1SGGRQF(1) LAPACK routine (version 3.2) SGGRQF(1)
2
6 SGGRQF - computes a generalized RQ factorization of an M-by-N matrix A
7 and a P-by-N matrix B
8
10 SUBROUTINE SGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK,
11 INFO )
12 INTEGER INFO, LDA, LDB, LWORK, M, N, P
14 REAL A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
16 WORK( * )
17
19 SGGRQF computes a generalized RQ factorization of an M-by-N matrix A
20 and a P-by-N matrix B:
21 A = R*Q, B = Z*T*Q,
22 where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
23 matrix, and R and T assume one of the forms:
24 if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
25 N-M M ( R21 ) N
26 N
27 where R12 or R21 is upper triangular, and
28 if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
29 ( 0 ) P-N P N-P
30 N
31 where T11 is upper triangular.
32 In particular, if B is square and nonsingular, the GRQ factorization of
33 A and B implicitly gives the RQ factorization of A*inv(B):
34 A*inv(B) = (R*inv(T))*Z'
35 where inv(B) denotes the inverse of the matrix B, and Z' denotes the
36 transpose of the matrix Z.
37
39 M (input) INTEGER
40 The number of rows of the matrix A. M >= 0.
41 P (input) INTEGER
43 The number of rows of the matrix B. P >= 0.
44 N (input) INTEGER
46 The number of columns of the matrices A and B. N >= 0.
47 A (input/output) REAL array, dimension (LDA,N)
49 On entry, the M-by-N matrix A. On exit, if M <= N, the upper
50 triangle of the subarray A(1:M,N-M+1:N) contains the M-by-M
51 upper triangular matrix R; if M > N, the elements on and above
52 the (M-N)-th subdiagonal contain the M-by-N upper trapezoidal
53 matrix R; the remaining elements, with the array TAUA, repre‐
54 sent the orthogonal matrix Q as a product of elementary reflec‐
55 tors (see Further Details).
56 LDA (input) INTEGER
58 The leading dimension of the array A. LDA >= max(1,M).
59 TAUA (output) REAL array, dimension (min(M,N))
61 The scalar factors of the elementary reflectors which represent
62 the orthogonal matrix Q (see Further Details). B
63 (input/output) REAL array, dimension (LDB,N) On entry, the P-
64 by-N matrix B. On exit, the elements on and above the diagonal
65 of the array contain the min(P,N)-by-N upper trapezoidal matrix
66 T (T is upper triangular if P >= N); the elements below the
67 diagonal, with the array TAUB, represent the orthogonal matrix
68 Z as a product of elementary reflectors (see Further Details).
69 LDB (input) INTEGER The leading dimension of the array B.
70 LDB >= max(1,P).
71 TAUB (output) REAL array, dimension (min(P,N))
73 The scalar factors of the elementary reflectors which represent
74 the orthogonal matrix Z (see Further Details). WORK
75 (workspace/output) REAL array, dimension (MAX(1,LWORK)) On
76 exit, if INFO = 0, WORK(1) returns the optimal LWORK.
77 LWORK (input) INTEGER
79 The dimension of the array WORK. LWORK >= max(1,N,M,P). For
80 optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3), where
81 NB1 is the optimal blocksize for the RQ factorization of an M-
82 by-N matrix, NB2 is the optimal blocksize for the QR factoriza‐
83 tion of a P-by-N matrix, and NB3 is the optimal blocksize for a
84 call of SORMRQ. If LWORK = -1, then a workspace query is
85 assumed; the routine only calculates the optimal size of the
86 WORK array, returns this value as the first entry of the WORK
87 array, and no error message related to LWORK is issued by
88 XERBLA.
89 INFO (output) INTEGER
91 = 0: successful exit
92 < 0: if INF0= -i, the i-th argument had an illegal value.
93
95 The matrix Q is represented as a product of elementary reflectors
96 Q = H(1) H(2) . . . H(k), where k = min(m,n).
97 Each H(i) has the form
98 H(i) = I - taua * v * v'
99 where taua is a real scalar, and v is a real vector with
100 v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
101 A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
102 To form Q explicitly, use LAPACK subroutine SORGRQ.
103 To use Q to update another matrix, use LAPACK subroutine SORMRQ. The
104 matrix Z is represented as a product of elementary reflectors
105 Z = H(1) H(2) . . . H(k), where k = min(p,n).
106 Each H(i) has the form
107 H(i) = I - taub * v * v'
108 where taub is a real scalar, and v is a real vector with
109 v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
110 and taub in TAUB(i).
111 To form Z explicitly, use LAPACK subroutine SORGQR.
112 To use Z to update another matrix, use LAPACK subroutine SORMQR.
113 LAPACK routine (version 3.2) November 2008 SGGRQF(1)