CTPQRT(3)

💜 CTPQRT(3) 💜

okay so bestie, welcome to the Library Functions Manual ✨ this one is CTPQRT, and she lives at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/ctpqrt.f — yes that's the whole path, she's thorough 💅

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🌸 NAME 🌸

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

like... that's it. that's the name. the file path IS the name, and honestly? iconic. no further notes 🎀

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🦋 SYNOPSIS 🦋

✨ Functions/Subroutines ✨

subroutine CTPQRT (m, n, l, nb, a, lda, b, ldb, t, ldt, work, info)

CTPQRT (yes she's named after herself, main character energy 👑)

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💖 Function/Subroutine Documentation 💖

subroutine CTPQRT (integer m, integer n, integer l, integer nb, complex,
    dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b,
    integer ldb, complex, dimension( ldt, * ) t, integer ldt, complex,
    dimension( * ) work, integer info)

CTPQRT 🫶

🎀 Purpose 🎀

okay so here's the vibe: CTPQRT computes a blocked QR factorization of a complex matrix C, and C is built out of two besties — a triangular block A and a pentagonal block B — using the compact WY representation for Q. slay 💅

🌈 Parameters 🌈

• M — INTEGER. the number of rows of the matrix B. and like, M >= 0, periodt.
• N — INTEGER. the number of columns of the matrix B, and also the order of the triangular matrix A. N >= 0 ngl.
• L — INTEGER. the number of rows of the upper trapezoidal part of B. constraint check bestie: MIN(M,N) >= L >= 0. see Further Details below for the whole story 🦋
• NB — INTEGER. the block size used in the blocked QR. N >= NB >= 1 — no zero blocks allowed, we don't do that here ✨
• A — COMPLEX array, dimension (LDA,N). on entry: the upper triangular N-by-N matrix A. on exit: the elements on and above the diagonal of the array contain the upper triangular matrix R. she transforms 💫
• LDA — INTEGER. the leading dimension of the array A. LDA >= max(1,N), non-negotiable.
• B — COMPLEX array, dimension (LDB,N). on entry: the pentagonal M-by-N matrix B. the first M-L rows are rectangular, and the last L rows are upper trapezoidal. on exit: B contains the pentagonal matrix V — see Further Details 🐛
• LDB — INTEGER. the leading dimension of the array B. LDB >= max(1,M), keep up bestie.
• T — COMPLEX array, dimension (LDT,N). the upper triangular block reflectors, stored in compact form as a sequence of upper triangular blocks. Further Details has the receipts 🎀
• LDT — INTEGER. the leading dimension of the array T. LDT >= NB. that's the rule.
• WORK — COMPLEX array, dimension (NB*N). she works 💅
• INFO — INTEGER. the vibe check:
  • = 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.

(academia said "we made this" and honestly, respect 🫶)

🔥 Further Details: 🔥

okay this is the part where it gets good, bestie. the input matrix C is a (N+M)-by-N matrix:

     C = [ A ]
         [ B ]

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

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

so the upper trapezoidal matrix B2 consists of the first L rows of an N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). if L=0, B is rectangular M-by-N; if M=L=N, B is upper triangular. cute how that works 🌸

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

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

so that W can be represented as

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

so like, all of the information needed for W is contained on exit in B, which we call V above (yes we renamed her mid-paragraph, keep up 💀). note that V has the same form as B; that is,

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

the columns of V represent the vectors which define the H(i)'s ✨

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

     T = [T1 T2 ... TB].

ngl if you got lost in the block math, that's valid — but fr fr every T is accounted for, no cap 🙈

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⭐ Definition at line 187 of file ctpqrt.f. ⭐

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💀 Author 💀

Generated automatically by Doxygen for LAPACK from the source code. (yes the docs wrote themselves, we love an autonomous queen 🧚)

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and that's the whole page bestie!! 💖 this was LAPACK Version 3.12.0, and you can find her at /home/abui...RC/ctpqrt.f(3) — go forth and factorize your blocked QR, slay, periodt ✨🦋🎀