geqr(3)

📦

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🦋 geqr(3) — QR factor, flexible

Library Functions Manual bestie, and we are so in our flexible QR factor era. ✨

🎀 NAME 🎀

geqr — geqr: QR factor, flexible

🌸 SYNOPSIS 🌸

🧚 Functions

subroutine cgeqr (m, n, a, lda, t, tsize, work, lwork, info) — CGEQR
subroutine dgeqr (m, n, a, lda, t, tsize, work, lwork, info) — DGEQR
subroutine sgeqr (m, n, a, lda, t, tsize, work, lwork, info) — SGEQR
subroutine zgeqr (m, n, a, lda, t, tsize, work, lwork, info) — ZGEQR

📚 Detailed Description

Okay so the original page literally leaves this section empty — no cap, nothing to see here. The actual lore starts in Function Documentation below. 💅

🔥 Function Documentation

💖 `subroutine cgeqr`

subroutine cgeqr (integer m, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) t, integer tsize, complex, dimension( * ) work, integer lwork, integer info)

CGEQR

✨ Purpose

CGEQR computes a QR factorization of a complex M-by-N matrix A:

A = Q * ( R ),
        ( 0 )

where:

Q is a M-by-M orthogonal matrix;
R is an upper-triangular N-by-N matrix;
0 is a (M-N)-by-N zero matrix, if M > N.

🎀 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.
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 used to store part of the data structure to represent Q.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
T — T is COMPLEX array, dimension (MAX(5,TSIZE)). On exit, if INFO = 0, T(1) returns optimal (or either minimal or optimal, if query is assumed) TSIZE. See TSIZE for details. Remaining T contains part of the data structure used to represent Q. If one wants to apply or construct Q, then one needs to keep T (in addition to A) and pass it to further subroutines.
TSIZE — TSIZE is INTEGER. If TSIZE >= 5, the dimension of the array T. If TSIZE = -1 or -2, then a workspace query is assumed. The routine only calculates the sizes of the T and WORK arrays, returns these values as the first entries of the T and WORK arrays, and no error message related to T or WORK is issued by XERBLA. If TSIZE = -1, the routine calculates optimal size of T for the optimum performance and returns this value in T(1). If TSIZE = -2, the routine calculates minimal size of T and returns this value in T(1).
WORK — (workspace) COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) contains optimal (or either minimal or optimal, if query was assumed) LWORK. See LWORK for details.
LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= 1. If LWORK = -1 or -2, then a workspace query is assumed. The routine only calculates the sizes of the T and WORK arrays, returns these values as the first entries of the T and WORK arrays, and no error message related to T or WORK is issued by XERBLA. If LWORK = -1, the routine calculates optimal size of WORK for the optimal performance and returns this value in WORK(1). If LWORK = -2, the routine calculates minimal size of WORK and returns this value in WORK(1).
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 whole point of this interface is maximum freedom for the developers — creating any QR factorization algorithm they wish. The triangular (trapezoidal) R has to be stored in the upper part of A. The lower part of A and the array T can be used to store any relevant information for applying or constructing the Q factor. The WORK array can safely be discarded after exit. ✨

⚠️ Caution: One should not expect the sizes of T and WORK to be the same from one LAPACK implementation to the other, or even from one execution to the other. A workspace query (for T and WORK) is needed at each execution. However, for a given execution, the size of T and WORK are fixed and will not change from one query to the next.

🧬 Further Details particular to this LAPACK implementation

These details are particular for this LAPACK implementation. Users should not take them for granted. These details may change in the future, and are not likely true for another LAPACK implementation. These details are relevant if one wants to try to understand the code. They are not part of the interface.

In this version,

T(2): row block size (MB)
T(3): column block size (NB)
T(6:TSIZE): data structure needed for Q, computed by
                 CLATSQR or CGEQRT

Depending on the matrix dimensions M and N, and row and column block sizes MB and NB returned by ILAENV, CGEQR will use either CLATSQR (if the matrix is tall-and-skinny) or CGEQRT to compute the QR factorization. 🫶

📍 Definition at line 174 of file cgeqr.f.

💙 `subroutine dgeqr`

subroutine dgeqr (integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) t, integer tsize, double precision, dimension( * ) work, integer lwork, integer info)

DGEQR

✨ Purpose

DGEQR computes a QR factorization of a real M-by-N matrix A:

A = Q * ( R ),
        ( 0 )

where:

Q is a M-by-M orthogonal matrix;
R is an upper-triangular N-by-N matrix;
0 is a (M-N)-by-N zero matrix, if M > N.

🎀 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.
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 used to store part of the data structure to represent Q.
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
T — T is DOUBLE PRECISION array, dimension (MAX(5,TSIZE)). On exit, if INFO = 0, T(1) returns optimal (or either minimal or optimal, if query is assumed) TSIZE. See TSIZE for details. Remaining T contains part of the data structure used to represent Q. If one wants to apply or construct Q, then one needs to keep T (in addition to A) and pass it to further subroutines.
TSIZE — TSIZE is INTEGER. If TSIZE >= 5, the dimension of the array T. If TSIZE = -1 or -2, then a workspace query is assumed. The routine only calculates the sizes of the T and WORK arrays, returns these values as the first entries of the T and WORK arrays, and no error message related to T or WORK is issued by XERBLA. If TSIZE = -1, the routine calculates optimal size of T for the optimum performance and returns this value in T(1). If TSIZE = -2, the routine calculates minimal size of T and returns this value in T(1).
WORK — (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) contains optimal (or either minimal or optimal, if query was assumed) LWORK. See LWORK for details.
LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= 1. If LWORK = -1 or -2, then a workspace query is assumed. The routine only calculates the sizes of the T and WORK arrays, returns these values as the first entries of the T and WORK arrays, and no error message related to T or WORK is issued by XERBLA. If LWORK = -1, the routine calculates optimal size of WORK for the optimal performance and returns this value in WORK(1). If LWORK = -2, the routine calculates minimal size of WORK and returns this value in WORK(1).
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

💖 Further Details (the dgeqr ones)

Okay so the whole point of this interface is maximum freedom for the devs — like, they get to build literally any QR factorization algorithm they vibe with. The triangular (trapezoidal) R has to live in the upper part of A. The lower part of A and the array T? Free real estate — use them to stash whatever info you need for applying or constructing the Q factor. And the WORK array? Bestie, you can throw it away after exit, it's done its job. 🫶

⚠️ Caution, no cap: do not expect the sizes of T and WORK to match from one LAPACK implementation to another — or even from one execution to the next. You need a workspace query (for T and WORK) at each execution. That said, for a given execution, the sizes of T and WORK are fixed and will not change from one query to the next. ✨

🌈 Further Details particular to this LAPACK implementation

Okay these deets are specific to this LAPACK implementation — do not take them for granted, bestie. They may change in the future, and they're lowkey not likely true for another LAPACK implementation. These are the deets that matter if you're trying to actually understand the code. They are not part of the interface. 💅

In this version:

         T(2): row block size (MB)
         T(3): column block size (NB)
         T(6:TSIZE): data structure needed for Q, computed by
                          DLATSQR or DGEQRT

Depending on the matrix dimensions M and N, and row and column block sizes MB and NB returned by ILAENV, DGEQR will use either DLATSQR (if the matrix is tall-and-skinny) or DGEQRT to compute the QR factorization. 🦋

Definition at line 174 of file dgeqr.f.

🌸 `subroutine sgeqr` — the single-precision girlie ✨

subroutine sgeqr (integer m, integer n, real, dimension( lda, * ) a, integer
    lda, real, dimension( * ) t, integer tsize, real, dimension( * ) work,
    integer lwork, integer info)

SGEQR

🎀 Purpose:

SGEQR computes a QR factorization of a real M-by-N matrix A:

   A = Q * ( R ),
           ( 0 )

where:

Q is a M-by-M orthogonal matrix;
R is an upper-triangular N-by-N matrix;
0 is a (M-N)-by-N zero matrix, if M > N.

🧚 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.
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 used to store part of the data structure to represent Q. 👑
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
T — T is REAL array, dimension (MAX(5,TSIZE)). On exit, if INFO = 0, T(1) returns optimal (or either minimal or optimal, if query is assumed) TSIZE. See TSIZE for details. Remaining T contains part of the data structure used to represent Q. If one wants to apply or construct Q, then one needs to keep T (in addition to A) and pass it to further subroutines. 🔥
TSIZE — TSIZE is INTEGER. If TSIZE >= 5, the dimension of the array T. If TSIZE = -1 or -2, then a workspace query is assumed. The routine only calculates the sizes of the T and WORK arrays, returns these values as the first entries of the T and WORK arrays, and no error message related to T or WORK is issued by XERBLA. If TSIZE = -1, the routine calculates optimal size of T for the optimum performance and returns this value in T(1). If TSIZE = -2, the routine calculates minimal size of T and returns this value in T(1).
WORK — (workspace) REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) contains optimal (or either minimal or optimal, if query was assumed) LWORK. See LWORK for details.
LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK >= 1. If LWORK = -1 or -2, then a workspace query is assumed. The routine only calculates the sizes of the T and WORK arrays, returns these values as the first entries of the T and WORK arrays, and no error message related to T or WORK is issued by XERBLA. If LWORK = -1, the routine calculates optimal size of WORK for the optimal performance and returns this value in WORK(1). If LWORK = -2, the routine calculates minimal size of WORK and returns this value in WORK(1).
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 goal of the interface is to give maximum freedom to the developers for creating any QR factorization algorithm they wish. The triangular (trapezoidal) R has to be stored in the upper part of A. The lower part of A and the array T can be used to store any relevant information for applying or constructing the Q factor. The WORK array can safely be discarded after exit.

⚠️ Caution: One should not expect the sizes of T and WORK to be the same from one LAPACK implementation to the other, or even from one execution to the other. A workspace query (for T and WORK) is needed at each execution. However, for a given execution, the size of T and WORK are fixed and will not change from one query to the next.

💜 Further Details particular to this LAPACK implementation:

In this version,

         T(2): row block size (MB)
         T(3): column block size (NB)
         T(6:TSIZE): data structure needed for Q, computed by
                          SLATSQR or SGEQRT

Depending on the matrix dimensions M and N, and row and column block sizes MB and NB returned by ILAENV, SGEQR will use either SLATSQR (if the matrix is tall-and-skinny) or SGEQRT to compute the QR factorization. 💫

Definition at line 174 of file sgeqr.f.

🦋 `subroutine zgeqr` — double complex, double the drama 💅

subroutine zgeqr (integer m, integer n, complex*16, dimension( lda, * ) a,
    integer lda, complex*16, dimension( * ) t, integer tsize, complex*16,
    dimension( * ) work, integer lwork, integer info)

ZGEQR

🎀 Purpose:

ZGEQR computes a QR factorization of a complex M-by-N matrix A:

   A = Q * ( R ),
           ( 0 )

where:

Q is a M-by-M orthogonal matrix;
R is an upper-triangular N-by-N matrix;
0 is a (M-N)-by-N zero matrix, if M > N.

🧚 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.
A — A is COMPLEX*16 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 used to store part of the data structure to represent Q. 👑
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
T — T is COMPLEX*16 array, dimension (MAX(5,TSIZE)). On exit, if INFO = 0, T(1) returns optimal (or either minimal or optimal, if query is assumed) TSIZE. See TSIZE for details. Remaining T contains part of the data structure used to represent Q. If one wants to apply or construct Q, then one needs to keep T (in addition to A) and pass it to further subroutines. 🔥
TSIZE — TSIZE is INTEGER. If TSIZE >= 5, the dimension of the array T. If TSIZE = -1 or -2, then a workspace query is assumed. The routine only calculates the sizes of the T and WORK arrays, returns these values as the first entries of the T and WORK arrays, and no error message related to T or WORK is issued by XERBLA. If TSIZE = -1, the routine calculates optimal size of T for the optimum performance and returns this value in T(1). If TSIZE = -2, the routine calculates minimal size of T and returns this value in T(1).
WORK — (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) contains optimal (or either minimal or optimal, if query was assumed) LWORK. See LWORK for details.
LWORK — INTEGER. It's the dimension of the array WORK, and the rule is LWORK >= 1. If LWORK = -1 or -2, then a workspace query is assumed. The routine only calculates the sizes of the T and WORK arrays, returns these values as the first entries of the T and WORK arrays, and no error message related to T or WORK is issued by XERBLA. If LWORK = -1, the routine calculates optimal size of WORK for the optimal performance and returns this value in WORK(1). If LWORK = -2, the routine calculates minimal size of WORK and returns this value in WORK(1). 💅
INFO — INTEGER, and here's the vibe check:
- = 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 whole point of this interface is to give developers maximum freedom to create any QR factorization algorithm they want. The triangular (trapezoidal) R has to be stored in the upper part of A. The lower part of A and the array T can be used to store any relevant information for applying or constructing the Q factor. The WORK array can safely be discarded after exit. ✨

Caution, bestie: one should not expect the sizes of T and WORK to be the same from one LAPACK implementation to the other, or even from one execution to the other. A workspace query (for T and WORK) is needed at each execution. However, for a given execution, the size of T and WORK are fixed and will not change from one query to the next. 🫶

Further Details particular to this LAPACK implementation:

These details are particular for this LAPACK implementation. Users should not take them for granted. These details may change in the future, and are not likely true for another LAPACK implementation. These details are relevant if one wants to try to understand the code. They are not part of the interface. 👑

In this version,

T(2): row block size (MB)
T(3): column block size (NB)
T(6:TSIZE): data structure needed for Q, computed by
                 ZLATSQR or ZGEQRT

Depending on the matrix dimensions M and N, and row and column block sizes MB and NB returned by ILAENV, ZGEQR will use either ZLATSQR (if the matrix is tall-and-skinny) or ZGEQRT to compute the QR factorization. 🌸

Definition at line 174 of file zgeqr.f.

Author

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geqr(3)

🦋 geqr(3) — QR factor, flexible

🎀 NAME 🎀

🌸 SYNOPSIS 🌸

🧚 Functions

📚 Detailed Description

🔥 Function Documentation

💖 subroutine cgeqr

✨ Purpose

🎀 Parameters

👑 Author

🔍 Further Details

🧬 Further Details particular to this LAPACK implementation

💙 subroutine dgeqr

✨ Purpose

🎀 Parameters

👑 Author

🔍 Further Details

💖 Further Details (the dgeqr ones)

🌈 Further Details particular to this LAPACK implementation

🌸 subroutine sgeqr — the single-precision girlie ✨

🎀 Purpose:

🧚 Parameters

🌟 Author

🍓 Further Details

💜 Further Details particular to this LAPACK implementation:

🦋 subroutine zgeqr — double complex, double the drama 💅

🎀 Purpose:

🧚 Parameters

Author

Further Details

Further Details particular to this LAPACK implementation:

Author

💖 `subroutine cgeqr`

💙 `subroutine dgeqr`

🌸 `subroutine sgeqr` — the single-precision girlie ✨

🦋 `subroutine zgeqr` — double complex, double the drama 💅