1SGGQRF(1) LAPACK routine (version 3.2) SGGQRF(1)
2
6 SGGQRF - computes a generalized QR factorization of an N-by-M matrix A
7 and an N-by-P matrix B
8
10 SUBROUTINE SGGQRF( N, M, P, 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 SGGQRF computes a generalized QR factorization of an N-by-M matrix A
20 and an N-by-P matrix B:
21 A = Q*R, B = Q*T*Z,
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 N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N,
25 ( 0 ) N-M N M-N
26 M
27 where R11 is upper triangular, and
28 if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P,
29 P-N N ( T21 ) P
30 P
31 where T12 or T21 is upper triangular.
32 In particular, if B is square and nonsingular, the GQR factorization of
33 A and B implicitly gives the QR factorization of inv(B)*A:
34 inv(B)*A = Z'*(inv(T)*R)
35 where inv(B) denotes the inverse of the matrix B, and Z' denotes the
36 transpose of the matrix Z.
37
39 N (input) INTEGER
40 The number of rows of the matrices A and B. N >= 0.
41 M (input) INTEGER
43 The number of columns of the matrix A. M >= 0.
44 P (input) INTEGER
46 The number of columns of the matrix B. P >= 0.
47 A (input/output) REAL array, dimension (LDA,M)
49 On entry, the N-by-M matrix A. On exit, the elements on and
50 above the diagonal of the array contain the min(N,M)-by-M upper
51 trapezoidal matrix R (R is upper triangular if N >= M); the
52 elements below the diagonal, with the array TAUA, represent the
53 orthogonal matrix Q as a product of min(N,M) elementary reflec‐
54 tors (see Further Details).
55 LDA (input) INTEGER
57 The leading dimension of the array A. LDA >= max(1,N).
58 TAUA (output) REAL array, dimension (min(N,M))
60 The scalar factors of the elementary reflectors which represent
61 the orthogonal matrix Q (see Further Details). B
62 (input/output) REAL array, dimension (LDB,P) On entry, the N-
63 by-P matrix B. On exit, if N <= P, the upper triangle of the
64 subarray B(1:N,P-N+1:P) contains the N-by-N upper triangular
65 matrix T; if N > P, the elements on and above the (N-P)-th sub‐
66 diagonal contain the N-by-P upper trapezoidal matrix T; the
67 remaining elements, with the array TAUB, represent the orthogo‐
68 nal matrix Z as a product of elementary reflectors (see Further
69 Details).
70 LDB (input) INTEGER
72 The leading dimension of the array B. LDB >= max(1,N).
73 TAUB (output) REAL array, dimension (min(N,P))
75 The scalar factors of the elementary reflectors which represent
76 the orthogonal matrix Z (see Further Details). WORK
77 (workspace/output) REAL array, dimension (MAX(1,LWORK)) On
78 exit, if INFO = 0, WORK(1) returns the optimal LWORK.
79 LWORK (input) INTEGER
81 The dimension of the array WORK. LWORK >= max(1,N,M,P). For
82 optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3), where
83 NB1 is the optimal blocksize for the QR factorization of an N-
84 by-M matrix, NB2 is the optimal blocksize for the RQ factoriza‐
85 tion of an N-by-P matrix, and NB3 is the optimal blocksize for
86 a call of SORMQR. If LWORK = -1, then a workspace query is
87 assumed; the routine only calculates the optimal size of the
88 WORK array, returns this value as the first entry of the WORK
89 array, and no error message related to LWORK is issued by
90 XERBLA.
91 INFO (output) INTEGER
93 = 0: successful exit
94 < 0: if INFO = -i, the i-th argument had an illegal value.
95
97 The matrix Q is represented as a product of elementary reflectors
98 Q = H(1) H(2) . . . H(k), where k = min(n,m).
99 Each H(i) has the form
100 H(i) = I - taua * v * v'
101 where taua is a real scalar, and v is a real vector with
102 v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
103 and taua in TAUA(i).
104 To form Q explicitly, use LAPACK subroutine SORGQR.
105 To use Q to update another matrix, use LAPACK subroutine SORMQR. The
106 matrix Z is represented as a product of elementary reflectors
107 Z = H(1) H(2) . . . H(k), where k = min(n,p).
108 Each H(i) has the form
109 H(i) = I - taub * v * v'
110 where taub is a real scalar, and v is a real vector with
111 v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
112 B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
113 To form Z explicitly, use LAPACK subroutine SORGRQ.
114 To use Z to update another matrix, use LAPACK subroutine SORMRQ.
115 LAPACK routine (version 3.2) November 2008 SGGQRF(1)