tpmlqt(3)

📦

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tpmlqt(3) — Library Functions Manual 🦋✨

okay so bestie, welcome to tpmlqt(3), the manual page that's literally all about applying Q. This is part 1 of 3, and we're covering ctpmlqt and dtpmlqt today. No cap, it's giving linear algebra realness. 💅

NAME

tpmlqt — tpmlqt: applies Q. That's it. That's the whole name. Periodt. 💖

SYNOPSIS ✨

Functions

ctpmlqt

subroutine ctpmlqt (side, trans, m, n, k, l, mb, v, ldv, t, ldt, a, lda,
    b, ldb, work, info)

CTPMLQT 🫶

dtpmlqt

subroutine dtpmlqt (side, trans, m, n, k, l, mb, v, ldv, t, ldt, a, lda,
    b, ldb, work, info)

DTPMLQT 💜

stpmlqt

subroutine stpmlqt (side, trans, m, n, k, l, mb, v, ldv, t, ldt, a, lda,
    b, ldb, work, info)

STPMLQT 🌸

ztpmlqt

subroutine ztpmlqt (side, trans, m, n, k, l, mb, v, ldv, t, ldt, a, lda,
    b, ldb, work, info)

ZTPMLQT ⭐

Detailed Description

(okay so this section is literally empty in the original — no tea here, moving on to the function docs 👇)

Function Documentation

`ctpmlqt` — CTPMLQT 🦋

subroutine ctpmlqt (character side, character trans, integer m, integer n,
    integer k, integer l, integer mb, complex, dimension( ldv, * ) v,
    integer ldv, complex, dimension( ldt, * ) t, integer ldt, complex,
    dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b,
    integer ldb, complex, dimension( * ) work, integer info)

CTPMLQT

Purpose:

okay so CTPMLQT applies a complex unitary matrix Q obtained from a complex block reflector H to a general complex matrix C, which consists of two blocks A and B. That's the whole vibe. 💅

Parameters:

SIDE — CHARACTER*1 tells you which side Q is applied from:
- 'L': apply Q or Q**H from the Left;
- 'R': apply Q or Q**H from the Right.
TRANS — CHARACTER*1
- 'N': No transpose, apply Q;
- 'C': Conjugate transpose, apply Q**H.
M — INTEGER The number of rows of the matrix B. M >= 0.
N — INTEGER The number of columns of the matrix B. N >= 0.
K — INTEGER The number of elementary reflectors whose product defines the matrix Q.
L — INTEGER The order of the trapezoidal part of V. K >= L >= 0. See Further Details.
MB — INTEGER The block size used for the storage of T. K >= MB >= 1. This must be the same value of MB used to generate T in CTPLQT.
V — COMPLEX array, dimension (LDV,K) The i-th row must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by CTPLQT in B. See Further Details.
LDV — INTEGER The leading dimension of the array V. LDV >= K.
T — COMPLEX array, dimension (LDT,K) The upper triangular factors of the block reflectors as returned by CTPLQT, stored as a MB-by-K matrix.
LDT — INTEGER The leading dimension of the array T. LDT >= MB.
A — COMPLEX array, dimension (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R' On entry, the K-by-N or M-by-K matrix A. On exit, A is overwritten by the corresponding block of Q*C or QH*C or C*Q or C*QH. See Further Details.
LDA — INTEGER The leading dimension of the array A. If SIDE = 'L', LDA >= max(1,K); If SIDE = 'R', LDA >= max(1,M).
B — COMPLEX array, dimension (LDB,N) On entry, the M-by-N matrix B. On exit, B is overwritten by the corresponding block of Q*C or QH*C or C*Q or C*QH. See Further Details.
LDB — INTEGER The leading dimension of the array B. LDB >= max(1,M).
WORK — COMPLEX array The dimension of WORK is N*MB if SIDE = 'L', or M*MB if SIDE = 'R'.
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 so the columns of the pentagonal matrix V contain the elementary reflectors H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a trapezoidal block V2:

V = [V1] [V2].

The size of the trapezoidal block V2 is determined by the parameter L, where 0 <= L <= K; V2 is lower trapezoidal, consisting of the first L rows of a K-by-K upper triangular matrix. If L=K, V2 is lower triangular; if L=0, there is no trapezoidal block, hence V = V1 is rectangular.

If SIDE = 'L':  C = [A]  where A is K-by-N,  B is M-by-N and V is K-by-M.
                    [B]

If SIDE = 'R':  C = [A B]  where A is M-by-K, B is M-by-N and V is K-by-N.

The complex unitary matrix Q is formed from V and T.

If TRANS='N' and SIDE='L', C is on exit replaced with Q * C.

If TRANS='C' and SIDE='L', C is on exit replaced with Q**H * C.

If TRANS='N' and SIDE='R', C is on exit replaced with C * Q.

If TRANS='C' and SIDE='R', C is on exit replaced with C * Q**H.

Definition at line 197 of file ctpmlqt.f.

`dtpmlqt` — DTPMLQT 💜

subroutine dtpmlqt (character side, character trans, integer m, integer n,
    integer k, integer l, integer mb, double precision, dimension( ldv, * )
    v, integer ldv, double precision, dimension( ldt, * ) t, integer ldt,
    double precision, dimension( lda, * ) a, integer lda, double precision,
    dimension( ldb, * ) b, integer ldb, double precision, dimension( * )
    work, integer info)

DTPMLQT

Purpose:

okay so the original says (and yes it literally says DTPMQRT, not DTPMLQT — we don't make the rules, bestie) — DTPMQRT applies a real orthogonal matrix Q obtained from a real block reflector H to a general real matrix C, which consists of two blocks A and B.

Parameters:

SIDE — CHARACTER*1 tells you which side Q is applied from:
- 'L': apply Q or Q**T from the Left;
- 'R': apply Q or Q**T from the Right.
TRANS — CHARACTER*1
- 'N': No transpose, apply Q;
- 'T': Transpose, apply Q**T.
M — INTEGER The number of rows of the matrix B. M >= 0.
N — INTEGER The number of columns of the matrix B. N >= 0.
K — INTEGER The number of elementary reflectors whose product defines the matrix Q.
L — INTEGER The order of the trapezoidal part of V. K >= L >= 0. See Further Details.
MB — INTEGER The block size used for the storage of T. K >= MB >= 1. This must be the same value of MB used to generate T in DTPLQT.
V — DOUBLE PRECISION array, dimension (LDV,K) The i-th row must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by DTPLQT in B. See Further Details.
LDV — INTEGER The leading dimension of the array V. LDV >= K.
T — DOUBLE PRECISION array, dimension (LDT,K) The upper triangular factors of the block reflectors as returned by DTPLQT, stored as a MB-by-K matrix.
LDT — INTEGER The leading dimension of the array T. LDT >= MB.
A — DOUBLE PRECISION array, dimension (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R' On entry, the K-by-N or M-by-K matrix A. On exit, A is overwritten by the corresponding block of Q*C or QT*C or C*Q or C*QT. See Further Details.
LDA — INTEGER The leading dimension of the array A. If SIDE = 'L', LDA >= max(1,K); If SIDE = 'R', LDA >= max(1,M).
B — DOUBLE PRECISION array, dimension (LDB,N) On entry, the M-by-N matrix B. On exit, B is overwritten by the corresponding block of Q*C or QT*C or C*Q or C*QT. See Further Details.
LDB — INTEGER The leading dimension of the array B. LDB >= max(1,M).
WORK — DOUBLE PRECISION array The dimension of WORK is N*MB if SIDE = 'L', or M*MB if SIDE = 'R'.
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:

(and that's where part 1 ends, bestie — part 2 is coming with stpmlqt and ztpmlqt, so don't go anywhere 💖)

Okay so real talk: the columns of that pentagonal matrix V? They contain the elementary reflectors H(1), H(2), ..., H(K). And V is literally composed of a rectangular block V1 and a trapezoidal block V2, like:

V = [V1] [V2].

The size of that trapezoidal block V2? It’s totally determined by the parameter L, where 0 <= L <= K. V2 is lower trapezoidal — it’s made of the first L rows of a K-by-K upper triangular matrix. If L=K, then V2 is lower triangular; if L=0, there’s no trapezoidal block at all, so V = V1 and it’s just rectangular. Periodt. 💅

Now check the SIDE vibes:

If SIDE = 'L':  C = [A]  where A is K-by-N,  B is M-by-N and V is K-by-M.
                    [B]

If SIDE = 'R':  C = [A B]  where A is M-by-K, B is M-by-N and V is K-by-N.

The real orthogonal matrix Q is formed from V and T.

If TRANS='N' and SIDE='L', C is on exit replaced with Q * C.
If TRANS='T' and SIDE='L', C is on exit replaced with Q**T * C.
If TRANS='N' and SIDE='R', C is on exit replaced with C * Q.
If TRANS='T' and SIDE='R', C is on exit replaced with C * Q**T.

Definition at line 212 of file dtpmlqt.f.

✨ `subroutine stpmlqt` ✨

   subroutine stpmlqt (character side, character trans, integer m, integer n,
       integer k, integer l, integer mb, real, dimension( ldv, * ) v, integer
       ldv, real, dimension( ldt, * ) t, integer ldt, real, dimension( lda, * )
       a, integer lda, real, dimension( ldb, * ) b, integer ldb, real,
       dimension( * ) work, integer info)
       STPMLQT

🎯 Purpose

Bestie, STPMLQT applies a real orthogonal matrix Q obtained from a real block reflector H to a general real matrix C, which consists of two blocks A and B. That’s the whole vibe. 🫶

🌸 Parameters

SIDE — CHARACTER*1. = 'L': apply Q or Q**T from the Left; = 'R': apply Q or Q**T from the Right.
TRANS — CHARACTER*1. = 'N': No transpose, apply Q; = 'T': Transpose, apply Q**T.
M — INTEGER. The number of rows of the matrix B. M >= 0.
N — INTEGER. The number of columns of the matrix B. N >= 0.
K — INTEGER. The number of elementary reflectors whose product defines the matrix Q.
L — INTEGER. The order of the trapezoidal part of V. K >= L >= 0. See Further Details.
MB — INTEGER. The block size used for the storage of T. K >= MB >= 1. This must be the same value of MB used to generate T in STPLQT.
V — REAL array, dimension (LDV,K). The i-th row must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by STPLQT in B. See Further Details.
LDV — INTEGER. The leading dimension of the array V. LDV >= K.
T — REAL array, dimension (LDT,K). The upper triangular factors of the block reflectors as returned by STPLQT, stored as a MB-by-K matrix.
LDT — INTEGER. The leading dimension of the array T. LDT >= MB.
A — REAL array, dimension (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R'. On entry, the K-by-N or M-by-K matrix A. On exit, A is overwritten by the corresponding block of Q*C or Q**T*C or C*Q or C*Q**T. See Further Details.
LDA — INTEGER. The leading dimension of the array A. If SIDE = 'L', LDA >= max(1,K); If SIDE = 'R', LDA >= max(1,M).
B — REAL array, dimension (LDB,N). On entry, the M-by-N matrix B. On exit, B is overwritten by the corresponding block of Q*C or Q**T*C or C*Q or C*Q**T. See Further Details.
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,M).
WORK — REAL array. The dimension of WORK is N*MB if SIDE = 'L', or M*MB if SIDE = 'R'.
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

V = [V1] [V2].

Now check the SIDE vibes:

If SIDE = 'L':  C = [A]  where A is K-by-N,  B is M-by-N and V is K-by-M.
                    [B]

If SIDE = 'R':  C = [A B]  where A is M-by-K, B is M-by-N and V is K-by-N.

The real orthogonal matrix Q is formed from V and T.

If TRANS='N' and SIDE='L', C is on exit replaced with Q * C.
If TRANS='T' and SIDE='L', C is on exit replaced with Q**T * C.
If TRANS='N' and SIDE='R', C is on exit replaced with C * Q.
If TRANS='T' and SIDE='R', C is on exit replaced with C * Q**T.

Definition at line 212 of file stpmlqt.f.

✨ `subroutine ztpmlqt` ✨

   subroutine ztpmlqt (character side, character trans, integer m, integer n,
       integer k, integer l, integer mb, complex*16, dimension( ldv, * ) v,
       integer ldv, complex*16, dimension( ldt, * ) t, integer ldt, complex*16,
       dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b,
       integer ldb, complex*16, dimension( * ) work, integer info)
       ZTPMLQT

🎯 Purpose

ZTPMLQT applies a complex unitary matrix Q obtained from a complex block reflector H to a general complex matrix C, which consists of two blocks A and B. It’s giving main character energy for complex matrices. 💜

🌈 Parameters

SIDE — CHARACTER*1. = 'L': apply Q or Q**H from the Left; = 'R': apply Q or Q**H from the Right.
TRANS — CHARACTER*1. = 'N': No transpose, apply Q; = 'C': Conjugate transpose, apply Q**H.
M — INTEGER. The number of rows of the matrix B. M >= 0.
N — INTEGER. The number of columns of the matrix B. N >= 0.
K — INTEGER. The number of elementary reflectors whose product defines the matrix Q.
L — INTEGER. The order of the trapezoidal part of V. K >= L >= 0. See Further Details.
MB — INTEGER. The block size used for the storage of T. K >= MB >= 1. This must be the same value of MB used to generate T in ZTPLQT.
V — COMPLEX*16 array, dimension (LDV,K). The i-th row must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by ZTPLQT in B. See Further Details.
LDV — INTEGER. The leading dimension of the array V. LDV >= K.
T — COMPLEX*16 array, dimension (LDT,K). The upper triangular factors of the block reflectors as returned by ZTPLQT, stored as a MB-by-K matrix.
LDT — INTEGER. The leading dimension of the array T. LDT >= MB.
A — COMPLEX*16 array, dimension (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R'. On entry, the K-by-N or M-by-K matrix A. On exit, A is overwritten by the corresponding block of Q*C or Q**H*C or C*Q or C*Q**H. See Further Details.
LDA — INTEGER. The leading dimension of the array A. If SIDE = 'L', LDA >= max(1,K); If SIDE = 'R', LDA >= max(1,M).
B — COMPLEX*16 array, dimension (LDB,N). On entry, the M-by-N matrix B. On exit, B is overwritten by the corresponding block of Q*C or Q**H*C or C*Q or C*Q**H. See Further Details.
LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,M).
WORK — COMPLEX*16 array. The dimension of WORK is N*MB if SIDE = 'L', or M*MB if SIDE = 'R'.
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 so the columns of the pentagonal matrix V contain the elementary reflectors H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a trapezoidal block V2:

V = [V1] [V2].

Not me being obsessed with a matrix that's literally two blocks in a trench coat 💅

The size of the trapezoidal block V2 is determined by the parameter L, where 0 <= L <= K; V2 is lower trapezoidal, consisting of the first L rows of a K-by-K upper triangular matrix. If L=K, V2 is lower triangular; if L=0, there is no trapezoidal block, hence V = V1 is rectangular.

tl;dr bestie: L is the slider. Max it out (L=K) and V2 goes full lower triangular ✨; zero it out (L=0) and the trapezoidal block just does not exist, so V = V1 and it's plain rectangular. The range is 0 <= L <= K, no exceptions, no vibes-based values 🎀

Then the shapes kick in, and honestly it's giving choose your own adventure 🦋

If SIDE = 'L': C gets stacked like this, where A is K-by-N, B is M-by-N and V is K-by-M:

If SIDE = 'L':  C = [A]  where A is K-by-N,  B is M-by-N and V is K-by-M.
                    [B]

If SIDE = 'R': now it's side-by-side instead, where A is M-by-K, B is M-by-N and V is K-by-N:

If SIDE = 'R':  C = [A B]  where A is M-by-K, B is M-by-N and V is K-by-N.

The complex unitary matrix Q is formed from V and T. That's the whole glow-up right there 👑 V and T walk in, Q walks out.

And then depending on your TRANS + SIDE combo, C gets replaced on exit like so — four flavours, one icon 🍓:

If TRANS='N' and SIDE='L', C is on exit replaced with Q * C.
If TRANS='C' and SIDE='L', C is on exit replaced with Q**H * C.
If TRANS='N' and SIDE='R', C is on exit replaced with C * Q.
If TRANS='C' and SIDE='R', C is on exit replaced with C * Q**H.

note the **H bestie 🫶 — the TRANS='C' cases are the conjugate-transpose girlies, not a copy-paste typo, iykyk.

Definition at line 212 of file ztpmlqt.f.

👑 Author

Generated automatically by Doxygen for LAPACK from the source code.

Not a human author, not a ghostwriter — literally a doc generator doing its little thing and honestly? it ate. ✨

💖 and that's the whole page, bestie 💖

This lil masterpiece is LAPACK Version 3.12.0, living at tpmlqt(3) — go forth, call it correctly, and may your L always be in range 🎀🦋🌸💜

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

tpmlqt(3)

tpmlqt(3) — Library Functions Manual 🦋✨

NAME

SYNOPSIS ✨

Functions

Detailed Description

Function Documentation

ctpmlqt — CTPMLQT 🦋

dtpmlqt — DTPMLQT 💜

✨ subroutine stpmlqt ✨

🎯 Purpose

🌸 Parameters

👑 Author

🦋 Further Details

✨ subroutine ztpmlqt ✨

🎯 Purpose

🌈 Parameters

👑 Author

🧚 Further Details

👑 Author

`ctpmlqt` — CTPMLQT 🦋

`dtpmlqt` — DTPMLQT 💜

✨ `subroutine stpmlqt` ✨

✨ `subroutine ztpmlqt` ✨