gbmv(3)

📦

version 3.12.1 · 9 versions

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🌸 NAME 🌸

gbmv(3) — from the Library Functions Manual — is the page for gbmv, aka gbmv: general matrix-vector multiply. Bestie, it’s literally the general matrix-vector multiply routine, and it’s lowkey iconic. 💅✨

✨ SYNOPSIS ✨

🎀 Functions 🎀

cgbmv — CGBMV

subroutine cgbmv (trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y,
    incy)

dgbmv — DGBMV

subroutine dgbmv (trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y,
    incy)

sgbmv — SGBMV

subroutine sgbmv (trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y,
    incy)

zgbmv — ZGBMV

subroutine zgbmv (trans, m, n, kl, ku, alpha, a, lda, x, incx, beta, y,
    incy)

🦋 Detailed Description 🦋

Okay so the detailed-description section is handing the mic straight to the function docs below. 🎤✨ No extra detour here, bestie — we’re going in.

🌈 Function Documentation 🌈

💜 `cgbmv` 💜

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 — yes bestie, that’s the label. 👑

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. Periodt. 💀
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. No cap. 🫶

👑 Author 👑

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
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. 💖

💙 `dgbmv` 💙

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 — the double-precision sibling is here, and yes, it ate. 👑

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. Literally not zero, bestie. 💀
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. Periodt. 🫶

👑 Author 👑

Univ. of Tennessee

👑 Author

Bestie, this one's got the whole academic avengers squad behind it:

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
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

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. That's the whole vibe. ✨

Parameters

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 — is INTEGER. On entry, M specifies the number of rows of the matrix A. M must be at least zero.
N — is INTEGER. On entry, N specifies the number of columns of the matrix A. N must be at least zero.
KL — is INTEGER. On entry, KL specifies the number of sub-diagonals of the matrix A. KL must satisfy 0 .le. KL.
KU — is INTEGER. On entry, KU specifies the number of super-diagonals of the matrix A. KU must satisfy 0 .le. KU.
ALPHA — is REAL. On entry, ALPHA specifies the scalar alpha.
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 — 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 — 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 — is INTEGER. On entry, INCX specifies the increment for the elements of X. INCX must not be zero. 🚫
BETA — is REAL. On entry, BETA specifies the scalar beta. When BETA is supplied as zero then Y need not be set on input. (Free pass, iykyk 💅)
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 — 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
Univ. of Colorado Denver
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

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. Three flavours, pick your fighter. ✨💅

Parameters

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.

Heads up bestie — for the complex one, 'C' gives you the conjugate transpose A**H, not just the plain transpose like in sgbmv. Different vibe entirely. 🫶

M — is INTEGER. On entry, M specifies the number of rows of the matrix A. M must be at least zero.
N — is INTEGER. On entry, N specifies the number of columns of the matrix A. N must be at least zero.
KL — is INTEGER. On entry, KL specifies the number of sub-diagonals of the matrix A. KL must satisfy 0 .le. KL.
KU — is INTEGER. On entry, KU specifies the number of super-diagonals of the matrix A. KU must satisfy 0 .le. KU.
ALPHA — is COMPLEX*16. On entry, ALPHA specifies the scalar alpha.
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 — 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 — 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 — is INTEGER. On entry, INCX specifies the increment for the elements of X. INCX must not be zero. 🚫
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 — 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 — 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
Univ. of Colorado Denver
NAG Ltd.

🔍 Further Details

Okay so the source file literally opens with this precious little Fortran comment block 💖 and it’s giving “receipts from the Unix elders” — here’s every crumb:

Level 2 Blas routine. — that’s the whole vibe check, no cap.
The vector and matrix arguments are not referenced when N = 0, or M = 0 🫶 (so if either one is zero, those args get zero attention, periodt).
-- Written on 22-October-1986. by:
- Jack Dongarra, Argonne National Lab.
- Jeremy Du Croz, Nag Central Office.
- Sven Hammarling, Nag Central Office.
- Richard Hanson, Sandia National Labs.

And the file itself? Definition at line 188 of file zgbmv.f. — screenshot that for the group chat 📸✨

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code. — yes bestie, the docs wrote themselves, and honestly? Slay.

That’s the whole page, bestie — brought to you by LAPACK Version 3.12.0 and the gbmv(3) manual-page energy. Mwah 💋✨

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

gbmv(3)

🌸 NAME 🌸

✨ SYNOPSIS ✨

🎀 Functions 🎀

🦋 Detailed Description 🦋

🌈 Function Documentation 🌈

💜 cgbmv 💜

🍓 Parameters 🍓

👑 Author 👑

🌟 Further Details: 🌟

💙 dgbmv 💙

🍓 Parameters 🍓

👑 Author 👑

👑 Author

🔍 Further Details

💜 subroutine sgbmv

🌸 SGBMV

Purpose

Parameters

👑 Author

🔍 Further Details

🦋 subroutine zgbmv

💜 ZGBMV

Purpose

Parameters

👑 Author

🔍 Further Details

👑 Author 👑

💜 `cgbmv` 💜

💙 `dgbmv` 💙