 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dorgrq()

 subroutine dorgrq ( integer M, integer N, integer K, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) TAU, double precision, dimension( * ) WORK, integer LWORK, integer INFO )

DORGRQ

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.```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix Q. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix Q. N >= M.``` [in] K ``` K is INTEGER The number of elementary reflectors whose product defines the matrix Q. M >= K >= 0.``` [in,out] A ``` A is 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.``` [in] LDA ``` LDA is INTEGER The first dimension of the array A. LDA >= max(1,M).``` [in] TAU ``` TAU is DOUBLE PRECISION array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by DGERQF.``` [out] WORK ``` WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is 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.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument has an illegal value```

Definition at line 127 of file dorgrq.f.

128*
129* -- LAPACK computational routine --
130* -- LAPACK is a software package provided by Univ. of Tennessee, --
131* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
132*
133* .. Scalar Arguments ..
134 INTEGER INFO, K, LDA, LWORK, M, N
135* ..
136* .. Array Arguments ..
137 DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
138* ..
139*
140* =====================================================================
141*
142* .. Parameters ..
143 DOUBLE PRECISION ZERO
144 parameter( zero = 0.0d+0 )
145* ..
146* .. Local Scalars ..
147 LOGICAL LQUERY
148 INTEGER I, IB, II, IINFO, IWS, J, KK, L, LDWORK,
149 \$ LWKOPT, NB, NBMIN, NX
150* ..
151* .. External Subroutines ..
152 EXTERNAL dlarfb, dlarft, dorgr2, xerbla
153* ..
154* .. Intrinsic Functions ..
155 INTRINSIC max, min
156* ..
157* .. External Functions ..
158 INTEGER ILAENV
159 EXTERNAL ilaenv
160* ..
161* .. Executable Statements ..
162*
163* Test the input arguments
164*
165 info = 0
166 lquery = ( lwork.EQ.-1 )
167 IF( m.LT.0 ) THEN
168 info = -1
169 ELSE IF( n.LT.m ) THEN
170 info = -2
171 ELSE IF( k.LT.0 .OR. k.GT.m ) THEN
172 info = -3
173 ELSE IF( lda.LT.max( 1, m ) ) THEN
174 info = -5
175 END IF
176*
177 IF( info.EQ.0 ) THEN
178 IF( m.LE.0 ) THEN
179 lwkopt = 1
180 ELSE
181 nb = ilaenv( 1, 'DORGRQ', ' ', m, n, k, -1 )
182 lwkopt = m*nb
183 END IF
184 work( 1 ) = lwkopt
185*
186 IF( lwork.LT.max( 1, m ) .AND. .NOT.lquery ) THEN
187 info = -8
188 END IF
189 END IF
190*
191 IF( info.NE.0 ) THEN
192 CALL xerbla( 'DORGRQ', -info )
193 RETURN
194 ELSE IF( lquery ) THEN
195 RETURN
196 END IF
197*
198* Quick return if possible
199*
200 IF( m.LE.0 ) THEN
201 RETURN
202 END IF
203*
204 nbmin = 2
205 nx = 0
206 iws = m
207 IF( nb.GT.1 .AND. nb.LT.k ) THEN
208*
209* Determine when to cross over from blocked to unblocked code.
210*
211 nx = max( 0, ilaenv( 3, 'DORGRQ', ' ', m, n, k, -1 ) )
212 IF( nx.LT.k ) THEN
213*
214* Determine if workspace is large enough for blocked code.
215*
216 ldwork = m
217 iws = ldwork*nb
218 IF( lwork.LT.iws ) THEN
219*
220* Not enough workspace to use optimal NB: reduce NB and
221* determine the minimum value of NB.
222*
223 nb = lwork / ldwork
224 nbmin = max( 2, ilaenv( 2, 'DORGRQ', ' ', m, n, k, -1 ) )
225 END IF
226 END IF
227 END IF
228*
229 IF( nb.GE.nbmin .AND. nb.LT.k .AND. nx.LT.k ) THEN
230*
231* Use blocked code after the first block.
232* The last kk rows are handled by the block method.
233*
234 kk = min( k, ( ( k-nx+nb-1 ) / nb )*nb )
235*
236* Set A(1:m-kk,n-kk+1:n) to zero.
237*
238 DO 20 j = n - kk + 1, n
239 DO 10 i = 1, m - kk
240 a( i, j ) = zero
241 10 CONTINUE
242 20 CONTINUE
243 ELSE
244 kk = 0
245 END IF
246*
247* Use unblocked code for the first or only block.
248*
249 CALL dorgr2( m-kk, n-kk, k-kk, a, lda, tau, work, iinfo )
250*
251 IF( kk.GT.0 ) THEN
252*
253* Use blocked code
254*
255 DO 50 i = k - kk + 1, k, nb
256 ib = min( nb, k-i+1 )
257 ii = m - k + i
258 IF( ii.GT.1 ) THEN
259*
260* Form the triangular factor of the block reflector
261* H = H(i+ib-1) . . . H(i+1) H(i)
262*
263 CALL dlarft( 'Backward', 'Rowwise', n-k+i+ib-1, ib,
264 \$ a( ii, 1 ), lda, tau( i ), work, ldwork )
265*
266* Apply H**T to A(1:m-k+i-1,1:n-k+i+ib-1) from the right
267*
268 CALL dlarfb( 'Right', 'Transpose', 'Backward', 'Rowwise',
269 \$ ii-1, n-k+i+ib-1, ib, a( ii, 1 ), lda, work,
270 \$ ldwork, a, lda, work( ib+1 ), ldwork )
271 END IF
272*
273* Apply H**T to columns 1:n-k+i+ib-1 of current block
274*
275 CALL dorgr2( ib, n-k+i+ib-1, ib, a( ii, 1 ), lda, tau( i ),
276 \$ work, iinfo )
277*
278* Set columns n-k+i+ib:n of current block to zero
279*
280 DO 40 l = n - k + i + ib, n
281 DO 30 j = ii, ii + ib - 1
282 a( j, l ) = zero
283 30 CONTINUE
284 40 CONTINUE
285 50 CONTINUE
286 END IF
287*
288 work( 1 ) = iws
289 RETURN
290*
291* End of DORGRQ
292*
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dlarft(DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
DLARFT forms the triangular factor T of a block reflector H = I - vtvH
Definition: dlarft.f:163
subroutine dlarfb(SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK, LDWORK)
DLARFB applies a block reflector or its transpose to a general rectangular matrix.
Definition: dlarfb.f:197
subroutine dorgr2(M, N, K, A, LDA, TAU, WORK, INFO)
DORGR2 generates all or part of the orthogonal matrix Q from an RQ factorization determined by sgerqf...
Definition: dorgr2.f:114
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