SRC/zggqrf.f(3)

📦
version 3.12.1 · 7 versions
, in arch rolling

💜 NAME

SRC/zggqrf.f — bestie, this is the LAPACK page for the ZGGQRF subroutine, and it's literally the glow-up machine for your generalized QR factorization ✨

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine zggqrf (n, m, p, a, lda, taua, b, ldb, taub, work, lwork,
    info)

ZGGQRF 🎀

🫶 Function/Subroutine Documentation 🫶

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 🌸

Okay so hear me out 💅 — ZGGQRF computes a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B. Like, both of them, together, one iconic little package:

            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 these 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. Not me drawing shapes with parentheses 😭 it's giving linear algebra art fr fr.

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. No cap, that's the whole serve 🔥

🦋 Parameters 🦋

  • N — INTEGER. The number of rows of the matrices A and B. N >= 0. (zero rows is valid, she's just chillin' 💀)
  • 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 — COMPLEX*16 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). She gets a whole makeover on the way out, obsessed 🧚
  • LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
  • TAUA — COMPLEX*16 array, dimension (min(N,M)). The scalar factors of the elementary reflectors which represent the unitary matrix Q (see Further Details). Little tau-a icons 💖
  • B — 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). ✨
  • LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N).
  • TAUB — COMPLEX*16 array, dimension (min(N,P)). The scalar factors of the elementary reflectors which represent the unitary matrix Z (see Further Details). 💫
  • WORK — COMPLEX*16 array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
  • LWORK — 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.
Heads up bestie ⚠️ 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. So like, don't come for XERBLA, she's off duty for that one 😌
  • 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 💜

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.

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.

🎀 Definition 🎀

Definition at line 213 of file zggqrf.f.

🫶 Author (the meta one) 🫶

Generated automatically by Doxygen for LAPACK from the source code. She didn't write herself, she was generated, and that's main character energy honestly 🌈

ok that's the whole page bestie, go forth and factorize 🦋✨

LAPACK — Version 3.12.0 — SRC/zggqrf.f(3) 💅 periodt, slay, bye 🎀

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