SRC/ctplqt.f(3)

okay so welcome to SRC/ctplqt.f(3), bestie — this is the LAPACK Library Functions Manual page for the CTPLQT routine, and we're gonna get through it together 💅✨

💅 NAME

SRC/ctplqt.f

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✨ SYNOPSIS

🧚 Functions/Subroutines

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

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🎀 Function/Subroutine Documentation

subroutine ctplqt (integer m, integer n, integer l, integer mb, complex,
    dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b,
    integer ldb, complex, dimension( ldt, * ) t, integer ldt, complex,
    dimension( * ) work, integer info)
    CTPLQT

🧚 Purpose

okay so CTPLQT computes a blocked LQ 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. iconic behavior, honestly 🫶

🍓 Parameters

• M — M is INTEGER. The number of rows of the matrix B, and the order of the triangular matrix A. Constraint: M >= 0.
• N — N is INTEGER. The number of columns of the matrix B. Constraint: N >= 0.
• L — L is INTEGER. The number of rows of the lower trapezoidal part of B. Constraint: MIN(M,N) >= L >= 0. See Further Details.
• MB — MB is INTEGER. The block size to be used in the blocked QR. Constraint: M >= MB >= 1.
• A — A is COMPLEX array, dimension (LDA,M). On entry, the lower triangular M-by-M matrix A. On exit, the elements on and below the diagonal of the array contain the lower triangular matrix L.
• LDA — LDA is INTEGER. The leading dimension of the array A. Constraint: LDA >= max(1,M).
• B — B is COMPLEX 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 — LDB is INTEGER. The leading dimension of the array B. Constraint: LDB >= max(1,M).
• T — T is COMPLEX array, dimension (LDT,N). The lower 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 >= MB.
• WORK — WORK is COMPLEX array, dimension (MB*M).
• 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 M-by-(M+N) matrix:

             C = [ A ] [ B ]

where A is an lower triangular M-by-M 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 lower trapezoidal.

The lower trapezoidal matrix B2 consists of the first L columns of a M-by-M 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 M-by-M
                 [ B ]  <- M-by-N pentagonal

so that W can be represented as:

          [ W ] = [ I ] [ V ]
                 [ I ]  <- identity, M-by-M
                 [ 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, a 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 172 of file ctplqt.f.

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🌸 Author

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