ggqrf(3)

📦

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okay so welcome to ggqrf(3) — Library Functions Manual — the generalized QR factor page, and honestly? iconic linear algebra behavior 🎀✨

NAME 🌸

ggqrf — ggqrf: Generalized QR factor (short name, big main-character energy 💅)

SYNOPSIS ✨

okay bestie, here's the function lineup. Four flavors, same general vibe:

Functions 💖

       subroutine cggqrf (n, m, p, a, lda, taua, b, ldb, taub, work, lwork,
           info)
           CGGQRF
       subroutine dggqrf (n, m, p, a, lda, taua, b, ldb, taub, work, lwork,
           info)
           DGGQRF
       subroutine sggqrf (n, m, p, a, lda, taua, b, ldb, taub, work, lwork,
           info)
           SGGQRF
       subroutine zggqrf (n, m, p, a, lda, taua, b, ldb, taub, work, lwork,
           info)
           ZGGQRF

Detailed Description 🦋

this section is literally just a title in the source, no extra prose, bestie — the real tea starts in Function Documentation 👑

Function Documentation 👑

subroutine `cggqrf` 💅

subroutine cggqrf (integer n, integer m, integer p, complex, dimension( lda,
* ) a, integer lda, complex, dimension( * ) taua, complex, dimension(
ldb, * ) b, integer ldb, complex, dimension( * ) taub, complex,
dimension( * ) work, integer lwork, integer info)

this one's labeled CGGQRF, and yes it's giving complex unitary realness ✨

Purpose 🎀

okay so CGGQRF computes a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B:

            A = Q*R,        B = Q*T*Z,

where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix, and R and T assume one of the forms:

if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
                (  0  ) N-M                         N   M-N
                   M

where R11 is upper triangular, and

if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
                 P-N  N                           ( T21 ) P
                                                     P

where T12 or T21 is upper triangular.

in particular, if B is square and nonsingular, the GQR factorization of A and B implicitly gives the QR factorization of inv(B)*A:

             inv(B)*A = Z**H * (inv(T)*R)

where inv(B) denotes the inverse of the matrix B, and Z' denotes the conjugate transpose of matrix Z.

Parameters 🫶

N — N is INTEGER. the number of rows of the matrices A and B. N >= 0.
M — M is INTEGER. the number of columns of the matrix A. M >= 0.
P — P is INTEGER. the number of columns of the matrix B. P >= 0.
A — A is COMPLEX array, dimension (LDA,M). On entry, the N-by-M matrix A. On exit, the elements on and above the diagonal of the array contain the min(N,M)-by-M upper trapezoidal matrix R (R is upper triangular if N >= M); the elements below the diagonal, with the array TAUA, represent the unitary matrix Q as a product of min(N,M) elementary reflectors (see Further Details).
LDA — LDA is INTEGER. the leading dimension of the array A. LDA >= max(1,N).
TAUA — TAUA is COMPLEX array, dimension (min(N,M)). the scalar factors of the elementary reflectors which represent the unitary matrix Q (see Further Details).
B — B is COMPLEX array, dimension (LDB,P). On entry, the N-by-P matrix B. On exit, if N <= P, the upper triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; if N > P, the elements on and above the (N-P)-th subdiagonal contain the N-by-P upper trapezoidal matrix T; the remaining elements, with the array TAUB, represent the unitary matrix Z as a product of elementary reflectors (see Further Details).
LDB — LDB is INTEGER. the leading dimension of the array B. LDB >= max(1,N).
TAUB — TAUB is COMPLEX array, dimension (min(N,P)). the scalar factors of the elementary reflectors which represent the unitary matrix Z (see Further Details).
WORK — WORK is COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — LWORK is INTEGER. the dimension of the array WORK. LWORK >= max(1,N,M,P). For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3), where NB1 is the optimal blocksize for the QR factorization of an N-by-M matrix, NB2 is the optimal blocksize for the RQ factorization of an N-by-P matrix, and NB3 is the optimal blocksize for a call of CUNMQR. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

Author 💖

the source says:

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

Further Details 🦋

the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(n,m).

each H(i) has the form

H(i) = I - taua * v * v**H

where taua is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and taua in TAUA(i). To form Q explicitly, use LAPACK subroutine CUNGQR. To use Q to update another matrix, use LAPACK subroutine CUNMQR.

the matrix Z is represented as a product of elementary reflectors

Z = H(1) H(2) . . . H(k), where k = min(n,p).

each H(i) has the form

H(i) = I - taub * v * v**H

where taub is a complex scalar, and v is a complex vector with v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in B(n-k+i,1:p-k+i-1), and taub in TAUB(i). To form Z explicitly, use LAPACK subroutine CUNGRQ. To use Z to update another matrix, use LAPACK subroutine CUNMRQ.

Definition at line 213 of file cggqrf.f.

subroutine `dggqrf` 💜

subroutine dggqrf (integer n, integer m, integer p, double precision,
dimension( lda, * ) a, integer lda, double precision, dimension( * )
taua, double precision, dimension( ldb, * ) b, integer ldb, double
precision, dimension( * ) taub, double precision, dimension( * ) work,
integer lwork, integer info)

this one's labeled DGGQRF, and it's the double-precision orthogonal sibling — no cap 🫶

Purpose 🎀

DGGQRF computes a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B:

            A = Q*R,        B = Q*T*Z,

where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal matrix, and R and T assume one of the forms:

if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
                (  0  ) N-M                         N   M-N
                   M

where R11 is upper triangular, and

if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
                 P-N  N                           ( T21 ) P
                                                     P

where T12 or T21 is upper triangular.

in particular, if B is square and nonsingular, the GQR factorization of A and B implicitly gives the QR factorization of inv(B)*A:

             inv(B)*A = Z**T*(inv(T)*R)

where inv(B) denotes the inverse of the matrix B, and Z**T denotes the transpose of the matrix Z.

Parameters 🫶

N — N is INTEGER. the number of rows of the matrices A and B. N >= 0.
M — M is INTEGER. the number of columns of the matrix A. M >= 0.
P — P is INTEGER. the number of columns of the matrix B. P >= 0.
A — A is DOUBLE PRECISION array, dimension (LDA,M). On entry, the N-by-M matrix A. On exit, the elements on and above the diagonal of the array contain the min(N,M)-by-M upper trapezoidal matrix R (R is upper triangular if N >= M); the elements below the diagonal, with the array TAUA, represent the orthogonal matrix Q as a product of min(N,M) elementary reflectors (see Further Details).
LDA — LDA is INTEGER. the leading dimension of the array A. LDA >= max(1,N).
TAUA — TAUA is DOUBLE PRECISION array, dimension (min(N,M)). the scalar factors of the elementary reflectors which represent the orthogonal matrix Q (see Further Details).
B — B is DOUBLE PRECISION array, dimension (LDB,P). On entry, the N-by-P matrix B. On exit, if N <= P, the upper triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; if N > P, the elements on and above the (N-P)-th subdiagonal contain the N-by-P upper trapezoidal matrix T; the remaining elements, with the array TAUB, represent the orthogonal matrix Z as a product of elementary reflectors (see Further Details).
LDB — LDB is INTEGER. the leading dimension of the array B. LDB >= max(1,N).
TAUB — TAUB is DOUBLE PRECISION array, dimension (min(N,P)). the scalar factors of the elementary reflectors which represent the orthogonal matrix Z (see Further Details).
WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

`LWORK` 💀

and this is exactly where this chunk of the page cuts off, bestie — part 2 picks up the rest of LWORK's lore and then INFO ✨

LWORK — INTEGER. Dimension of the WORK array, and bestie, she has requirements. ✨
- LWORK >= max(1,N,M,P) — the absolute bare minimum, the "it'll run, don't @ me" setting.
- For optimum performance, LWORK >= max(N,M,P)*max(NB1,NB2,NB3), where NB1 is the optimal blocksize for the QR factorization of an N-by-M matrix, NB2 is the optimal blocksize for the RQ factorization of an N-by-P matrix, and NB3 is the optimal blocksize for a call of DORMQR. That's the full glam, the blowout, the "I booked the whole salon" vibe. 💅
- If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. It's literally just window shopping — no commitment, no scolding. 🛍️
INFO — INTEGER. The little report card at the end. 🎀
- = 0: successful exit. Slay, it worked.
- < 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 Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(n,m).

Each H(i) has the form

H(i) = I - taua * v * v**T

where taua is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and taua in TAUA(i). To form Q explicitly, use LAPACK subroutine DORGQR. To use Q to update another matrix, use LAPACK subroutine DORMQR.

The matrix Z is represented as a product of elementary reflectors

Z = H(1) H(2) . . . H(k), where k = min(n,p).

Each H(i) has the form

H(i) = I - taub * v * v**T

where taub is a real scalar, and v is a real vector with v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in B(n-k+i,1:p-k+i-1), and taub in TAUB(i). To form Z explicitly, use LAPACK subroutine DORGRQ. To use Z to update another matrix, use LAPACK subroutine DORMRQ.

Definition at line 213 of file dggqrf.f.

🔥 `sggqrf` — the single-precision girlie

subroutine sggqrf (integer n, integer m, integer p, real, dimension( lda, * ) a, integer lda, real, dimension( * ) taua, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) taub, real, dimension( * ) work, integer lwork, integer info)

SGGQRF 💜

Purpose:

SGGQRF computes a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B:

A = Q*R,        B = Q*T*Z,

where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal matrix, and R and T assume one of the forms:

if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
                (  0  ) N-M                         N   M-N
                   M

where R11 is upper triangular, and

if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
                 P-N  N                           ( T21 ) P
                                                    P

where T12 or T21 is upper triangular.

In particular, if B is square and nonsingular, the GQR factorization of A and B implicitly gives the QR factorization of inv(B)*A:

inv(B)*A = Z**T*(inv(T)*R)

where inv(B) denotes the inverse of the matrix B, and Z**T denotes the transpose of the matrix Z. It's giving ✨free factorization✨.

Parameters

N — INTEGER. The number of rows of the matrices A and B. N >= 0. 🦋
M — INTEGER. The number of columns of the matrix A. M >= 0.
P — INTEGER. The number of columns of the matrix B. P >= 0.
A — REAL array, dimension (LDA,M). On entry, the N-by-M matrix A.

On exit, the elements on and above the diagonal of the array contain the min(N,M)-by-M upper trapezoidal matrix R (R is upper triangular if N >= M); the elements below the diagonal, with the array TAUA, represent the orthogonal matrix Q as a product of min(N,M) elementary reflectors (see Further Details). 🌸

LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
TAUA — REAL array, dimension (min(N,M)). The scalar factors of the elementary reflectors which represent the orthogonal matrix Q (see Further Details).
B — REAL array, dimension (LDB,P). On entry, the N-by-P matrix B.

On exit, if N <= P, the upper triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; if N > P, the elements on and above the (N-P)-th subdiagonal contain the N-by-P upper trapezoidal matrix T; the remaining elements, with the array TAUB, represent the orthogonal matrix Z as a product of elementary reflectors (see Further Details). 💫

LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
TAUB — REAL array, dimension (min(N,P)). The scalar factors of the elementary reflectors which represent the orthogonal matrix Z (see Further Details).
WORK — REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. She tells you what she needs, and honestly? Listen to her. 🙈
LWORK — INTEGER. Dimension of WORK, same whole deal as before:
- LWORK >= max(1,N,M,P) — the floor, the "good enough" era.
- For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3), where NB1 is the optimal blocksize for the QR factorization of an N-by-M matrix, NB2 is the optimal blocksize for the RQ factorization of an N-by-P matrix, and NB3 is the optimal blocksize for a call of SORMQR. 🔥
- If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.
INFO — INTEGER. The verdict.
- = 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 Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(n,m).

Each H(i) has the form

H(i) = I - taua * v * v**T

where taua is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and taua in TAUA(i). To form Q explicitly, use LAPACK subroutine SORGQR. To use Q to update another matrix, use LAPACK subroutine SORMQR.

The matrix Z is represented as a product of elementary reflectors

Z = H(1) H(2) . . . H(k), where k = min(n,p).

Each H(i) has the form

H(i) = I - taub * v * v**T

where taub is a real scalar, and v is a real vector with v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in B(n-k+i,1:p-k+i-1), and taub in TAUB(i). To form Z explicitly, use LAPACK subroutine SORGRQ. To use Z to update another matrix, use LAPACK subroutine SORMRQ.

Definition at line 213 of file sggqrf.f.

💎 `zggqrf` — double complex, double the drama

subroutine zggqrf (integer n, integer m, integer p, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) taua, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) taub, complex*16, dimension( * ) work, integer lwork, integer info)

ZGGQRF 🧚

Purpose:

ZGGQRF computes a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B:

A = Q*R,        B = Q*T*Z,

where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix, and R and T assume one of the forms:

if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
                (  0  ) N-M                         N   M-N
                   M

where R11 is upper triangular, and

if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
                 P-N  N                           ( T21 ) P
                                                    P

where T12 or T21 is upper triangular.

In particular, if B is square and nonsingular, the GQR factorization of A and B implicitly gives the QR factorization of inv(B)*A:

inv(B)*A = Z**H * (inv(T)*R)

where inv(B) denotes the inverse of the matrix B, and Z**H denotes the conjugate transpose of matrix Z. Note it's **H here, not **T — she's complex, she conjugates, that's just her vibe. 💅

Parameters

N — INTEGER. The number of rows of the matrices A and B. N >= 0. ⭐
M — INTEGER. The number of columns of the matrix A. M >= 0.
P — INTEGER. The number of columns of the matrix B. P >= 0.
A —

Okay bestie, back to the argument glow-up list ✨ — these are the storage besties carrying the whole factorization vibe:

A — a COMPLEX*16 array, dimension (LDA,M). On entry, it's the N-by-M matrix A — main character energy 💅. On exit, the elements on and above the diagonal of the array hold the min(N,M)-by-M upper trapezoidal matrix R (and yes, R is upper triangular if N >= M); the elements below the diagonal, with the array TAUA, represent the unitary matrix Q as a product of min(N,M) elementary reflectors (peep Further Details below, iykyk).
LDA — is INTEGER. The leading dimension of the array A. LDA >= max(1,N). No cap, just dimension rules ✨.
TAUA — is a COMPLEX*16 array, dimension (min(N,M)). The scalar factors of the elementary reflectors which represent the unitary matrix Q (see Further Details). These little icons are important 🫶.
B — is a COMPLEX*16 array, dimension (LDB,P). On entry, the N-by-P matrix B. On exit, if N <= P, the upper triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; if N > P, the elements on and above the (N-P)-th subdiagonal contain the N-by-P upper trapezoidal matrix T; the remaining elements, with the array TAUB, represent the unitary matrix Z as a product of elementary reflectors (see Further Details). It's giving "same matrix, different era" 😭🔥.
LDB — is INTEGER. The leading dimension of the array B. LDB >= max(1,N). Periodt.
TAUB — is a COMPLEX*16 array, dimension (min(N,P)). The scalar factors of the elementary reflectors which represent the unitary matrix Z (see Further Details). Slay, tiny scalar factors, slay 💜.
WORK — is a COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. The workspace bestie has entered the chat 🌸.
LWORK — is INTEGER. The dimension of the array WORK. LWORK >= max(1,N,M,P). For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3), where NB1 is the optimal blocksize for the QR factorization of an N-by-M matrix, NB2 is the optimal blocksize for the RQ factorization of an N-by-P matrix, and NB3 is the optimal blocksize for a call of ZUNMQR.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. That's the "just asking, not committing" vibe 👀.

INFO — is INTEGER.
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value. Ruh-roh 💀.

👑 Author 👑

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

🌈 Further Details: 🌈

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(n,m).

Each H(i) has the form

H(i) = I - taua * v * v**H

where taua is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and taua in TAUA(i). To form Q explicitly, use LAPACK subroutine ZUNGQR. To use Q to update another matrix, use LAPACK subroutine ZUNMQR. Iconic behavior honestly 💅✨.

The matrix Z is represented as a product of elementary reflectors

Z = H(1) H(2) . . . H(k), where k = min(n,p).

Each H(i) has the form

H(i) = I - taub * v * v**H

where taub is a complex scalar, and v is a complex vector with v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in B(n-k+i,1:p-k+i-1), and taub in TAUB(i). To form Z explicitly, use LAPACK subroutine ZUNGRQ. To use Z to update another matrix, use LAPACK subroutine ZUNMRQ. Chef's kiss, no notes 🧚💖.

Definition at line 213 of file zggqrf.f.

🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code.

Source/version check, bestie: LAPACK Version 3.12.0, page ggqrf(3). The footer doesn't give us a date, so I'm not inventing one — no fake-date energy here, periodt 🎀💜.

📦 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

ggqrf(3)

NAME 🌸

SYNOPSIS ✨

Functions 💖

Detailed Description 🦋

Function Documentation 👑

subroutine cggqrf 💅

Purpose 🎀

Parameters 🫶

Author 💖

Further Details 🦋

subroutine dggqrf 💜

Purpose 🎀

Parameters 🫶

LWORK 💀

🫶 Author

🌈 Further Details:

🔥 sggqrf — the single-precision girlie

Purpose:

Parameters

🌷 Author

⭐ Further Details:

💎 zggqrf — double complex, double the drama

Purpose:

Parameters

👑 Author 👑

🌈 Further Details: 🌈

🧚 Author 🧚

subroutine `cggqrf` 💅

subroutine `dggqrf` 💜

`LWORK` 💀

🔥 `sggqrf` — the single-precision girlie

💎 `zggqrf` — double complex, double the drama