TESTING/LIN/dtplqt.f(3)

✨ NAME ✨

TESTING/LIN/dtplqt.f — bestie, this is the TESTING/LIN/dtplqt.f(3) page from the Library Functions Manual, and it's all about DTPLQT. 🦋

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🌸 SYNOPSIS 🌸

Functions/Subroutines

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

DTPLQT 👑

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💖 Function/Subroutine Documentation 💖

subroutine dtplqt (integer m, integer n, integer l, integer mb, 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)

DTPLQT is the main character here, no cap. 💅

✨ Purpose ✨

Okay so DTPLQT computes a blocked LQ factorization of a real matrix C, and C is composed of a triangular block A and pentagonal block B, using the compact WY representation for Q. That's the whole vibe. 🫶

🎀 Parameters 🎀

• M — INTEGER. The number of rows of the matrix B, and the order of the triangular matrix A. M >= 0.
• N — INTEGER. The number of columns of the matrix B. N >= 0.
• L — INTEGER. The number of rows of the lower trapezoidal part of B. MIN(M,N) >= L >= 0. See Further Details.
• MB — INTEGER. The block size to be used in the blocked QR. M >= MB >= 1.
• A — DOUBLE PRECISION array, dimension (LDA,N). On entry, the lower triangular N-by-N matrix A. On exit, the elements on and below the diagonal of the array contain the lower triangular matrix L.
• LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• B — DOUBLE PRECISION 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.
• LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,M).
• T — DOUBLE PRECISION array, dimension (LDT,N). The lower triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details.
• LDT — INTEGER. The leading dimension of the array T. LDT >= MB.
• WORK — DOUBLE PRECISION array, dimension (MB*M).
• INFO — 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 🦋

Okay so picture this: the input matrix C is a M-by-(M+N) matrix

C = [ A ] [ B ]

where A is a lower triangular N-by-N 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 upper trapezoidal.

The lower trapezoidal matrix B2 consists of the first L columns of a N-by-N 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. ✨

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 N-by-N
         [ B ]  <- M-by-N pentagonal

so that W can be represented as

[ W ] = [ I ] [ V ]
         [ 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 ] [ 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, an 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].

Definition at line 187 of file dtplqt.f.

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⭐ Author ⭐

Generated automatically by Doxygen for LAPACK from the source code. 🫶

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LAPACK Version 3.12.0 · TESTING/LIN/dtplqt.f(3) — that's the footer, bestie, and the source/version vibe is certified iconic. Byeee 💖✨