SRC/ctpqrt2.f(3)

hey bestie 💅 this is SRC/ctpqrt2.f(3) from the Library Functions Manual — yes, the whole page, and it's giving big linear-algebra energy. 💖

🎀 NAME 🎀

SRC/ctpqrt2.f — that's the page name, short and iconic.

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine ctpqrt2 (m, n, l, a, lda, b, ldb, t, ldt, info)

CTPQRT2 computes a QR factorization of a real or complex "triangular-pentagonal" matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

The original synopsis line has this exact formatting gremlin, preserved for the record: triangular-pentagonal"matrix,whichiscomposedofatriangularblockandapentagonalblock,usingthecompactWYrepresentationforQ. 😭 No cap, that's what it says.

💖 Function/Subroutine Documentation 💖

subroutine ctpqrt2 (integer m, integer n, integer l, complex, dimension(
     lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb,
     complex, dimension( ldt, * ) t, integer ldt, integer info)

CTPQRT2 computes a QR factorization of a real or complex "triangular-pentagonal" matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q. (Yes, the synopsis said real or complex; the Purpose section below gets specific about complex. Both receipts are kept. 🧾)

🧚 Purpose

CTPQRT2 computes a 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.

🌸 Parameters

• M — INTEGER. The total number of rows of the matrix B. M >= 0.
• N — INTEGER. The number of columns of the matrix B, and the order of the triangular matrix A. N >= 0.
• L — INTEGER. The number of rows of the upper trapezoidal part of B. MIN(M,N) >= L >= 0. See Further Details.
• A — COMPLEX 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 — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• B — COMPLEX 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 — INTEGER. The leading dimension of the array B. LDB >= max(1,M).
• T — COMPLEX array, dimension (LDT,N). The N-by-N upper triangular factor T of the block reflector. See Further Details.
• LDT — INTEGER. The leading dimension of the array T. LDT >= max(1,N)
• 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 bestie, this is the part where the math girlies shine — stay with me, it's all shape and vibes. 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 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 (M+N)-by-(M+N) block reflector H is then given by:

             H = I - W * T * W**H

where W**H is the conjugate transpose of W and T is the upper triangular factor of the block reflector.

Definition at line 172 of file ctpqrt2.f.

🤖 Author

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later, bestie — this page is SRC/ctpqrt2.f(3) from LAPACK, Version 3.12.0. 💖🫶