ZTPLQT(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 ZTPLQT(3) — like, the manual page, bestie ✨

okay so this is a page from the Library Functions Manual — it lives at /home/abui...RC/ztplqt.f(3), which is giving "deep in the Fortran trenches" energy and honestly? iconic. it's a LAPACK thing. it does linear algebra. it does it blocked. let's get into it 🫶

✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/ztplqt.f

like... that's the whole name. that's the path. that's where the source lives. no cap, that's just what it is 💀

🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine ZTPLQT (m, n, l, mb, a, lda, b, ldb, t, ldt, work, info)

ZTPLQT — yes that's the name, yes it's all caps, yes it's a whole vibe 👑

🦋 Function/Subroutine Documentation 🦋

the actual signature, in full, in all its glory:

subroutine ZTPLQT (integer m, integer n, integer l, integer mb, complex*16,
     dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b,
     integer ldb, complex*16, dimension( ldt, * ) t, integer ldt, complex*16,
     dimension( * ) work, integer info)

ZTPLQT 🎀

💖 Purpose 💖

okay so listen up:

ZTPLQT computes a blocked LQ factorization of a complex matrix C, which is composed of a triangular block A and pentagonal block B, using the compact WY representation for Q.

that's it. that's the whole job. blocked LQ factorization, triangular + pentagonal blocks, compact WY representation for Q. slay 💅

🍓 Parameters 🍓

M

M is INTEGER. it's the number of rows of the matrix B, and the order of the triangular matrix A. and: M >= 0. simple as that ✨

N

N is INTEGER. the number of columns of the matrix B. constraint: N >= 0. periodt 🎀

L

L is INTEGER. the number of rows of the lower trapezoidal part of B. constraint: MIN(M,N) >= L >= 0. and — important — See Further Details. iykyk, it's down there 👇

MB

MB is INTEGER. the block size to be used in the blocked QR. constraint: M >= MB >= 1. tiny but mighty block size energy 🌈

A

A is a COMPLEX*16 array, dimension (LDA,M). on entry: the lower triangular M-by-M matrix A. on exit: the elements on and below the diagonal of the array contain the lower triangular matrix L. so it transforms — glow-up arc fr fr 🦋

LDA

LDA is INTEGER. the leading dimension of the array A. constraint: LDA >= max(1,M). don't skimp on it bestie 💀

B

B is a COMPLEX*16 array, dimension (LDB,N). on entry: the pentagonal M-by-N matrix B — the first N-L columns are rectangular, and the last L columns are lower trapezoidal. on exit: B contains the pentagonal matrix V. See Further Details — the whole V situation is explained down below 👀

LDB

LDB is INTEGER. the leading dimension of the array B. constraint: LDB >= max(1,M). 🎀

T

T is a COMPLEX*16 array, dimension (LDT,N). the lower triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details, because there is lore 😭

LDT

LDT is INTEGER. the leading dimension of the array T. constraint: LDT >= MB. 💜

WORK

WORK is a COMPLEX*16 array, dimension (MB*M). it works. that's its whole personality ✨

INFO

INFO is INTEGER. the receipts:

  • = 0: successful exit 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value 😬

👑 Author 👑

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

absolute legends, all of them 🫶

🔥 Further Details: 🔥

okay this is the part where it gets good. buckle up.

the input matrix C is a M-by-(M+N) matrix

C = [ A ] [ B ]

where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal matrix consisting of a M-by-(N-L) rectangular matrix B1 on left of a M-by-L upper trapezoidal matrix B2:

[B] = [ B1 ] [ B2 ]
         [ B1 ]  <- M-by-(N-L) rectangular
         [ B2 ]  <-     M-by-L lower trapezoidal.

the lower trapezoidal matrix B2 consists of the first L columns of a M-by-M lower triangular matrix, where 0 <= L <= MIN(M,N). if L=0, B is rectangular M-by-N; if M=L=N, B is lower triangular. see how it all clicks?? it's giving puzzle pieces clicking into place 🧚

the matrix W stores the elementary reflectors H(i) in the i-th row above the diagonal (of A) in the M-by-(M+N) input matrix C

[C] = [ A ] [ B ]
         [ A ]  <- lower triangular M-by-M
         [ B ]  <- M-by-N pentagonal

so that W can be represented as

[W] = [ I ] [ V ]
         [ I ]  <- identity, M-by-M
         [ V ]  <- M-by-N, same form as B.

thus, all of information needed for W is contained on exit in B, which we call V above. note that V has the same form as B; that is,

[V] = [ V1 ] [ V2 ]
         [ V1 ] <- M-by-(N-L) rectangular
         [ V2 ] <-     M-by-L lower trapezoidal.

the rows of V represent the vectors which define the H(i)'s. 💫

the number of blocks is B = ceiling(M/MB), where each block is of order MB except for the last block, which is of order IB = M - (M-1)*MB. for each of the B blocks, a upper triangular block reflector factor is computed: T1, T2, ..., TB. the MB-by-MB (and IB-by-IB for the last block) T's are stored in the MB-by-N matrix T as

T = [T1 T2 ... TB].

genuinely obsessed with this. the blocking?? the compact WY representation?? the way T is just a whole row of block reflectors?? ate and left no crumbs 🔥👑

⭐ Definition ⭐

definition at line 187 of file ztplqt.f. 📍

😭 Author (again, yes really) 😭

generated automatically by Doxygen for LAPACK from the source code. so like, a robot wrote this about a robot's math. meta fr fr 🤖💀

and that's the page!! brought to you by LAPACK, Version 3.12.0, living at /home/abui...RC/ztplqt.f(3) — mwah, go factor something blocky bestie 💖✨🎀

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