1DGGRQF(1) LAPACK routine (version 3.2) DGGRQF(1)
2
6 DGGRQF - computes a generalized RQ factorization of an M-by-N matrix A
7 and a P-by-N matrix B
8
10 SUBROUTINE DGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK,
11 INFO )
12 INTEGER INFO, LDA, LDB, LWORK, M, N, P
14 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB(
16 * ), WORK( * )
17
19 DGGRQF 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDB,N) On
64 entry, the P-by-N matrix B. On exit, the elements on and above
65 the diagonal of the array contain the min(P,N)-by-N upper
66 trapezoidal matrix T (T is upper triangular if P >= N); the
67 elements below the diagonal, with the array TAUB, represent the
68 orthogonal matrix Z as a product of elementary reflectors (see
69 Further Details). LDB (input) INTEGER The leading dimen‐
70 sion of the array B. LDB >= max(1,P).
71 TAUB (output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension
76 (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the opti‐
77 mal LWORK.
78 LWORK (input) INTEGER
80 The dimension of the array WORK. LWORK >= max(1,N,M,P). For
81 optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3), where
82 NB1 is the optimal blocksize for the RQ factorization of an M-
83 by-N matrix, NB2 is the optimal blocksize for the QR factoriza‐
84 tion of a P-by-N matrix, and NB3 is the optimal blocksize for a
85 call of DORMRQ. If LWORK = -1, then a workspace query is
86 assumed; the routine only calculates the optimal size of the
87 WORK array, returns this value as the first entry of the WORK
88 array, and no error message related to LWORK is issued by
89 XERBLA.
90 INFO (output) INTEGER
92 = 0: successful exit
93 < 0: if INF0= -i, the i-th argument had an illegal value.
94
96 The matrix Q is represented as a product of elementary reflectors
97 Q = H(1) H(2) . . . H(k), where k = min(m,n).
98 Each H(i) has the form
99 H(i) = I - taua * v * v'
100 where taua is a real scalar, and v is a real vector with
101 v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
102 A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
103 To form Q explicitly, use LAPACK subroutine DORGRQ.
104 To use Q to update another matrix, use LAPACK subroutine DORMRQ. The
105 matrix Z is represented as a product of elementary reflectors
106 Z = H(1) H(2) . . . H(k), where k = min(p,n).
107 Each H(i) has the form
108 H(i) = I - taub * v * v'
109 where taub is a real scalar, and v is a real vector with
110 v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
111 and taub in TAUB(i).
112 To form Z explicitly, use LAPACK subroutine DORGQR.
113 To use Z to update another matrix, use LAPACK subroutine DORMQR.
114 LAPACK routine (version 3.2) November 2008 DGGRQF(1)