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