SDBTF2(3)

🎀 NAME 🎀

SDBTF2 — bestie, this one computes an LU factorization of a real m-by-n band matrix A without using partial pivoting with row interchanges ✨

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🌸 SYNOPSIS 🌸

SUBROUTINE SDBTF2( M, N, KL, KU, AB, LDAB, INFO )
    INTEGER INFO, KL, KU, LDAB, M, N
    REAL  AB( LDAB, * )

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💅 DESCRIPTION 💅

Okay so, no cap: Sdbtrf computes an LU factorization of a real m-by-n band matrix A without using partial pivoting with row interchanges. That's the whole vibe. 💖

And listen — this is the unblocked version of the algorithm, calling Level 2 BLAS. She's doing it the straightforward way, one step at a time, and honestly? Iconic. 👑

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✨ ARGUMENTS ✨

• M (input) INTEGER — the number of rows of the matrix A. Gotta be M >= 0, fr fr.
• N (input) INTEGER — the number of columns of the matrix A. Also N >= 0. We love a non-negative count. 🫶
• 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. (Yes queen, you have to count them yourself.)
• AB (input/output) REAL array, dimension (LDAB,N) — the main character of this whole page, so let's break it down. 🦋

On entry, this holds the matrix A in band storage, sitting 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, you get the details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details — and by "below" we mean literally right below, keep scrolling bestie. 😌

• LDAB (input) INTEGER — the leading dimension of the array AB. And she has standards: LDAB >= 2*KL+KU+1. 💅
• INFO (output) INTEGER — the vibe check at the end. Here's the tea:
  • = 0: successful exit. Slay. 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value. Not the move. 😭
  • > 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.

⚠️ That last one is not a drill, bestie. The factorization finished, but U is singular and you will get division by zero if you try to solve a system with it. No cap.

🧚 THE BAND STORAGE SCHEME 🧚

Here's the band storage scheme illustrated by the following 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; elements marked + need not be set on entry, but are required by the routine to store elements of U, because of fill-in resulting from the row interchanges. 📝

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And that's the whole page, bestie — SDBTF2, the unblocked Level 2 BLAS band-matrix LU, from the LAPACK crew. Go forth and factor responsibly. 🫶✨