ZDBTRF(3)

📦 version 1.5, in ubuntu jammy

💖 ZDBTRF 💖

okay bestie, pull up a chair, because we're about to talk about Zdbtrf (that's the name, she's from ZDBTRF(3) in your friendly neighbourhood manual) and honestly? she ate. 💅

Zdbtrf computes an LU factorization of a real m-by-n band matrix A without using partial pivoting or row interchanges. no cap, that's the whole job description.

✨ SYNOPSIS ✨

SUBROUTINE ZDBTRF( M, N, KL, KU, AB, LDAB, INFO )
    INTEGER INFO, KL, KU, LDAB, M, N
    COMPLEX*16 AB( LDAB, * )

🌸 DESCRIPTION 🌸

so. Zdbtrf computes an LU factorization of a real m-by-n band matrix A without using partial pivoting or row interchanges. yes, you read that right, no pivoting, no swapping, we're just rawdogging this factorization with the diagonal we were given. 🫶

and like — this is the blocked version of the algorithm, calling Level 3 BLAS. blocked! level 3! she's efficient, she's got main character energy, she's doing the most with the least. ✨

🎀 THE ARGUMENTS 🎀

one per line, no skips, we don't gatekeep here:

  • M (input) INTEGER — the number of rows of the matrix A. M >= 0.
  • N (input) INTEGER — the number of columns of the matrix A. N >= 0.
  • KL (input) INTEGER — the number of subdiagonals within the band of A. KL >= 0.
  • KU (input) INTEGER — the number of superdiagonals within the band of A. KU >= 0.
  • AB (input/output) REAL array, dimension (LDAB,N).

okay the AB one is the big one, so let's give her her own lil section. 🦋

🧚 AB, in detail 🧚

On entry, AB holds the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set.

the j-th column of A is stored in the j-th column of the array AB as follows:

AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

On exit, AB holds details of the factorization:

  • U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1
  • the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1

see below for further details. 👀 (the example is down there, keep scrolling, it's worth it.)

💜 the rest of the arguments 💜

  • LDAB (input) INTEGER — the leading dimension of the array AB. LDAB >= 2*KL+KU+1.
  • INFO (output) INTEGER — the vibe check, and she has three moods:
    • = 0: successful exit 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value 😭
    • > 0: if INFO = +i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations. 💀
bestie. if INFO > 0 you have a singular U. the factorization finished, she's done, she's posted, but she's singular and if you try to solve a system with her you're getting division by zero. that's not a drill. 👑⚠️

🌈 THE BAND STORAGE SCHEME, ILLUSTRATED 🌈

this is the part where it all clicks, i promise. here's the example, when M = N = 6, KL = 2, KU = 1:

On entry:                       On exit:

    *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
   a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
   a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
   a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *

array elements marked * are not used by the routine. they're just there. decorative. background characters. 🙈

like — look at her. left side is you walking in with A, right side is you walking out with U up top and the multipliers m down bottom. the glow up is real. ✨

⭐ SEE ALSO ⭐

okay so the original page doesn't hand us any SEE ALSO friends here — this one is a solo act, she's the main character, no supporting cast listed. iykyk. 💅

and that's the whole page, bestie. go forth and factorize responsibly. stay banded, stay blocked, stay un-pivoted. 🎀🦋💖

source/version and date: the classic LAPACK manual-page sign-off energy, straight from the docs themselves. mwah. 😘

📦 where this page comes from