tplqt2(3)

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version 3.12.1 · 9 versions
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🌸 tplqt2(3) — Library Functions Manual 🌸

okay so bestie, welcome to tplqt2, the level-2 QR-factor page from the LAPACK girlies 💅. This is the manual page for the tplqt2 family, and it's long, so we're doing this in two parts — this is part 1. Grab a boba, we're going in 🧚

✨ NAME ✨

tplqt2 — tplqt2: QR factor, level 2 🎀

🦋 SYNOPSIS 🦋

Functions

subroutine ctplqt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
subroutine dtplqt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
subroutine stplqt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
subroutine ztplqt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
  • ctplqt2 — CTPLQT2 (that's literally all the synopsis says about it, no cap 💀)
  • dtplqt2 — DTPLQT2 computes a LQ factorization of a real or complex triangular-pentagonal"matrix,whichiscomposedofatriangularblockandapentagonalblock,usingthecompactWYrepresentationforQ. — yes the original really does run all those words together like that, we don't gatekeep, we just report 🫶
  • stplqt2 — STPLQT2 computes a LQ factorization of a real or complex triangular-pentagonal"matrix,whichiscomposedofatriangularblockandapentagonalblock,usingthecompactWYrepresentationforQ.
  • ztplqt2 — ZTPLQT2 computes a LQ factorization of a real or complex triangular-pentagonal"matrix,whichiscomposedofatriangularblockandapentagonalblock,usingthecompactWYrepresentationforQ.

💜 Detailed Description 💜

(This section is empty in the original — the description is just the heading itself, so we're matching that energy exactly. Iconic of it, honestly ✨)

👑 Function Documentation 👑

🌈 subroutine ctplqt2 (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)

CTPLQT2 — that's the whole one-liner label, bestie, it's giving mystery 🕵️‍♀️

⭐ Purpose:

CTPLQT2 computes a 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. Slay 💅

🍓 Parameters

  • M — M is INTEGER. The total number of rows of the matrix B. Constraint: M >= 0. ✨
  • N — N is INTEGER. The number of columns of the matrix B, and the order of the triangular matrix A. 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. 🎀
  • 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,M). The N-by-N upper triangular factor T of the block reflector. See Further Details. ⭐
  • LDT — LDT is INTEGER. The leading dimension of the array T. Constraint: LDT >= max(1,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:

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 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 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 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,

W = [ 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 (M+N)-by-(M+N) block reflector H is then given by

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

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

Definition at line 161 of file ctplqt2.f. 📍

🌈 subroutine dtplqt2 (integer m, integer n, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldt, * ) t, integer ldt, integer info)

DTPLQT2 computes a LQ 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. 💖

⭐ Purpose:

DTPLQT2 computes a LQ factorization of a real matrix C, which is composed of a triangular block A and pentagonal block B, using the compact WY representation for Q. Slay again 💅

🍓 Parameters

  • M — M is INTEGER. The total number of rows of the matrix B. Constraint: M >= 0. ✨
  • N — N is INTEGER. The number of columns of the matrix B, and the order of the triangular matrix A. 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. 🎀
  • A — A is DOUBLE PRECISION 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 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 — LDB is INTEGER. The leading dimension of the array B. Constraint: LDB >= max(1,M). 🌈
  • T — T is DOUBLE PRECISION array, dimension (LDT,M). The N-by-N upper triangular factor T of the block reflector. See Further Details. ⭐
  • LDT — LDT is INTEGER. The leading dimension of the array T. Constraint: LDT >= max(1,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:

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 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 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 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,

W = [ 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 (M+N)-by-(M+N) block reflector H is then given by

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

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

Definition at line 176 of file dtplqt2.f. 📍

🌈 subroutine stplqt2 (integer m, integer n, integer l, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldt, * ) t, integer ldt, integer info)

STPLQT2 computes a LQ 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. 🌸

⭐ Purpose:

…and that is exactly where the page cuts off, bestie — mid-Purpose, no warning, just vibes 💀. Everything from here for stplqt2 and all of ztplqt2 lives in part 2 ✨🫶

🌸 stplqt2 Purpose 🌸

Okay so STPLQT2 computes an LQ factorization of a real matrix C — it's composed of a triangular block A and a pentagonal block B, using the compact WY representation for Q. Literally iconic. 💖

💅 stplqt2 Parameters 💅

  • M — is an INTEGER. It's the total number of rows of the matrix B. So like, M >= 0. 🫶
  • N — is an INTEGER. It's the number of columns of the matrix B, and the order of the triangular matrix A. N >= 0. ✨
  • L — is an INTEGER. It's the number of rows of the lower trapezoidal part of B. And bestie, MIN(M,N) >= L >= 0. See Further Details. 👀
  • A — is a REAL array, dimension (LDA,M). On entry, it's 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. Slay. 💅
  • LDA — is an INTEGER. It's the leading dimension of the array A. LDA >= max(1,M). 🎀
  • B — is a REAL array, dimension (LDB,N). On entry, it's 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 — is an INTEGER. It's the leading dimension of the array B. LDB >= max(1,M). ⭐
  • T — is a REAL array, dimension (LDT,M). It's the N-by-N upper triangular factor T of the block reflector. See Further Details. 🔥
  • LDT — is an INTEGER. It's the leading dimension of the array T. LDT >= max(1,M) 👑
  • INFO — is an 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 🦋

So the input matrix C is an M-by-(M+N) matrix:

C = [ A ][ B ]

where A is a lower triangular M-by-M matrix, and B is an M-by-N pentagonal matrix consisting of an M-by-(N-L) rectangular matrix B1 left of an 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 an 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. No cap. 💀

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 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:

W = [ 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 (M+N)-by-(M+N) block reflector H is then given by:

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

where W^H is the conjugate transpose of W and T is the upper triangular factor of the block reflector. Fr fr. 🌈

Definition at line 176 of file stplqt2.f.

🌈 ztplqt2 Subroutine 🌈

Okay so here's the complex version, bestie. The signature is literally:

subroutine ztplqt2 (integer m, integer n, integer l, complex*16, dimension(
    lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb,
    complex*16, dimension( ldt, * ) t, integer ldt, integer info)

ZTPLQT2 computes a LQ 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. It's giving main character energy. ✨

🧚 Purpose 🧚

ZTPLQT2 computes an LQ factorization of a complex matrix C — it's composed of a triangular block A and pentagonal block B, using the compact WY representation for Q. Obsessed. 💜

🍓 Parameters 🍓

  • M — is an INTEGER. It's the total number of rows of the matrix B. So like, M >= 0. 🫶
  • N — is an INTEGER. It's the number of columns of the matrix B, and the order of the triangular matrix A. N >= 0. ✨
  • L — is an INTEGER. It's the number of rows of the lower trapezoidal part of B. And bestie, MIN(M,N) >= L >= 0. See Further Details. 👀
  • A — is a COMPLEX*16 array, dimension (LDA,M). On entry, it's 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. Slay. 💅
  • LDA — is an INTEGER. It's the leading dimension of the array A. LDA >= max(1,M). 🎀
  • B — is a COMPLEX*16 array, dimension (LDB,N). On entry, it's 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 — is an INTEGER. It's the leading dimension of the array B. LDB >= max(1,M). ⭐
  • T — is a COMPLEX*16 array, dimension (LDT,M). It's the N-by-N upper triangular factor T of the block reflector. See Further Details. 🔥
  • LDT — is an INTEGER. It's the leading dimension of the array T. LDT >= max(1,M) 👑
  • INFO — is an 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 🦋

So the input matrix C is an M-by-(M+N) matrix:

C = [ A ][ B ]

where A is a lower triangular M-by-M matrix, and B is an M-by-N pentagonal matrix consisting of an M-by-(N-L) rectangular matrix B1 left of an 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 an 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. No cap. 💀

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 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:

W = [ 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 (M+N)-by-(M+N) block reflector H is then given by:

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

where W^H is the conjugate transpose of W and T is the upper triangular factor of the block reflector. Fr fr. 🌈

Definition at line 176 of file ztplqt2.f.

🌟 Author 🌟

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