SRC/cggrqf.f(3)

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version 3.12.1 · 7 versions
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💅 SRC/cggrqf.f — the cggrqf manual page, bestie edition 🎀

okay so this is the man page for SRC/cggrqf.f, straight out of the LAPACK library functions manual. yes it's Fortran, no we're not scared of it 💖

✨ SYNOPSIS ✨

🌸 Functions/Subroutines

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

CGGRQF — that's the vibe, that's the name 🦋

👑 Function/Subroutine Documentation 👑

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

CGGRQF 🫶

🌈 Purpose:

okay so what does it DO. CGGRQF 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 the forms:

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. slay 💅

and like, 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. no cap, that's the whole point 🧚

🍓 Parameters

  • M — M is INTEGER. the number of rows of the matrix A. M >= 0. periodt.
  • 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 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 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 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 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 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 CUNMRQ.
🚨 bestie listen: 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 don't come crying to XERBLA about it, she's not gonna help 😭
  • 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 CUNGRQ.
  • To use Q to update another matrix, use LAPACK subroutine CUNMRQ.

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 CUNGQR.
  • To use Z to update another matrix, use LAPACK subroutine CUNMQR.

⭐ Definition

Definition at line 212 of file cggrqf.f. iykyk 🫶

🎀 Author

Generated automatically by Doxygen for LAPACK from the source code. not me being auto-generated and still serving looks 💅

okay that's the whole page bestie, go forth and factorize your generalized RQ like the main character you are 👑

LAPACK — Version 3.12.0 — SRC/cggrqf.f(3) 🌈💖🌸

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