ZGGRQF(3)

🌸 NAME 🌸

okay so bestie, welcome to zggrqf.f(3) — it's a Library Functions Manual page, and it's giving LAPACK main character energy. ✨💅

The NAME section is literally this path, no cap:

/home/abuild/rpmbuild/BUILD/lapack-
static-3.12.0-build/lapack-3.12.0/SRC/zggrqf.f

---

✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

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

---

🧚 Function/Subroutine Documentation 🧚

   subroutine ZGGRQF (integer m, integer p, integer n, 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)
       ZGGRQF

🎀 Purpose 🎀

okay so here's the tea: ZGGRQF computes a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B:

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

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 — periodt:

if M <= N,  R = ( 0  R12 ) M,   or if M > N,  R = ( R11 ) M-N,
                 N-M  M                           ( R21 ) N
                                                     N

where R12 or R21 is upper triangular, and:

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

where T11 is upper triangular.

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

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

where inv(B) denotes the inverse of the matrix B, and Z**H denotes the conjugate transpose of the matrix Z. iconic. 💅

💅 Parameters 💅

• M — M is INTEGER. The number of rows of the matrix A. M >= 0.
• P — P is INTEGER. The number of rows of the matrix B. P >= 0.
• N — N is INTEGER. The number of columns of the matrices A and B. N >= 0.
• A — A is COMPLEX*16 array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, if M <= N, the upper triangle of the subarray A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R; if M > N, the elements on and above the (M-N)-th subdiagonal contain the M-by-N upper trapezoidal matrix R; the remaining elements, with the array TAUA, represent the unitary matrix Q as a product of elementary reflectors (see Further Details).
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
• TAUA — TAUA is COMPLEX*16 array, dimension (min(M,N)). The scalar factors of the elementary reflectors which represent the unitary matrix Q (see Further Details).
• B — B is COMPLEX*16 array, dimension (LDB,N). On entry, the P-by-N matrix B. On exit, the elements on and above the diagonal of the array contain the min(P,N)-by-N upper trapezoidal matrix T (T is upper triangular if P >= N); the elements below the diagonal, 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,P).
• TAUB — TAUB is COMPLEX*16 array, dimension (min(P,N)). The scalar factors of the elementary reflectors which represent the unitary matrix Z (see Further Details).
• WORK — WORK is COMPLEX*16 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 RQ factorization of an M-by-N matrix, NB2 is the optimal blocksize for the QR factorization of a P-by-N matrix, and NB3 is the optimal blocksize for a call of ZUNMRQ.

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 🌈

• 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(m,n).

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(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in A(m-k+i,1:n-k+i-1), and taua in TAUA(i).

To form Q explicitly, use LAPACK subroutine ZUNGRQ. To use Q to update another matrix, use LAPACK subroutine ZUNMRQ.

The matrix Z is represented as a product of elementary reflectors:

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

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(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i), and taub in TAUB(i).

To form Z explicitly, use LAPACK subroutine ZUNGQR. To use Z to update another matrix, use LAPACK subroutine ZUNMQR.

⭐ Definition at line 212 of file zggrqf.f. ⭐

---

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

---

Cute sign-off, bestie: this footer is LAPACK Version 3.12.0, page path /home/abui...RC/zggrqf.f(3), and it doesn't show a date so I'm not making one up — no cap. 💖🫶