SRC/dggqrf.f(3)

💖 NAME 💖

SRC/dggqrf.f — okay bestie, this is the SRC/dggqrf.f(3) manual page from the Library Functions Manual, and it's all about the DGGQRF routine. 💅

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

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

🦋 Function/Subroutine Documentation 🦋

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)

DGGQRF

🎀 Purpose 🎀

Okay so this one is a whole generalized QR factorization moment: 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 — 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 DORMQR.

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(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.

👑 Author 👑

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