1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
|
SUBROUTINE DORGRQ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER INFO, K, LDA, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DORGRQ generates an M-by-N real matrix Q with orthonormal rows,
* which is defined as the last M rows of a product of K elementary
* reflectors of order N
*
* Q = H(1) H(2) . . . H(k)
*
* as returned by DGERQF.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix Q. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix Q. N >= M.
*
* K (input) INTEGER
* The number of elementary reflectors whose product defines the
* matrix Q. M >= K >= 0.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the (m-k+i)-th row must contain the vector which
* defines the elementary reflector H(i), for i = 1,2,...,k, as
* returned by DGERQF in the last k rows of its array argument
* A.
* On exit, the M-by-N matrix Q.
*
* LDA (input) INTEGER
* The first dimension of the array A. LDA >= max(1,M).
*
* TAU (input) DOUBLE PRECISION array, dimension (K)
* TAU(i) must contain the scalar factor of the elementary
* reflector H(i), as returned by DGERQF.
*
* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,M).
* For optimum performance LWORK >= M*NB, where NB is the
* optimal blocksize.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument has an illegal value
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IB, II, IINFO, IWS, J, KK, L, LDWORK,
$ LWKOPT, NB, NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL DLARFB, DLARFT, DORGR2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.M ) THEN
INFO = -2
ELSE IF( K.LT.0 .OR. K.GT.M ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -5
END IF
*
IF( INFO.EQ.0 ) THEN
IF( M.LE.0 ) THEN
LWKOPT = 1
ELSE
NB = ILAENV( 1, 'DORGRQ', ' ', M, N, K, -1 )
LWKOPT = M*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.MAX( 1, M ) .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DORGRQ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.LE.0 ) THEN
RETURN
END IF
*
NBMIN = 2
NX = 0
IWS = M
IF( NB.GT.1 .AND. NB.LT.K ) THEN
*
* Determine when to cross over from blocked to unblocked code.
*
NX = MAX( 0, ILAENV( 3, 'DORGRQ', ' ', M, N, K, -1 ) )
IF( NX.LT.K ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = M
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: reduce NB and
* determine the minimum value of NB.
*
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'DORGRQ', ' ', M, N, K, -1 ) )
END IF
END IF
END IF
*
IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
*
* Use blocked code after the first block.
* The last kk rows are handled by the block method.
*
KK = MIN( K, ( ( K-NX+NB-1 ) / NB )*NB )
*
* Set A(1:m-kk,n-kk+1:n) to zero.
*
DO 20 J = N - KK + 1, N
DO 10 I = 1, M - KK
A( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
KK = 0
END IF
*
* Use unblocked code for the first or only block.
*
CALL DORGR2( M-KK, N-KK, K-KK, A, LDA, TAU, WORK, IINFO )
*
IF( KK.GT.0 ) THEN
*
* Use blocked code
*
DO 50 I = K - KK + 1, K, NB
IB = MIN( NB, K-I+1 )
II = M - K + I
IF( II.GT.1 ) THEN
*
* Form the triangular factor of the block reflector
* H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL DLARFT( 'Backward', 'Rowwise', N-K+I+IB-1, IB,
$ A( II, 1 ), LDA, TAU( I ), WORK, LDWORK )
*
* Apply H' to A(1:m-k+i-1,1:n-k+i+ib-1) from the right
*
CALL DLARFB( 'Right', 'Transpose', 'Backward', 'Rowwise',
$ II-1, N-K+I+IB-1, IB, A( II, 1 ), LDA, WORK,
$ LDWORK, A, LDA, WORK( IB+1 ), LDWORK )
END IF
*
* Apply H' to columns 1:n-k+i+ib-1 of current block
*
CALL DORGR2( IB, N-K+I+IB-1, IB, A( II, 1 ), LDA, TAU( I ),
$ WORK, IINFO )
*
* Set columns n-k+i+ib:n of current block to zero
*
DO 40 L = N - K + I + IB, N
DO 30 J = II, II + IB - 1
A( J, L ) = ZERO
30 CONTINUE
40 CONTINUE
50 CONTINUE
END IF
*
WORK( 1 ) = IWS
RETURN
*
* End of DORGRQ
*
END
|
