tbmv(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💅 tbmv(3) — Library Functions Manual ✨

okay bestie, buckle up — this is tbmv from the tbmv(3) manual page, and it's a whole family of routines, not just one. Four of them, actually. We're doing triangular matrix-vector multiply and it's lowkey iconic. Let's get into it 🫶

🌸 NAME 🌸

tbmv — tbmv: triangular matrix-vector multiply

✨ SYNOPSIS ✨

🎀 Functions

  • subroutine ctbmv (uplo, trans, diag, n, k, a, lda, x, incx) — CTBMV
  • subroutine dtbmv (uplo, trans, diag, n, k, a, lda, x, incx) — DTBMV
  • subroutine stbmv (uplo, trans, diag, n, k, a, lda, x, incx) — STBMV
  • subroutine ztbmv (uplo, trans, diag, n, k, a, lda, x, incx) — ZTBMV

🦋 Detailed Description 🦋

(no cap, that's genuinely all it says here — the real tea is below)

👑 Function Documentation 👑

🌈 subroutine ctbmv (character uplo, character trans, character diag, integer n, integer k, complex, dimension(lda,*) a, integer lda, complex, dimension(*) x, integer incx) 🌈

CTBMV

💖 Purpose:

!>
!> CTBMV  performs one of the matrix-vector operations
!>
!>    x := A*x,   or   x := A**T*x,   or   x := A**H*x,
!>
!> where x is an n element vector and  A is an n by n unit, or non-unit,
!> upper or lower triangular band matrix, with ( k + 1 ) diagonals.
!>

🎀 Parameters

UPLO

!>          UPLO is CHARACTER*1
!>           On entry, UPLO specifies whether the matrix is an upper or
!>           lower triangular matrix as follows:
!>
!>              UPLO = 'U' or 'u'   A is an upper triangular matrix.
!>
!>              UPLO = 'L' or 'l'   A is a lower triangular matrix.
!>

TRANS

!>          TRANS is CHARACTER*1
!>           On entry, TRANS specifies the operation to be performed as
!>           follows:
!>
!>              TRANS = 'N' or 'n'   x := A*x.
!>
!>              TRANS = 'T' or 't'   x := A**T*x.
!>
!>              TRANS = 'C' or 'c'   x := A**H*x.
!>

DIAG

!>          DIAG is CHARACTER*1
!>           On entry, DIAG specifies whether or not A is unit
!>           triangular as follows:
!>
!>              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
!>
!>              DIAG = 'N' or 'n'   A is not assumed to be unit
!>                                  triangular.
!>

N

!>          N is INTEGER
!>           On entry, N specifies the order of the matrix A.
!>           N must be at least zero.
!>

K

!>          K is INTEGER
!>           On entry with UPLO = 'U' or 'u', K specifies the number of
!>           super-diagonals of the matrix A.
!>           On entry with UPLO = 'L' or 'l', K specifies the number of
!>           sub-diagonals of the matrix A.
!>           K must satisfy  0 .le. K.
!>

A

!>          A is COMPLEX 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 matrix of coefficients, 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 will transfer an upper
!>           triangular 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 matrix of coefficients, 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 will transfer a lower
!>           triangular 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
!>
!>           Note that when DIAG = 'U' or 'u' the elements of the array A
!>           corresponding to the diagonal elements of the matrix are not
!>           referenced, but are assumed to be unity.
!>

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
!>           ( k + 1 ).
!>

X

!>          X is COMPLEX array, dimension at least
!>           ( 1 + ( n - 1 )*abs( INCX ) ).
!>           Before entry, the incremented array X must contain the n
!>           element vector x. On exit, X is overwritten with the
!>           transformed vector x.
!>

INCX

!>          INCX is INTEGER
!>           On entry, INCX specifies the increment for the elements of
!>           X. INCX 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 185 of file ctbmv.f.

🌈 subroutine dtbmv (character uplo, character trans, character diag, integer n, integer k, double precision, dimension(lda,*) a, integer lda, double precision, dimension(*) x, integer incx) 🌈

DTBMV

💖 Purpose:

!>
!> DTBMV  performs one of the matrix-vector operations
!>
!>    x := A*x,   or   x := A**T*x,
!>
!> where x is an n element vector and  A is an n by n unit, or non-unit,
!> upper or lower triangular band matrix, with ( k + 1 ) diagonals.
!>

🎀 Parameters

UPLO

!>          UPLO is CHARACTER*1
!>           On entry, UPLO specifies whether the matrix is an upper or
!>           lower triangular matrix as follows:
!>
!>              UPLO = 'U' or 'u'   A is an upper triangular matrix.
!>
!>              UPLO = 'L' or 'l'   A is a lower triangular matrix.
!>

TRANS

!>          TRANS is CHARACTER*1
!>           On entry, TRANS specifies the operation to be performed as
!>           follows:
!>
!>              TRANS = 'N' or 'n'   x := A*x.
!>
!>              TRANS = 'T' or 't'   x := A**T*x.
!>
!>              TRANS = 'C' or 'c'   x := A**T*x.
!>

DIAG

!>          DIAG is CHARACTER*1
!>           On entry, DIAG specifies whether or not A is unit
!>           triangular as follows:
!>
!>              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
!>
!>              DIAG = 'N' or 'n'   A is not assumed to be unit
!>                                  triangular.
!>

N

!>          N is INTEGER
!>           On entry, N specifies the order of the matrix A.
!>           N must be at least zero.
!>

K

!>          K is INTEGER
!>           On entry with UPLO = 'U' or 'u', K specifies the number of
!>           super-diagonals of the matrix A.
!>           On entry with UPLO = 'L' or 'l', K specifies the number of
!>           sub-diagonals of the matrix A.
!>           K must satisfy  0 .le. K.
!>

A

!>          A is DOUBLE PRECISION 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 matrix of coefficients, 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 will transfer an upper
!>           triangular 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 matrix of coefficients, 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 will transfer a lower
!>           triangular 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
!>
!>           Note that when DIAG = 'U' or 'u' the elements of the array A
!>           corresponding to the diagonal elements of the matrix are not
!>           referenced, but are assumed to be unity.
!>

LDA

…and bestie, the LDA deets for DTBMV are literally where this part cuts off 💀 keep scrolling into part 2, it picks right back up 🧚

LDA

LDA is INTEGER. On entry, LDA tells the routine the first dimension of A as declared in the calling (sub) program. LDA must be at least ( k + 1 ). like, that's the floor, bestie, don't go under it 💅

X

X is a DOUBLE PRECISION array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element vector x. On exit, X is overwritten with the transformed vector x. ✨ so yeah, it comes back different — that's the whole point, that's the transformation slay

INCX

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

🫶 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.
!>

okay so Level 2 Blas routine — that's the classification, and the vector and matrix arguments are not referenced when N = 0, or M = 0. And these absolute icons wrote it 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). legends, all of them 👑

Definition at line 185 of file dtbmv.f.

🌸 subroutine stbmv 🌸

subroutine stbmv (character uplo, character trans, character diag, integer
    n, integer k, real, dimension(lda,*) a, integer lda, real, dimension(*)
    x, integer incx)

⭐ STBMV

💖 Purpose:

!>
!> STBMV  performs one of the matrix-vector operations
!>
!>    x := A*x,   or   x := A**T*x,
!>
!> where x is an n element vector and  A is an n by n unit, or non-unit,
!> upper or lower triangular band matrix, with ( k + 1 ) diagonals.
!>

so STBMV does one of these matrix-vector ops — x := A*x, or x := A**T*x — where x is an n element vector and A is an n by n unit, or non-unit, upper or lower triangular band matrix, with ( k + 1 ) diagonals. that's the whole gig 💫

✨ Parameters

UPLO

UPLO is CHARACTER*1. On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix, like this:

  • UPLO = 'U' or 'u' — A is an upper triangular matrix.
  • UPLO = 'L' or 'l' — A is a lower triangular matrix.

TRANS

TRANS is CHARACTER*1. On entry, TRANS specifies the operation to be performed, as follows:

  • TRANS = 'N' or 'n' — x := A*x.
  • TRANS = 'T' or 't' — x := A**T*x.
  • TRANS = 'C' or 'c' — x := A**T*x.

note that for the real version, 'C' gives you A**T too — no conjugate here, it's all real, there's nothing to conjugate 💀

DIAG

DIAG is CHARACTER*1. On entry, DIAG specifies whether or not A is unit triangular, as follows:

  • DIAG = 'U' or 'u' — A is assumed to be unit triangular.
  • DIAG = 'N' or 'n' — A is not assumed to be unit triangular.

N

N is INTEGER. On entry, N specifies the order of the matrix A. N must be at least zero. (zero is allowed, she's flexible 🫶)

K

K is INTEGER. On entry with UPLO = 'U' or 'u', K specifies the number of super-diagonals of the matrix A. On entry with UPLO = 'L' or 'l', K specifies the number of sub-diagonals of the matrix A. K must satisfy 0 .le. K. ⭐

A

A is a REAL 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 matrix of coefficients, 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 will transfer an upper triangular 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 matrix of coefficients, 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 will transfer a lower triangular 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
!>
⚠️ Note that when DIAG = 'U' or 'u' the elements of the array A corresponding to the diagonal elements of the matrix are not referenced, but are assumed to be unity. so like... don't stress about writing them, they're lowkey imaginary ones 🍓

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 ( k + 1 ). same rule as before, bestie, it's a hard floor 💅

X

X is a REAL array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element vector x. On exit, X is overwritten with the transformed vector x. ✨ icon behaviour honestly

INCX

INCX is INTEGER. On entry, INCX specifies the increment for the elements of X. INCX 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.
!>

again with the Level 2 Blas routine tag, and again: the vector and matrix arguments are not referenced when N = 0, or M = 0. 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). 💜

Definition at line 185 of file stbmv.f.

💜 subroutine ztbmv 💜

subroutine ztbmv (character uplo, character trans, character diag, integer
    n, integer k, complex*16, dimension(lda,*) a, integer lda, complex*16,
    dimension(*) x, integer incx)

⭐ ZTBMV

💖 Purpose:

!>
!> ZTBMV  performs one of the matrix-vector operations
!>
!>    x := A*x,   or   x := A**T*x,   or   x := A**H*x,
!>
!> where x is an n element vector and  A is an n by n unit, or non-unit,
!> upper or lower triangular band matrix, with ( k + 1 ) diagonals.
!>

okay the complex one, she's the main character 👑 — ZTBMV performs one of the matrix-vector operations x := A*x, or x := A**T*x, or x := A**H*x, where x is an n element vector and A is an n by n unit, or non-unit, upper or lower triangular band matrix, with ( k + 1 ) diagonals. three flavours instead of two because complex is extra like that 🦋💅

✨ Parameters

UPLO

UPLO is CHARACTER*1. On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix, as follows:

  • UPLO = 'U' or 'u' — A is an upper triangular matrix.
  • UPLO = 'L' or 'l' — A is a lower triangular matrix.

TRANS

TRANS is CHARACTER*1. On entry, TRANS specifies the operation to be performed, as follows:

  • TRANS = 'N' or 'n' — x := A*x.
  • TRANS = 'T' or 't' — x := A**T*x.
  • TRANS = 'C' or 'c' — x := A**H*x.

and this is where she differs, bestie — for the complex version, 'C' actually gives you the conjugate transpose A**H. the glow-up is real 😭🔥

DIAG

DIAG is CHARACTER*1. On entry, DIAG specifies whether or not A is unit triangular, as follows:

  • DIAG = 'U' or 'u' — A is assumed to be unit triangular.
  • DIAG = 'N' or 'n' — A is not assumed to be unit triangular.

N

N is INTEGER. On entry, N specifies the order of the matrix A. N must be at least zero. ⭐

K

K is INTEGER. On entry with UPLO = 'U' or 'u', K specifies the number of super-diagonals of the matrix A. On entry with UPLO = 'L' or 'l', K specifies the number of sub-diagonals of the matrix A. K must satisfy 0 .le. K. 💫

A

🧮 A

okay so A is a COMPLEX*16 array, dimension ( LDA, N ). No cap. 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 matrix of coefficients, 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 will transfer an upper triangular 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

and then 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 matrix of coefficients, 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 will transfer a lower triangular 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
🫶 Note that when DIAG = 'U' or 'u' the elements of the array A corresponding to the diagonal elements of the matrix are not referenced, but are assumed to be unity.

🧱 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 ( k + 1 ). Like, don't skimp on it.

🦋 X

X is COMPLEX*16 array, dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element vector x. On exit, X is overwritten with the transformed vector x. She gets a whole glow-up on the way out, lowkey.

🎀 INCX

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

👑 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 185 of file ztbmv.f.

🤖 Author

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