1*> \brief \b CLASWLQ
2*
3*  Definition:
4*  ===========
5*
6*       SUBROUTINE CLASWLQ( M, N, MB, NB, A, LDA, T, LDT, WORK,
7*                            LWORK, INFO)
8*
9*       .. Scalar Arguments ..
10*       INTEGER           INFO, LDA, M, N, MB, NB, LDT, LWORK
11*       ..
12*       .. Array Arguments ..
13*       COMPLEX           A( LDA, * ), T( LDT, * ), WORK( * )
14*       ..
15*
16*
17*> \par Purpose:
18*  =============
19*>
20*> \verbatim
21*>
22*> CLASWLQ computes a blocked Tall-Skinny LQ factorization of
23*> a complex M-by-N matrix A for M <= N:
24*>
25*>    A = ( L 0 ) *  Q,
26*>
27*> where:
28*>
29*>    Q is a n-by-N orthogonal matrix, stored on exit in an implicit
30*>    form in the elements above the diagonal of the array A and in
31*>    the elements of the array T;
32*>    L is a lower-triangular M-by-M matrix stored on exit in
33*>    the elements on and below the diagonal of the array A.
34*>    0 is a M-by-(N-M) zero matrix, if M < N, and is not stored.
35*>
36*> \endverbatim
37*
38*  Arguments:
39*  ==========
40*
41*> \param[in] M
42*> \verbatim
43*>          M is INTEGER
44*>          The number of rows of the matrix A.  M >= 0.
45*> \endverbatim
46*>
47*> \param[in] N
48*> \verbatim
49*>          N is INTEGER
50*>          The number of columns of the matrix A.  N >= M >= 0.
51*> \endverbatim
52*>
53*> \param[in] MB
54*> \verbatim
55*>          MB is INTEGER
56*>          The row block size to be used in the blocked QR.
57*>          M >= MB >= 1
58*> \endverbatim
59*> \param[in] NB
60*> \verbatim
61*>          NB is INTEGER
62*>          The column block size to be used in the blocked QR.
63*>          NB > M.
64*> \endverbatim
65*>
66*> \param[in,out] A
67*> \verbatim
68*>          A is COMPLEX array, dimension (LDA,N)
69*>          On entry, the M-by-N matrix A.
70*>          On exit, the elements on and below the diagonal
71*>          of the array contain the N-by-N lower triangular matrix L;
72*>          the elements above the diagonal represent Q by the rows
73*>          of blocked V (see Further Details).
74*>
75*> \endverbatim
76*>
77*> \param[in] LDA
78*> \verbatim
79*>          LDA is INTEGER
80*>          The leading dimension of the array A.  LDA >= max(1,M).
81*> \endverbatim
82*>
83*> \param[out] T
84*> \verbatim
85*>          T is COMPLEX array,
86*>          dimension (LDT, N * Number_of_row_blocks)
87*>          where Number_of_row_blocks = CEIL((N-M)/(NB-M))
88*>          The blocked upper triangular block reflectors stored in compact form
89*>          as a sequence of upper triangular blocks.
90*>          See Further Details below.
91*> \endverbatim
92*>
93*> \param[in] LDT
94*> \verbatim
95*>          LDT is INTEGER
96*>          The leading dimension of the array T.  LDT >= MB.
97*> \endverbatim
98*>
99*>
100*> \param[out] WORK
101*> \verbatim
102*>         (workspace) COMPLEX array, dimension (MAX(1,LWORK))
103*>
104*> \endverbatim
105*> \param[in] LWORK
106*> \verbatim
107*>          The dimension of the array WORK.  LWORK >= MB*M.
108*>          If LWORK = -1, then a workspace query is assumed; the routine
109*>          only calculates the optimal size of the WORK array, returns
110*>          this value as the first entry of the WORK array, and no error
111*>          message related to LWORK is issued by XERBLA.
112*>
113*> \endverbatim
114*> \param[out] INFO
115*> \verbatim
116*>          INFO is INTEGER
117*>          = 0:  successful exit
118*>          < 0:  if INFO = -i, the i-th argument had an illegal value
119*> \endverbatim
120*
121*  Authors:
122*  ========
123*
124*> \author Univ. of Tennessee
125*> \author Univ. of California Berkeley
126*> \author Univ. of Colorado Denver
127*> \author NAG Ltd.
128*
129*> \par Further Details:
130*  =====================
131*>
132*> \verbatim
133*> Short-Wide LQ (SWLQ) performs LQ by a sequence of orthogonal transformations,
134*> representing Q as a product of other orthogonal matrices
135*>   Q = Q(1) * Q(2) * . . . * Q(k)
136*> where each Q(i) zeros out upper diagonal entries of a block of NB rows of A:
137*>   Q(1) zeros out the upper diagonal entries of rows 1:NB of A
138*>   Q(2) zeros out the bottom MB-N rows of rows [1:M,NB+1:2*NB-M] of A
139*>   Q(3) zeros out the bottom MB-N rows of rows [1:M,2*NB-M+1:3*NB-2*M] of A
140*>   . . .
141*>
142*> Q(1) is computed by GELQT, which represents Q(1) by Householder vectors
143*> stored under the diagonal of rows 1:MB of A, and by upper triangular
144*> block reflectors, stored in array T(1:LDT,1:N).
145*> For more information see Further Details in GELQT.
146*>
147*> Q(i) for i>1 is computed by TPLQT, which represents Q(i) by Householder vectors
148*> stored in columns [(i-1)*(NB-M)+M+1:i*(NB-M)+M] of A, and by upper triangular
149*> block reflectors, stored in array T(1:LDT,(i-1)*M+1:i*M).
150*> The last Q(k) may use fewer rows.
151*> For more information see Further Details in TPQRT.
152*>
153*> For more details of the overall algorithm, see the description of
154*> Sequential TSQR in Section 2.2 of [1].
155*>
156*> [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
157*>     J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
158*>     SIAM J. Sci. Comput, vol. 34, no. 1, 2012
159*> \endverbatim
160*>
161*  =====================================================================
162      SUBROUTINE CLASWLQ( M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK,
163     $                  INFO)
164*
165*  -- LAPACK computational routine --
166*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
167*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd. --
168*
169*     .. Scalar Arguments ..
170      INTEGER           INFO, LDA, M, N, MB, NB, LWORK, LDT
171*     ..
172*     .. Array Arguments ..
173      COMPLEX           A( LDA, * ), WORK( * ), T( LDT, *)
174*     ..
175*
176*  =====================================================================
177*
178*     ..
179*     .. Local Scalars ..
180      LOGICAL    LQUERY
181      INTEGER    I, II, KK, CTR
182*     ..
183*     .. EXTERNAL FUNCTIONS ..
184      LOGICAL            LSAME
185      EXTERNAL           LSAME
186*     .. EXTERNAL SUBROUTINES ..
187      EXTERNAL           CGELQT, CTPLQT, XERBLA
188*     .. INTRINSIC FUNCTIONS ..
189      INTRINSIC          MAX, MIN, MOD
190*     ..
191*     .. EXTERNAL FUNCTIONS ..
192      INTEGER            ILAENV
193      EXTERNAL           ILAENV
194*     ..
195*     .. EXECUTABLE STATEMENTS ..
196*
197*     TEST THE INPUT ARGUMENTS
198*
199      INFO = 0
200*
201      LQUERY = ( LWORK.EQ.-1 )
202*
203      IF( M.LT.0 ) THEN
204        INFO = -1
205      ELSE IF( N.LT.0 .OR. N.LT.M ) THEN
206        INFO = -2
207      ELSE IF( MB.LT.1 .OR. ( MB.GT.M .AND. M.GT.0 )) THEN
208        INFO = -3
209      ELSE IF( NB.LE.M ) THEN
210        INFO = -4
211      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
212        INFO = -5
213      ELSE IF( LDT.LT.MB ) THEN
214        INFO = -8
215      ELSE IF( ( LWORK.LT.M*MB) .AND. (.NOT.LQUERY) ) THEN
216        INFO = -10
217      END IF
218      IF( INFO.EQ.0)  THEN
219      WORK(1) = MB*M
220      END IF
221*
222      IF( INFO.NE.0 ) THEN
223        CALL XERBLA( 'CLASWLQ', -INFO )
224        RETURN
225      ELSE IF (LQUERY) THEN
226       RETURN
227      END IF
228*
229*     Quick return if possible
230*
231      IF( MIN(M,N).EQ.0 ) THEN
232          RETURN
233      END IF
234*
235*     The LQ Decomposition
236*
237       IF((M.GE.N).OR.(NB.LE.M).OR.(NB.GE.N)) THEN
238        CALL CGELQT( M, N, MB, A, LDA, T, LDT, WORK, INFO)
239        RETURN
240       END IF
241*
242       KK = MOD((N-M),(NB-M))
243       II=N-KK+1
244*
245*      Compute the LQ factorization of the first block A(1:M,1:NB)
246*
247       CALL CGELQT( M, NB, MB, A(1,1), LDA, T, LDT, WORK, INFO)
248       CTR = 1
249*
250       DO I = NB+1, II-NB+M , (NB-M)
251*
252*      Compute the QR factorization of the current block A(1:M,I:I+NB-M)
253*
254         CALL CTPLQT( M, NB-M, 0, MB, A(1,1), LDA, A( 1, I ),
255     $                  LDA, T(1,CTR*M+1),
256     $                  LDT, WORK, INFO )
257         CTR = CTR + 1
258       END DO
259*
260*     Compute the QR factorization of the last block A(1:M,II:N)
261*
262       IF (II.LE.N) THEN
263        CALL CTPLQT( M, KK, 0, MB, A(1,1), LDA, A( 1, II ),
264     $                  LDA, T(1,CTR*M+1), LDT,
265     $                  WORK, INFO )
266       END IF
267*
268      WORK( 1 ) = M * MB
269      RETURN
270*
271*     End of CLASWLQ
272*
273      END
274