1SGGRQF ‐ a generalized RQ factorization of an M‐by‐N matrix A and
2a P‐by‐N matrix B SUBROUTINE SGGRQF( M, P, N, A,  LDA,  TAUA,  B,
3LDB, TAUB, WORK, LWORK, INFO )
4    INTEGER INFO, LDA, LDB, LWORK, M, N, P
5    REAL  A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), WORK( *
6) SGGRQF computes a generalized RQ factorization of an M‐by‐N ma‐
7trix 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 orthogonal matrix, Z is a P‐by‐P orthogonal
12matrix, 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
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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 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) REAL  array,  di‐
39mension  (LDA,N) On entry, the M‐by‐N matrix A.  On exit, if M <=
40N, the upper triangle of the subarray A(1:M,N‐M+1:N) contains the
41M‐by‐M  upper  triangular matrix R; if M > N, the elements on and
42above the (M‐N)‐th subdiagonal contain the  M‐by‐N  upper  trape‐
43zoidal  matrix  R;  the  remaining elements, with the array TAUA,
44represent the orthogonal matrix Q as a product of elementary  re‐
45flectors  (see  Further  Details).   LDA      (input) INTEGER The
46leading dimension of the array A. LDA >= max(1,M).  TAUA    (out‐
47put)  REAL  array, dimension (min(M,N)) The scalar factors of the
48elementary reflectors which represent  the  orthogonal  matrix  Q
49(see Further Details).  B       (input/output) REAL array, dimen‐
50sion (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 orthogonal matrix Z as a product of elementary  re‐
55flectors  (see  Further  Details).   LDB      (input) INTEGER The
56leading dimension of the array B. LDB >= max(1,P).  TAUB    (out‐
57put)  REAL  array, dimension (min(P,N)) The scalar factors of the
58elementary reflectors which represent  the  orthogonal  matrix  Z
59(see  Further  Details).   WORK    (workspace/output) REAL array,
60dimension (MAX(1,LWORK)) On exit, if INFO =  0,  WORK(1)  returns
61the  optimal LWORK.  LWORK   (input) INTEGER The dimension of the
62array WORK. LWORK >= max(1,N,M,P).  For optimum performance LWORK
63>=  max(N,M,P)*max(NB1,NB2,NB3),  where NB1 is the optimal block‐
64size for the RQ factorization of an M‐by‐N matrix, NB2 is the op‐
65timal  blocksize for the QR factorization of a P‐by‐N matrix, and
66NB3 is the optimal blocksize for a call of SORMRQ.
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 INF0= ‐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 real scalar, and v is a real 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 SORGRQ.
86To use Q to update another matrix, use LAPACK subroutine SORMRQ.
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 real scalar, and v is a real 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 SORGQR.
100To use Z to update another matrix, use LAPACK subroutine SORMQR.
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