SRC/dtpqrt.f(3)

✨ NAME ✨

SRC/dtpqrt.f — this is the DTPQRT page from the Library Functions Manual, bestie. It's the LAPACK routine docs for DTPQRT, and we're gonna get into every single detail 💅

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📚 SYNOPSIS 📚

🧚 Functions/Subroutines 🧚

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

DTPQRT — that's the function, no cap.

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💖 Function/Subroutine Documentation 💖

   subroutine dtpqrt (integer m, integer n, integer l, integer nb, double
       precision, dimension( lda, * ) a, integer lda, double precision,
       dimension( ldb, * ) b, integer ldb, double precision, dimension( ldt, *
       ) t, integer ldt, double precision, dimension( * ) work, integer info)

DTPQRT

Purpose:

Okay so here's the moment: DTPQRT computes a blocked QR factorization of a real matrix C, which is composed of a triangular block A and pentagonal block B, using the compact WY representation for Q. Periodt. ✨

Parameters

• M — M is INTEGER; this is the number of rows of the matrix B. Constraint: M >= 0. 🫶
• N — N is INTEGER; this is the number of columns of the matrix B, and the order of the triangular matrix A. Constraint: N >= 0. 🌸
• L — L is INTEGER; this is the number of rows of the upper trapezoidal part of B. Constraint: MIN(M,N) >= L >= 0. See Further Details. 👀
• NB — NB is INTEGER; this is the block size to be used in the blocked QR. Constraint: N >= NB >= 1. 🎀
• A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, it's 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. ✨
• LDA — LDA is INTEGER; this is the leading dimension of the array A. Constraint: LDA >= max(1,N). 💜
• B — B is DOUBLE PRECISION array, dimension (LDB,N). On entry, it's 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 — LDB is INTEGER; this is the leading dimension of the array B. Constraint: LDB >= max(1,M). 💖
• T — T is DOUBLE PRECISION array, dimension (LDT,N). These are the upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details. ⭐
• LDT — LDT is INTEGER; this is the leading dimension of the array T. Constraint: LDT >= NB. 🌈
• WORK — WORK is DOUBLE PRECISION array, dimension (NB*N). 🍓
• INFO — INFO is INTEGER; = 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.

Further Details:

Bestie, here's the whole picture: 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. That B is made of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N upper trapezoidal matrix B2:

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

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. 👑

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.

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

The columns of V represent the vectors which define the H(i)'s. 🧚

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, a 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].

📜 Definition at line 187 of file dtpqrt.f. 📜

Definition at line 187 of file dtpqrt.f.

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🌸 Author 🌸

Generated automatically by Doxygen for LAPACK from the source code.

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Ok that's the whole page, bestie 💅 — this one is SRC/dtpqrt.f(3) from LAPACK Version 3.12.0, and the footer line is literally LAPACK / Version 3.12.0 / SRC/dtpqrt.f(3). No separate calendar date printed on this one, so I'm not inventing one ✨🫶