SRC/ztpqrt.f(3)

okay bestie, welcome to SRC/ztpqrt.f from the Library Functions Manual — this is the ztpqrt era and we're learning it together 💅✨

💖 NAME 💖

SRC/ztpqrt.f — that's the name, that's the file, periodt. 🫶

✨ SYNOPSIS ✨

Functions/Subroutines

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

🌸 Function/Subroutine Documentation 🌸

       subroutine ztpqrt (integer m, integer n, integer l, integer nb, 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)
       ZTPQRT

🎀 Purpose 🎀

okay so ZTPQRT computes a blocked QR 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. It's giving linear algebra main character energy, no cap. 🔥

💖 Parameters 💖

• M — M is INTEGER. It's the number of rows of the matrix B. Constraint: M >= 0. 🫶
• N — N is INTEGER. It's the number of columns of the matrix B, and the order of the triangular matrix A. Constraint: N >= 0. ✨
• L — L is INTEGER. It's the number of rows of the upper trapezoidal part of B. Constraint: MIN(M,N) >= L >= 0. See Further Details. 👀
• NB — NB is INTEGER. The block size to be used in the blocked QR. Constraint: N >= NB >= 1. 🎀
• A — A is COMPLEX*16 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. 💖
• LDA — LDA is INTEGER. The leading dimension of the array A. Constraint: LDA >= max(1,N). 🦋
• B — B is COMPLEX*16 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 — LDB is INTEGER. The leading dimension of the array B. Constraint: LDB >= max(1,M). 💅
• T — T is COMPLEX*16 array, dimension (LDT,N). The upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details. 🔥
• LDT — LDT is INTEGER. The leading dimension of the array T. Constraint: LDT >= NB. 👑
• WORK — WORK is COMPLEX*16 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 🧚

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 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 a 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 the 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, 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].

📍 Definition at line 187 of file ztpqrt.f.

🖥️ Author 🖥️

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