1*> \brief \b SGELQ
2*
3*  Definition:
4*  ===========
5*
6*       SUBROUTINE SGELQ( M, N, A, LDA, T, TSIZE, WORK, LWORK,
7*                         INFO )
8*
9*       .. Scalar Arguments ..
10*       INTEGER           INFO, LDA, M, N, TSIZE, LWORK
11*       ..
12*       .. Array Arguments ..
13*       REAL              A( LDA, * ), T( * ), WORK( * )
14*       ..
15*
16*
17*> \par Purpose:
18*  =============
19*>
20*> \verbatim
21*>
22*> SGELQ computes an LQ factorization of a real M-by-N matrix A:
23*>
24*>    A = ( L 0 ) *  Q
25*>
26*> where:
27*>
28*>    Q is a N-by-N orthogonal matrix;
29*>    L is a lower-triangular M-by-M matrix;
30*>    0 is a M-by-(N-M) zero matrix, if M < N.
31*>
32*> \endverbatim
33*
34*  Arguments:
35*  ==========
36*
37*> \param[in] M
38*> \verbatim
39*>          M is INTEGER
40*>          The number of rows of the matrix A.  M >= 0.
41*> \endverbatim
42*>
43*> \param[in] N
44*> \verbatim
45*>          N is INTEGER
46*>          The number of columns of the matrix A.  N >= 0.
47*> \endverbatim
48*>
49*> \param[in,out] A
50*> \verbatim
51*>          A is REAL array, dimension (LDA,N)
52*>          On entry, the M-by-N matrix A.
53*>          On exit, the elements on and below the diagonal of the array
54*>          contain the M-by-min(M,N) lower trapezoidal matrix L
55*>          (L is lower triangular if M <= N);
56*>          the elements above the diagonal are used to store part of the
57*>          data structure to represent Q.
58*> \endverbatim
59*>
60*> \param[in] LDA
61*> \verbatim
62*>          LDA is INTEGER
63*>          The leading dimension of the array A.  LDA >= max(1,M).
64*> \endverbatim
65*>
66*> \param[out] T
67*> \verbatim
68*>          T is REAL array, dimension (MAX(5,TSIZE))
69*>          On exit, if INFO = 0, T(1) returns optimal (or either minimal
70*>          or optimal, if query is assumed) TSIZE. See TSIZE for details.
71*>          Remaining T contains part of the data structure used to represent Q.
72*>          If one wants to apply or construct Q, then one needs to keep T
73*>          (in addition to A) and pass it to further subroutines.
74*> \endverbatim
75*>
76*> \param[in] TSIZE
77*> \verbatim
78*>          TSIZE is INTEGER
79*>          If TSIZE >= 5, the dimension of the array T.
80*>          If TSIZE = -1 or -2, then a workspace query is assumed. The routine
81*>          only calculates the sizes of the T and WORK arrays, returns these
82*>          values as the first entries of the T and WORK arrays, and no error
83*>          message related to T or WORK is issued by XERBLA.
84*>          If TSIZE = -1, the routine calculates optimal size of T for the
85*>          optimum performance and returns this value in T(1).
86*>          If TSIZE = -2, the routine calculates minimal size of T and
87*>          returns this value in T(1).
88*> \endverbatim
89*>
90*> \param[out] WORK
91*> \verbatim
92*>          (workspace) REAL array, dimension (MAX(1,LWORK))
93*>          On exit, if INFO = 0, WORK(1) contains optimal (or either minimal
94*>          or optimal, if query was assumed) LWORK.
95*>          See LWORK for details.
96*> \endverbatim
97*>
98*> \param[in] LWORK
99*> \verbatim
100*>          LWORK is INTEGER
101*>          The dimension of the array WORK.
102*>          If LWORK = -1 or -2, then a workspace query is assumed. The routine
103*>          only calculates the sizes of the T and WORK arrays, returns these
104*>          values as the first entries of the T and WORK arrays, and no error
105*>          message related to T or WORK is issued by XERBLA.
106*>          If LWORK = -1, the routine calculates optimal size of WORK for the
107*>          optimal performance and returns this value in WORK(1).
108*>          If LWORK = -2, the routine calculates minimal size of WORK and
109*>          returns this value in WORK(1).
110*> \endverbatim
111*>
112*> \param[out] INFO
113*> \verbatim
114*>          INFO is INTEGER
115*>          = 0:  successful exit
116*>          < 0:  if INFO = -i, the i-th argument had an illegal value
117*> \endverbatim
118*
119*  Authors:
120*  ========
121*
122*> \author Univ. of Tennessee
123*> \author Univ. of California Berkeley
124*> \author Univ. of Colorado Denver
125*> \author NAG Ltd.
126*
127*> \par Further Details
128*  ====================
129*>
130*> \verbatim
131*>
132*> The goal of the interface is to give maximum freedom to the developers for
133*> creating any LQ factorization algorithm they wish. The triangular
134*> (trapezoidal) L has to be stored in the lower part of A. The lower part of A
135*> and the array T can be used to store any relevant information for applying or
136*> constructing the Q factor. The WORK array can safely be discarded after exit.
137*>
138*> Caution: One should not expect the sizes of T and WORK to be the same from one
139*> LAPACK implementation to the other, or even from one execution to the other.
140*> A workspace query (for T and WORK) is needed at each execution. However,
141*> for a given execution, the size of T and WORK are fixed and will not change
142*> from one query to the next.
143*>
144*> \endverbatim
145*>
146*> \par Further Details particular to this LAPACK implementation:
147*  ==============================================================
148*>
149*> \verbatim
150*>
151*> These details are particular for this LAPACK implementation. Users should not
152*> take them for granted. These details may change in the future, and are not likely
153*> true for another LAPACK implementation. These details are relevant if one wants
154*> to try to understand the code. They are not part of the interface.
155*>
156*> In this version,
157*>
158*>          T(2): row block size (MB)
159*>          T(3): column block size (NB)
160*>          T(6:TSIZE): data structure needed for Q, computed by
161*>                           SLASWLQ or SGELQT
162*>
163*>  Depending on the matrix dimensions M and N, and row and column
164*>  block sizes MB and NB returned by ILAENV, SGELQ will use either
165*>  SLASWLQ (if the matrix is short-and-wide) or SGELQT to compute
166*>  the LQ factorization.
167*> \endverbatim
168*>
169*  =====================================================================
170      SUBROUTINE SGELQ( M, N, A, LDA, T, TSIZE, WORK, LWORK,
171     $                  INFO )
172*
173*  -- LAPACK computational routine --
174*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
175*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd. --
176*
177*     .. Scalar Arguments ..
178      INTEGER            INFO, LDA, M, N, TSIZE, LWORK
179*     ..
180*     .. Array Arguments ..
181      REAL               A( LDA, * ), T( * ), WORK( * )
182*     ..
183*
184*  =====================================================================
185*
186*     ..
187*     .. Local Scalars ..
188      LOGICAL            LQUERY, LMINWS, MINT, MINW
189      INTEGER            MB, NB, MINTSZ, NBLCKS, LWMIN, LWOPT, LWREQ
190*     ..
191*     .. External Functions ..
192      LOGICAL            LSAME
193      EXTERNAL           LSAME
194*     ..
195*     .. External Subroutines ..
196      EXTERNAL           SGELQT, SLASWLQ, XERBLA
197*     ..
198*     .. Intrinsic Functions ..
199      INTRINSIC          MAX, MIN, MOD
200*     ..
201*     .. External Functions ..
202      INTEGER            ILAENV
203      EXTERNAL           ILAENV
204*     ..
205*     .. Executable statements ..
206*
207*     Test the input arguments
208*
209      INFO = 0
210*
211      LQUERY = ( TSIZE.EQ.-1 .OR. TSIZE.EQ.-2 .OR.
212     $           LWORK.EQ.-1 .OR. LWORK.EQ.-2 )
213*
214      MINT = .FALSE.
215      MINW = .FALSE.
216      IF( TSIZE.EQ.-2 .OR. LWORK.EQ.-2 ) THEN
217        IF( TSIZE.NE.-1 ) MINT = .TRUE.
218        IF( LWORK.NE.-1 ) MINW = .TRUE.
219      END IF
220*
221*     Determine the block size
222*
223      IF( MIN( M, N ).GT.0 ) THEN
224        MB = ILAENV( 1, 'SGELQ ', ' ', M, N, 1, -1 )
225        NB = ILAENV( 1, 'SGELQ ', ' ', M, N, 2, -1 )
226      ELSE
227        MB = 1
228        NB = N
229      END IF
230      IF( MB.GT.MIN( M, N ) .OR. MB.LT.1 ) MB = 1
231      IF( NB.GT.N .OR. NB.LE.M ) NB = N
232      MINTSZ = M + 5
233      IF ( NB.GT.M .AND. N.GT.M ) THEN
234        IF( MOD( N - M, NB - M ).EQ.0 ) THEN
235          NBLCKS = ( N - M ) / ( NB - M )
236        ELSE
237          NBLCKS = ( N - M ) / ( NB - M ) + 1
238        END IF
239      ELSE
240        NBLCKS = 1
241      END IF
242*
243*     Determine if the workspace size satisfies minimal size
244*
245      IF( ( N.LE.M ) .OR. ( NB.LE.M ) .OR. ( NB.GE.N ) ) THEN
246         LWMIN = MAX( 1, N )
247         LWOPT = MAX( 1, MB*N )
248      ELSE
249         LWMIN = MAX( 1, M )
250         LWOPT = MAX( 1, MB*M )
251      END IF
252      LMINWS = .FALSE.
253      IF( ( TSIZE.LT.MAX( 1, MB*M*NBLCKS + 5 ) .OR. LWORK.LT.LWOPT )
254     $    .AND. ( LWORK.GE.LWMIN ) .AND. ( TSIZE.GE.MINTSZ )
255     $    .AND. ( .NOT.LQUERY ) ) THEN
256        IF( TSIZE.LT.MAX( 1, MB*M*NBLCKS + 5 ) ) THEN
257          LMINWS = .TRUE.
258          MB = 1
259          NB = N
260        END IF
261        IF( LWORK.LT.LWOPT ) THEN
262          LMINWS = .TRUE.
263          MB = 1
264        END IF
265      END IF
266      IF( ( N.LE.M ) .OR. ( NB.LE.M ) .OR. ( NB.GE.N ) ) THEN
267         LWREQ = MAX( 1, MB*N )
268      ELSE
269         LWREQ = MAX( 1, MB*M )
270      END IF
271*
272      IF( M.LT.0 ) THEN
273        INFO = -1
274      ELSE IF( N.LT.0 ) THEN
275        INFO = -2
276      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
277        INFO = -4
278      ELSE IF( TSIZE.LT.MAX( 1, MB*M*NBLCKS + 5 )
279     $   .AND. ( .NOT.LQUERY ) .AND. ( .NOT.LMINWS ) ) THEN
280        INFO = -6
281      ELSE IF( ( LWORK.LT.LWREQ ) .AND .( .NOT.LQUERY )
282     $   .AND. ( .NOT.LMINWS ) ) THEN
283        INFO = -8
284      END IF
285*
286      IF( INFO.EQ.0 ) THEN
287        IF( MINT ) THEN
288          T( 1 ) = MINTSZ
289        ELSE
290          T( 1 ) = MB*M*NBLCKS + 5
291        END IF
292        T( 2 ) = MB
293        T( 3 ) = NB
294        IF( MINW ) THEN
295          WORK( 1 ) = LWMIN
296        ELSE
297          WORK( 1 ) = LWREQ
298        END IF
299      END IF
300      IF( INFO.NE.0 ) THEN
301        CALL XERBLA( 'SGELQ', -INFO )
302        RETURN
303      ELSE IF( LQUERY ) THEN
304        RETURN
305      END IF
306*
307*     Quick return if possible
308*
309      IF( MIN( M, N ).EQ.0 ) THEN
310        RETURN
311      END IF
312*
313*     The LQ Decomposition
314*
315      IF( ( N.LE.M ) .OR. ( NB.LE.M ) .OR. ( NB.GE.N ) ) THEN
316        CALL SGELQT( M, N, MB, A, LDA, T( 6 ), MB, WORK, INFO )
317      ELSE
318        CALL SLASWLQ( M, N, MB, NB, A, LDA, T( 6 ), MB, WORK,
319     $                LWORK, INFO )
320      END IF
321*
322      WORK( 1 ) = LWREQ
323      RETURN
324*
325*     End of SGELQ
326*
327      END
328