tpqrt2(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

🎀 tpqrt2 🎀

bestie, welcome to the tpqrt2 manual page — it’s giving “linear algebra but make it a group chat.” 💅

💖 NAME 💖

tpqrt2 — tpqrt2: QR factor, level 2

✨ SYNOPSIS ✨

Functions

  • 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. iconic.
  • subroutine dtpqrt2 (m, n, l, a, lda, b, ldb, t, ldt, info) — DTPQRT2 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. slay.
  • subroutine stpqrt2 (m, n, l, a, lda, b, ldb, t, ldt, info) — STPQRT2 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. periodt.
  • subroutine ztpqrt2 (m, n, l, a, lda, b, ldb, t, ldt, info) — ZTPQRT2 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. no cap.

🌸 Detailed Description 🌸

This is the “Detailed Description” section, bestie — it’s literally just vibes here in the original, and the real function tea starts in Function Documentation right below. 🫶

🦋 Function Documentation 🦋

🧚 ctpqrt2

   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. 💖

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

!>          M is INTEGER
!>          The total number of rows of the matrix B.
!>          M >= 0.
!>

N

!>          N is INTEGER
!>          The number of columns of the matrix B, and the order of
!>          the triangular matrix A.
!>          N >= 0.
!>

L

!>          L is INTEGER
!>          The number of rows of the upper trapezoidal part of B.
!>          MIN(M,N) >= L >= 0.  See Further Details.
!>

A

!>          A is 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

!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,N).
!>

B

!>          B is 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

!>          LDB is INTEGER
!>          The leading dimension of the array B.  LDB >= max(1,M).
!>

T

!>          T is COMPLEX array, dimension (LDT,N)
!>          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.  LDT >= max(1,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 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 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.

🧚 dtpqrt2

   subroutine dtpqrt2 (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)

DTPQRT2 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. 💫

Purpose

!>
!> DTPQRT2 computes a QR 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.
!>

Parameters

M

!>          M is INTEGER
!>          The total number of rows of the matrix B.
!>          M >= 0.
!>

N

!>          N is INTEGER
!>          The number of columns of the matrix B, and the order of
!>          the triangular matrix A.
!>          N >= 0.
!>

L

!>          L is INTEGER
!>          The number of rows of the upper trapezoidal part of B.
!>          MIN(M,N) >= L >= 0.  See Further Details.
!>

A

!>          A is DOUBLE PRECISION 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.  LDA >= max(1,N).
!>

B

!>          B is DOUBLE PRECISION 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.  LDB >= max(1,M).
!>

T

!>          T is DOUBLE PRECISION array, dimension (LDT,N)
!>          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.  LDT >= max(1,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 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 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**T
!>
!>  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 dtpqrt2.f.

🧚 stpqrt2

   subroutine stpqrt2 (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)

STPQRT2 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. 🍓

Purpose

🌸 Parameters 🌸

okay bestie, these are the inputs — every single one, no skipping 💖

  • M — INTEGER. it's the total number of rows of the matrix B, and it's gotta satisfy M >= 0. that's the vibe check ✨
  • N — INTEGER. this one is doing double duty: it's the number of columns of the matrix B and the order of the triangular matrix A. requirement: N >= 0.
  • L — INTEGER. the number of rows of the upper trapezoidal part of B. valid range is MIN(M,N) >= L >= 0. curious about what "trapezoidal" even means here? 👀 peep Further Details, it's all laid out.
  • A — REAL 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 hold the upper triangular matrix R. glow-up ✨
  • LDA — INTEGER. the leading dimension of the array A. needs LDA >= max(1,N).
  • B — REAL 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 holds the pentagonal matrix V. 👑 see Further Details for the full tea.
  • LDB — INTEGER. the leading dimension of the array B. needs LDB >= max(1,M).
  • T — REAL array, dimension (LDT,N). this is the N-by-N upper triangular factor T of the block reflector 🫶 see Further Details.
  • LDT — INTEGER. the leading dimension of the array T. needs LDT >= max(1,N) (note: no period after this one in the original, and we respect that energy 💅).
  • INFO — INTEGER. the receipts 📜:
    • = 0: successful exit, we ate and left no crumbs 🎀
    • < 0: if INFO = -i, the i-th argument had an illegal value 😭 (rude, but at least it tells you which one)

🧚 Author 🧚

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

like a whole squad of icons collaborated on this, we love a collab 💜

🦋 Further Details 🦋

okay so this is the part where the shapes actually matter — stay with me, it's lowkey beautiful 🌈

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 made of a (M-L)-by-N rectangular matrix B1 sitting on top of an L-by-N upper trapezoidal matrix B2:

B = [ B1 ]  <- (M-L)-by-N rectangular
    [ B2 ]  <-     L-by-N upper trapezoidal.

stacked like a little matrix sandwich, iconic honestly 🥪

the upper trapezoidal matrix B2 is made of the first L rows of an N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). so: if L=0, then B is just rectangular M-by-N; if M=L=N, then B is upper triangular. shapes shifting like a glow-up montage ✨

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 W can be represented as:

W = [ I ]  <- identity, N-by-N
    [ V ]  <- M-by-N, same form as B.

so literally all the info needed for W is contained on exit in B, which we call V up there. note that V has the same form as B, periodt:

V = [ V1 ] <- (M-L)-by-N rectangular
    [ V2 ] <-     L-by-N upper trapezoidal.

the columns of V represent the vectors that define the H(i)'s. and 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 🫶 that's the whole recipe, no cap.

Definition at line 172 of file stpqrt2.f.

⭐ subroutine ztpqrt2 (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) ⭐

subroutine ztpqrt2 (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)

ZTPQRT2 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 bestie, you read that right — it works for real or complex here 💅🔥

🌸 Purpose: 🌸

okay so fr fr: ZTPQRT2 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. same energy, complex edition 💜

🎀 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*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 — INTEGER. the leading dimension of the array A. LDA >= max(1,N).
  • B — 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 — INTEGER. the leading dimension of the array B. LDB >= max(1,M).
  • T — COMPLEX*16 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) (again, no trailing period, we keep it real 💅)
  • INFO — INTEGER. the status report:
    • = 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 🦋

same shapes, same slay — 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, then B is rectangular M-by-N; if M=L=N, B is upper triangular. the glow-up options are endless 🌈

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 💖 (and yes, in the complex case it's written W**H — copy it exactly like that, bestie, don't get cute with it).

Definition at line 172 of file ztpqrt2.f.

ok so quick heads up: this whole page was generated automatically by Doxygen for LAPACK straight from the source code — it's not hand-written, it's ✨machine-made✨, which honestly explains the !> comment markers everywhere 💀

and that's a wrap, bestie 🎀 this gorgeous thing is from LAPACK, Version 3.12.0, and it's the tpqrt2(3) page — go forth and factorize responsibly, you absolute icon 👑💜

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