CDBTRF(3)

📦 version 1.5, in ubuntu jammy

💖 CDBTRF 💖

okay bestie, gather round, because today we're computing 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. she's brave. she's chaotic. she's iconic. 🎀

✨ SYNOPSIS ✨

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

🌸 DESCRIPTION 🌸

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

This is the blocked version of the algorithm, calling Level 3 BLAS. no cap, she's optimized, she's fast, she ate. 🔥

🦋 ARGUMENTS 🦋

  • M (input) INTEGER — the number of rows of the matrix A. M >= 0. slay, rows.
  • N (input) INTEGER — the number of columns of the matrix A. N >= 0. yes queen, columns too.
  • KL (input) INTEGER — the number of subdiagonals within the band of A. KL >= 0. the under-the-main-diagonal girlies.
  • KU (input) INTEGER — the number of superdiagonals within the band of A. KU >= 0. and the over-the-main-diagonal girlies.
  • AB (input/output) REAL array, dimension (LDAB,N) — bestie this is the whole moment. 💅

On entry, 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, 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. 👀

  • LDAB (input) INTEGER — the leading dimension of the array AB. LDAB >= 2*KL+KU+1. respect the leading dimension or perish, fr fr.
  • INFO (output) INTEGER — the vibe check, and she has three moods:
    • = 0: successful exit 🎉 slay, no notes.
    • < 0: if INFO = -i, the i-th argument had an illegal value. 😭 somebody fumbled the input.
    • > 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. 💀 this is not a drill, bestie.

🌈 BAND STORAGE SCHEME 🌈

the band storage scheme is 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. 🙈 they're just there for the ~aesthetic~, do not perceive them.

✨ lowkey reminder: no partial pivoting, no row interchanges. she trusts you. don't make her regret it.

and that's the whole page, bestie!! 🫶 this one's a LAPACK routine, so the source/version and the date are all chillin' down in the footer of the original — go peek at the fine print if you need the receipts, because this girl right here was definitely not built in a day. mwah. 💖🌸✨

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