SRC/ctpqrt.f(3)

✨ NAME ✨

SRC/ctpqrt.f — this is the LAPACK Library Functions Manual page for the CTPQRT subroutine, bestie. It’s giving blocked QR factorization with the compact WY representation, and honestly? Iconic. 💅

💖 SYNOPSIS 💖

📚 Functions/Subroutines 📚

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

🎀 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 🌸

CTPQRT computes a blocked QR factorization of a complex matrix C. C is made of a triangular block A and a pentagonal block B, and the routine uses the compact WY representation for Q. Slay. ✨

🌈 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. It’s the block size used in the blocked QR. Constraint: N >= NB >= 1.
• A — A is COMPLEX 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. It’s the leading dimension of the array A. Constraint: LDA >= max(1,N).
• B — B is COMPLEX 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. It’s the leading dimension of the array B. Constraint: LDB >= max(1,M).
• T — T is COMPLEX 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. It’s the leading dimension of the array T. Constraint: LDT >= NB.
• WORK — WORK is COMPLEX 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 🔍

Okay so 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.

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 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 ctpqrt.f.

🤖 Author 🤖

Generated automatically by Doxygen for LAPACK from the source code.

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💖 Cute little sign-off: this is LAPACK Version 3.12.0, source/page SRC/ctpqrt.f(3), and the footer doesn’t list a date, so we’re not inventing one, bestie. Bye! 🫶✨