpose:

``` SGBMV  performs one of the matrix-vector operations

y := alpha*A*x + beta*y,   or   y := alpha*A**T*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 sub-diagonals and ku super-diagonals.```
Parameters

TRANS

```          TRANS is CHARACTER*1
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**T*x + beta*y.

TRANS = 'C' or 'c'   y := alpha*A**T*x + beta*y.```

M

```          M is INTEGER
On entry, M specifies the number of rows of the matrix A.
M must be at least zero.```

N

```          N is INTEGER
On entry, N specifies the number of columns of the matrix A.
N must be at least zero.```

KL

```          KL is INTEGER
On entry, KL specifies the number of sub-diagonals of the
matrix A. KL must satisfy  0 .le. KL.```

KU

```          KU is INTEGER
On entry, KU specifies the number of super-diagonals of the
matrix A. KU must satisfy  0 .le. KU.```

ALPHA

```          ALPHA is REAL
On entry, ALPHA specifies the scalar alpha.```

A

```          A is REAL array, dimension ( LDA, N )
Before 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 super-diagonal
starting at position 2 in row ku, the first sub-diagonal
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 will transfer 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```

LDA

```          LDA is INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
( kl + ku + 1 ).```

X

```          X is REAL array, dimension at least
( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
and at least
( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
Before entry, the incremented array X must contain the
vector x.```

INCX

```          INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.```

BETA

```          BETA is REAL
On entry, BETA specifies the scalar beta. When BETA is
supplied as zero then Y need not be set on input.```

Y

```          Y is REAL array, dimension at least
( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
and at least
( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
Before entry, the incremented array Y must contain the
vector y. On exit, Y is overwritten by the updated vector y.
If either m or n is zero, then Y not referenced and the function
performs a quick return.```

INCY

```          INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y. INCY must not be zero.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

```  Level 2 Blas routine.
The vector and matrix arguments are not referenced when N = 0, or M = 0

-- 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.```

Definition at line 186 of file sgbmv.f.

### subroutine zgbmv (character trans, integer m, integer n, integer kl, integer ku, complex*16 alpha, complex*16, dimension(lda,*) a, integer lda, complex*16, dimension(*) x, integer incx, complex*16 beta, complex*16, dimension(*) y, integer incy)

ZGBMV

Purpose:

``` ZGBMV  performs one of the matrix-vector operations

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

y := alpha*A**H*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 sub-diagonals and ku super-diagonals.```
Parameters

TRANS

```          TRANS is CHARACTER*1
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**T*x + beta*y.

TRANS = 'C' or 'c'   y := alpha*A**H*x + beta*y.```

M

```          M is INTEGER
On entry, M specifies the number of rows of the matrix A.
M must be at least zero.```

N

```          N is INTEGER
On entry, N specifies the number of columns of the matrix A.
N must be at least zero.```

KL

```          KL is INTEGER
On entry, KL specifies the number of sub-diagonals of the
matrix A. KL must satisfy  0 .le. KL.```

KU

```          KU is INTEGER
On entry, KU specifies the number of super-diagonals of the
matrix A. KU must satisfy  0 .le. KU.```

ALPHA

```          ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha.```

A

```          A is COMPLEX*16 array, dimension ( LDA, N )
Before 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 super-diagonal
starting at position 2 in row ku, the first sub-diagonal
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 will transfer 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```

LDA

```          LDA is INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
( kl + ku + 1 ).```

X

```          X is COMPLEX*16 array, dimension at least
( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
and at least
( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
Before entry, the incremented array X must contain the
vector x.```

INCX

```          INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.```

BETA

```          BETA is COMPLEX*16
On entry, BETA specifies the scalar beta. When BETA is
supplied as zero then Y need not be set on input.```

Y

```          Y is COMPLEX*16 array, dimension at least
( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
and at least
( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
Before entry, the incremented array Y must contain the
vector y. On exit, Y is overwritten by the updated vector y.
If either m or n is zero, then Y not referenced and the function
performs a quick return.```

INCY

```          INCY is INTEGER
On entry, INCY specifies the increment for the elements of
Y. INCY must not be zero.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

```  Level 2 Blas routine.
The vector and matrix arguments are not referenced when N = 0, or M = 0

-- 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.```

Definition at line 188 of file zgbmv.f.

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Tue Nov 28 2023 12:08:43 Version 3.12.0 LAPACK