1CGGRQF ‐ a generalized RQ factorization of an M‐by‐N matrix A and
2a P‐by‐N matrix B SUBROUTINE CGGRQF( M, P, N, A,  LDA,  TAUA,  B,
3LDB, TAUB, WORK, LWORK, INFO )
4    INTEGER INFO, LDA, LDB, LWORK, M, N, P
5    COMPLEX A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), WORK(
6* ) CGGRQF computes a generalized RQ factorization of  an  M‐by‐N
7matrix A and a P‐by‐N matrix B:
8
9            A = R*Q,        B = Z*T*Q,
10
11where  Q  is  an N‐by‐N unitary matrix, Z is a P‐by‐P unitary ma‐
12trix, and R and T assume one of the forms:
13
14if M <= N,  R = ( 0  R12 ) M,   or if M > N,  R = ( R11 ) M‐N,
15                 N‐M  M                           ( R21 ) N
16                                                     N
17
18where R12 or R21 is upper triangular, and
19
20if P >= N,  T = ( T11 ) N  ,   or if P < N,  T = ( T11  T12 ) P,
21                (  0  ) P‐N                         P   N‐P
22                   N
23
24where T11 is upper triangular.
25
26In particular, if B is square and nonsingular, the GRQ factoriza‐
27tion  of  A  and  B  implicitly  gives  the  RQ  factorization of
28A*inv(B):
29
30             A*inv(B) = (R*inv(T))*Z'
31
32where inv(B) denotes the inverse of the matrix B, and Z'  denotes
33the conjugate transpose of the matrix Z.
34
35M       (input) INTEGER The number of rows of the matrix A.  M >=
360.  P       (input) INTEGER The number of rows of the  matrix  B.
37P >= 0.  N       (input) INTEGER The number of columns of the ma‐
38trices A and B. N >= 0.  A        (input/output)  COMPLEX  array,
39dimension  (LDA,N)  On entry, the M‐by‐N matrix A.  On exit, if M
40<= N, the upper triangle of the subarray A(1:M,N‐M+1:N)  contains
41the  M‐by‐M  upper triangular matrix R; if M > N, the elements on
42and above the  (M‐N)‐th  subdiagonal  contain  the  M‐by‐N  upper
43trapezoidal  matrix  R;  the  remaining  elements, with the array
44TAUA, represent the unitary matrix Q as a product  of  elementary
45reflectors  (see  Further  Details).  LDA     (input) INTEGER The
46leading dimension of the array A. LDA >= max(1,M).  TAUA    (out‐
47put)  COMPLEX  array,  dimension (min(M,N)) The scalar factors of
48the elementary reflectors which represent the  unitary  matrix  Q
49(see Further Details).  B       (input/output) COMPLEX array, di‐
50mension (LDB,N) On entry, the P‐by‐N matrix B.  On exit, the ele‐
51ments  on  and  above  the  diagonal  of  the  array  contain the
52min(P,N)‐by‐N upper trapezoidal matrix T (T is  upper  triangular
53if P >= N); the elements below the diagonal, with the array TAUB,
54represent the unitary matrix Z as a product of elementary reflec‐
55tors  (see Further Details).  LDB     (input) INTEGER The leading
56dimension of the array B. LDB >= max(1,P).  TAUB    (output) COM‐
57PLEX  array,  dimension (min(P,N)) The scalar factors of the ele‐
58mentary reflectors which represent the unitary matrix Z (see Fur‐
59ther  Details).  WORK    (workspace/output) COMPLEX array, dimen‐
60sion (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the op‐
61timal  LWORK.  LWORK   (input) INTEGER The dimension of the array
62WORK. LWORK >= max(1,N,M,P).  For optimum  performance  LWORK  >=
63max(N,M,P)*max(NB1,NB2,NB3),  where  NB1 is the optimal blocksize
64for the RQ factorization of an M‐by‐N matrix, NB2 is the  optimal
65blocksize for the QR factorization of a P‐by‐N matrix, and NB3 is
66the optimal blocksize for a call of CUNMRQ.
67
68If LWORK = ‐1, then a workspace query is assumed; the routine on‐
69ly  calculates  the  optimal size of the WORK array, returns this
70value as the first entry of the WORK array, and no error  message
71related to LWORK is issued by XERBLA.  INFO    (output) INTEGER =
720:  successful exit
73< 0:  if INFO=‐i, the i‐th argument had an  illegal  value.   The
74matrix Q is represented as a product of elementary reflectors
75
76   Q = H(1) H(2) . . . H(k), where k = min(m,n).
77
78Each H(i) has the form
79
80   H(i) = I ‐ taua * v * v'
81
82where  taua  is  a complex scalar, and v is a complex vector with
83v(n‐k+i+1:n) = 0 and v(n‐k+i) = 1; v(1:n‐k+i‐1) is stored on exit
84in A(m‐k+i,1:n‐k+i‐1), and taua in TAUA(i).
85To form Q explicitly, use LAPACK subroutine CUNGRQ.
86To use Q to update another matrix, use LAPACK subroutine CUNMRQ.
87
88The matrix Z is represented as a product of elementary reflectors
89
90   Z = H(1) H(2) . . . H(k), where k = min(p,n).
91
92Each H(i) has the form
93
94   H(i) = I ‐ taub * v * v'
95
96where  taub  is  a complex scalar, and v is a complex vector with
97v(1:i‐1) = 0 and  v(i)  =  1;  v(i+1:p)  is  stored  on  exit  in
98B(i+1:p,i), and taub in TAUB(i).
99To form Z explicitly, use LAPACK subroutine CUNGQR.
100To use Z to update another matrix, use LAPACK subroutine CUNMQR.
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