PZPBTRF(l)

💖 PZPBTRF — bestie, it's the banded Cholesky girlie 💖

okay so. this is PZPBTRF, a LAPACK routine (section l of the manual, iykyk), and she's here to do one specific thing and do it beautifully. sit down, grab a strawberry 🍓, and let's talk banded distributed matrices.

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✨ NAME ✨

PZPBTRF — computes a Cholesky factorization of an N-by-N complex banded symmetric positive definite distributed matrix with bandwidth BW 🎀

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🌸 SYNOPSIS 🌸

SUBROUTINE PZPBTRF( UPLO, N, BW, A, JA, DESCA, AF, LAF, WORK, LWORK,
                    INFO )

    CHARACTER       UPLO

    INTEGER         BW, INFO, JA, LAF, LWORK, N

    INTEGER         DESCA( * )

    COMPLEX*16      A( * ), AF( * ), WORK( * )

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🦋 PURPOSE 🦋

so PZPBTRF computes a Cholesky factorization of an N-by-N complex banded symmetric positive definite distributed matrix with bandwidth BW: A(1:N, JA:JA+N-1). and here's the tea — reordering is used to increase parallelism in the factorization. which means the factors you get back are DIFFERENT from the ones equivalent sequential codes would hand you. 💀

not gonna lie, that's a big deal, so let me say it again but louder: these factors cannot be used directly by users. nope. not for you directly, bestie. however — and this is the slay part — they can be used in subsequent calls to PZPBTRS to solve linear systems. 🫶 so the reordering isn't a problem, it's a whole pipeline moment.

the factorization has the form

P A(1:N, JA:JA+N-1) P^T = U' U ,  if UPLO = 'U', or

P A(1:N, JA:JA+N-1) P^T = L L', if UPLO = 'L'

where U is a banded upper triangular matrix and L is banded lower triangular, and P is a permutation matrix. main character energy from all three of them honestly. 👑

fr fr, remember: UPLO = 'U' gets you the U' U situation, UPLO = 'L' gets you the L L' situation. don't mix them up, that's not the vibe check you want to fail. 😭

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okay that's the whole page, no notes, she's short and she's iconic. this was LAPACK version 1.5, dated 12 May 1997 — yeah, vintage, but she still eats. 💅 ✨ 🎀