SDBTRF(3)

💖 SDBTRF(3) — bestie edition 💖

okay so. you're here because you need an LU factorization of a real m-by-n band matrix A, and you specifically do not want partial pivoting or row interchanges happening behind your back. no cap. SDBTRF has got you. 🫶

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✨ NAME ✨

SDBTRF — compute an LU factorization of a real m-by-n band matrix A without using partial pivoting or row interchanges.

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🌸 SYNOPSIS 🌸

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

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🦋 PURPOSE 🦋

Sdbtrf computes an LU factorization of a real m-by-n band matrix A without using partial pivoting or row interchanges. that's the whole job, and she does it. 💅

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🎀 DESCRIPTION 🎀

This is the blocked version of the algorithm, calling Level 3 BLAS. so like, the fast one. the one that actually respects your cache. icon behaviour fr fr. 🔥

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👑 ARGUMENTS 👑

• 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 this one's the main character, 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? they need not be set. do not stress about them. 🙈

The j-th column of A is stored in the j-th column of the array AB like this:

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, and here's the tea ☕: 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, bestie, I'm not gatekeeping.

• LDAB (input) INTEGER — the leading dimension of the array AB. LDAB >= 2*KL+KU+1. she said what she said. 💅
• INFO (output) INTEGER — the vibe report. three possible moods:
  • = 0: successful exit. slay. ✨
  • < 0: if INFO = -i, the i-th argument had an illegal value. somebody typed something wrong, and it was not me. 😭
  • > 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. so like... she finished, but she's singular, and dividing by zero is a whole situation. 💀

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🌈 THE BAND STORAGE SCHEME 🌈

here's the picture, 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. ignore them, they're just standing there. 🧚

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and that's the whole page, periodt. 💜 this one comes out of the library's manual set, with the version and date tucked down at the very bottom of the original like every good manpage keeps them — so mwah, go factor something banded, and remember: no pivoting, no row swaps, no drama. 🌸💅🦋