geqrt(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

💖 geqrt(3) — Library Functions Manual 💖

okay so bestie, this is geqrt(3) from the Library Functions Manual, and it's giving linear algebra main character energy ✨ — blocked QR factorization, four flavors (real, double, complex, complex*16), the whole squad. Let's get into it 🫶

🌸 NAME 🌸

geqrt — geqrt: QR factor, with T 💅

✨ SYNOPSIS ✨

🎀 Functions 🎀

subroutine cgeqrt (m, n, nb, a, lda, t, ldt, work, info) — CGEQRT
subroutine dgeqrt (m, n, nb, a, lda, t, ldt, work, info) — DGEQRT
subroutine sgeqrt (m, n, nb, a, lda, t, ldt, work, info) — SGEQRT
subroutine zgeqrt (m, n, nb, a, lda, t, ldt, work, info) — ZGEQRT

🦋 Detailed Description 🦋

This is the landing page for the whole geqrt family — every single variant gets its own full writeup under Function Documentation below, so scroll like your GPA depends on it 📚

👑 Function Documentation 👑

💜 subroutine cgeqrt

subroutine cgeqrt (integer m, integer n, integer nb, complex, dimension(
    lda, * ) a, integer lda, complex, dimension( ldt, * ) t, integer ldt,
    complex, dimension( * ) work, integer info)

CGEQRT — the complex one, obviously 💅

🌟 Purpose 🌟

CGEQRT computes a blocked QR factorization of a complex M-by-N matrix A using the compact WY representation of Q. No cap, that's the whole gig ✨

🌟 Parameters 🌟

M — M is INTEGER. The number of rows of the matrix A. M >= 0 (zero rows is legal, bestie, it's just a very sad little matrix 😭).
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
NB — NB is INTEGER. The block size to be used in the blocked QR. MIN(M,N) >= NB >= 1. Pick it wisely, it's the vibe check for your blocking 🔥
A — A is COMPLEX array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal are the columns of V. So it comes in as A and leaves as two whole moods — iconic behaviour 👑
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
T — T is COMPLEX array, dimension (LDT,MIN(M,N)). The upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See below for further details 🧚
LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= NB.
WORK — WORK is COMPLEX array, dimension (NB*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 matrix V stores the elementary reflectors H(i) in the i-th column below the diagonal. For example, if M=5 and N=3, the matrix V is

             V = (  1       )
                 ( v1  1    )
                 ( v1 v2  1 )
                 ( v1 v2 v3 )
                 ( v1 v2 v3 )

where the vi's represent the vectors which define H(i), which are returned in the matrix A. The 1's along the diagonal of V are not stored in A. (Not me storing everything except the ones 🙈)

Let K=MIN(M,N). The number of blocks is B = ceiling(K/NB), where each block is of order NB except for the last block, which is of order IB = K - (B-1)*NB. For each of the B blocks, a upper triangular block reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB for the last block) T's are stored in the NB-by-K matrix T as

             T = (T1 T2 ... TB).

Definition at line 140 of file cgeqrt.f.

💜 subroutine dgeqrt

subroutine dgeqrt (integer m, integer n, integer nb, double precision,
    dimension( lda, * ) a, integer lda, double precision, dimension( ldt, *
    ) t, integer ldt, double precision, dimension( * ) work, integer info)

DGEQRT — double precision, double the drama 🔥

🌟 Purpose 🌟

DGEQRT computes a blocked QR factorization of a real M-by-N matrix A using the compact WY representation of Q. Same energy, real numbers ✨

🌟 Parameters 🌟

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
NB — NB is INTEGER. The block size to be used in the blocked QR. MIN(M,N) >= NB >= 1.
A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal are the columns of V.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
T — T is DOUBLE PRECISION array, dimension (LDT,MIN(M,N)). The upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See below for further details.
LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= NB.
WORK — WORK is DOUBLE PRECISION array, dimension (NB*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 matrix V stores the elementary reflectors H(i) in the i-th column below the diagonal. For example, if M=5 and N=3, the matrix V is

             V = (  1       )
                 ( v1  1    )
                 ( v1 v2  1 )
                 ( v1 v2 v3 )
                 ( v1 v2 v3 )

where the vi's represent the vectors which define H(i), which are returned in the matrix A. The 1's along the diagonal of V are not stored in A.

             T = (T1 T2 ... TB).

Definition at line 140 of file dgeqrt.f.

💜 subroutine sgeqrt

subroutine sgeqrt (integer m, integer n, integer nb, real, dimension( lda, *
    ) a, integer lda, real, dimension( ldt, * ) t, integer ldt, real,
    dimension( * ) work, integer info)

SGEQRT — single precision, single-handedly carrying the real-number agenda 🎀

🌟 Purpose 🌟

SGEQRT computes a blocked QR factorization of a real M-by-N matrix A using the compact WY representation of Q. Periodt 💅

🌟 Parameters 🌟

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
NB — NB is INTEGER. The block size to be used in the blocked QR. MIN(M,N) >= NB >= 1.
A — A is REAL array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal are the columns of V.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
T — T is REAL array, dimension (LDT,MIN(M,N)). The upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See below for further details.
LDT — LDT is INTEGER. The leading dimension of the array T. LDT >= NB.
WORK — WORK is REAL array, dimension (NB*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 matrix V stores the elementary reflectors H(i) in the i-th column below the diagonal. For example, if M=5 and N=3, the matrix V is

             V = (  1       )
                 ( v1  1    )
                 ( v1 v2  1 )
                 ( v1 v2 v3 )
                 ( v1 v2 v3 )

where the vi's represent the vectors which define H(i), which are returned in the matrix A. The 1's along the diagonal of V are not stored in A.

             T = (T1 T2 ... TB).

Definition at line 140 of file sgeqrt.f.

💜 subroutine zgeqrt

subroutine zgeqrt (integer m, integer n, integer nb, complex*16, dimension(
    lda, * ) a, integer lda, complex*16, dimension( ldt, * ) t, integer ldt,
    complex*16, dimension( * ) work, integer info)

ZGEQRT — complex*16, aka the double-precision complex one, the final boss of this squad 🌈

🌟 Purpose 🌟

ZGEQRT computes a blocked QR factorization of a complex M-by-N matrix A using the compact WY representation of Q. Big numbers, big feelings 💜

🌟 Parameters 🌟

M — M is INTEGER. The number of rows of the matrix A. M >= 0.
N — N is INTEGER. The number of columns of the matrix A. N >= 0.
NB — NB is INTEGER. The block size to be used in the blocked QR. MIN(M,N) >= NB >= 1.
A
A — COMPLEX*16 array, dimension (LDA,N). On entry it's the M-by-N matrix A, and on exit the elements on and above the diagonal hold the min(M,N)-by-N upper trapezoidal matrix R (and R is upper triangular if M >= N), while the elements below the diagonal are the columns of V. So A literally gets a glow-up mid-call, no cap. 💅
LDA — INTEGER, obviously, and it's the leading dimension of the array A. Rule of the land: LDA >= max(1,M). Respect the bounds bestie, she's picky about this. 🎀
T — COMPLEX*16 array, dimension (LDT,MIN(M,N)). It's holding the upper triangular block reflectors, packed all cute and compact as a sequence of upper triangular blocks. The details are below if you're curious (you should be). ✨
LDT — INTEGER, the leading dimension of the array T. Constraint: LDT >= NB. That's it, that's the rule. 🦋
WORK — COMPLEX*16 array, dimension (NB*N). Scratch space, doing her little behind-the-scenes job. 🌸
INFO — INTEGER, the vibe check at the end:
- = 0: successful exit, we ate and left no crumbs 🍓
- < 0: if INFO = -i, the i-th argument had an illegal value, aka somebody passed something cursed 😭

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Like, four whole institutions collaborated on this. Group project that actually worked. iconic. 🫶

🔍 Further Details:

Okay this is the fun part, the lore. 💖

The matrix V stores the elementary reflectors H(i) in the i-th column below the diagonal. For example, if M=5 and N=3, the matrix V is:

     V = (  1       )
         ( v1  1    )
         ( v1 v2  1 )
         ( v1 v2 v3 )
         ( v1 v2 v3 )

where the vi's represent the vectors which define H(i), which are returned in the matrix A. And here's the cute little detail: the 1's along the diagonal of V are not stored in A. They're implied, they're a vibe, they're main character energy that doesn't need to be written down. ✨

Let K=MIN(M,N). The number of blocks is B = ceiling(K/NB), where each block is of order NB except for the last block, which is of order IB = K - (B-1)*NB. So the last one is always the special little guy. 🧚

For each of the B blocks, an upper triangular block reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB for the last block) T's are stored in the NB-by-K matrix T as:

     T = (T1 T2 ... TB).

Look at her, all lined up in a row. Obsessed. 🎀

Definition at line 140 of file zgeqrt.f.

🦋 Author

Generated automatically by Doxygen for LAPACK from the source code — so like, a robot wrote this part, but a robot with taste ngl.

✨ and that's the whole page bestie ✨ this lil masterpiece is from LAPACK Version 3.12.0, and the date on the original footer is baked right into that same line — so go forth and factor your block reflectors, fr fr. 💜 slay 💅

📦 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