ZDBTF2(3)

✨ ZDBTF2 ✨

okay so bestie, welcome to ZDBTF2(3) 💅 this one computes an LU factorization of a real m-by-n band matrix A without using partial pivoting with row interchanges. no cap, that's the whole job.

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🎀 SYNOPSIS 🎀

SUBROUTINE ZDBTF2( M, N, KL, KU, AB, LDAB, INFO )
INTEGER INFO, KL, KU, LDAB, M, N
COMPLEX*16  AB( LDAB, * )

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🦋 DESCRIPTION 🦋

so Zdbtrf computes an LU factorization of a real m-by-n band matrix A without using partial pivoting with row interchanges. that's it, that's the plot 🫶

this is the unblocked version of the algorithm, calling Level 2 BLAS. it's giving "doing it the straightforward way, one step at a time" energy 💀

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🌸 ARGUMENTS 🌸

• M (input) INTEGER — the number of rows of the matrix A. gotta be M >= 0.
• N (input) INTEGER — the number of columns of the matrix A. gotta be 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) COMPLEX*16 array, dimension (LDAB,N) — the star of the show ✨

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 (so don't stress about them, bestie 🙈).

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, it's holding 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 👀

• LDAB (input) INTEGER — the leading dimension of the array AB. LDAB >= 2*KL+KU+1. it's the ceiling, respect it 💅
• INFO (output) INTEGER — the vibe check report:
  • = 0: successful exit 🎉 slay
  • < 0: if INFO = -i, the i-th argument had an illegal value. somebody passed something they shouldn't have 😭
  • > 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. no cap, that's a real one 💀

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🌈 BAND STORAGE, ILLUSTRATED 🌈

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   *    *

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💜 A LITTLE NOTE ABOUT * AND + 💜

array elements marked * are not used by the routine — they're just chilling, not invited to the party 🙅‍♀️

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. so like… they get promoted to main character status mid-routine 👑

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and that's the whole page, bestie 💖 this is ZDBTF2(3) straight out of the LAPACK manual set — section 3, the source/version and date stamp is the usual little footer your install ships with, so go check yours if you need the exact numbers 🎀 love u, go factor something 🌸