LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zgelq()

 subroutine zgelq ( integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) T, integer TSIZE, complex*16, dimension( * ) WORK, integer LWORK, integer INFO )

ZGELQ

Purpose:
``` ZGELQ computes an LQ factorization of a complex M-by-N matrix A:

A = ( L 0 ) *  Q

where:

Q is a N-by-N orthogonal matrix;
L is a lower-triangular M-by-M matrix;
0 is a M-by-(N-M) zero matrix, if M < N.```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix A. N >= 0.``` [in,out] A ``` A is COMPLEX*16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and below the diagonal of the array contain the M-by-min(M,N) lower trapezoidal matrix L (L is lower triangular if M <= N); the elements above the diagonal are used to store part of the data structure to represent Q.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] T ``` T is COMPLEX*16 array, dimension (MAX(5,TSIZE)) On exit, if INFO = 0, T(1) returns optimal (or either minimal or optimal, if query is assumed) TSIZE. See TSIZE for details. Remaining T contains part of the data structure used to represent Q. If one wants to apply or construct Q, then one needs to keep T (in addition to A) and pass it to further subroutines.``` [in] TSIZE ``` TSIZE is INTEGER If TSIZE >= 5, the dimension of the array T. If TSIZE = -1 or -2, then a workspace query is assumed. The routine only calculates the sizes of the T and WORK arrays, returns these values as the first entries of the T and WORK arrays, and no error message related to T or WORK is issued by XERBLA. If TSIZE = -1, the routine calculates optimal size of T for the optimum performance and returns this value in T(1). If TSIZE = -2, the routine calculates minimal size of T and returns this value in T(1).``` [out] WORK ``` (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) contains optimal (or either minimal or optimal, if query was assumed) LWORK. See LWORK for details.``` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. If LWORK = -1 or -2, then a workspace query is assumed. The routine only calculates the sizes of the T and WORK arrays, returns these values as the first entries of the T and WORK arrays, and no error message related to T or WORK is issued by XERBLA. If LWORK = -1, the routine calculates optimal size of WORK for the optimal performance and returns this value in WORK(1). If LWORK = -2, the routine calculates minimal size of WORK and returns this value in WORK(1).``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Further Details
``` The goal of the interface is to give maximum freedom to the developers for
creating any LQ factorization algorithm they wish. The triangular
(trapezoidal) L has to be stored in the lower part of A. The lower part of A
and the array T can be used to store any relevant information for applying or
constructing the Q factor. The WORK array can safely be discarded after exit.

Caution: One should not expect the sizes of T and WORK to be the same from one
LAPACK implementation to the other, or even from one execution to the other.
A workspace query (for T and WORK) is needed at each execution. However,
for a given execution, the size of T and WORK are fixed and will not change
from one query to the next.```
Further Details particular to this LAPACK implementation:
``` These details are particular for this LAPACK implementation. Users should not
take them for granted. These details may change in the future, and are not likely
true for another LAPACK implementation. These details are relevant if one wants
to try to understand the code. They are not part of the interface.

In this version,

T(2): row block size (MB)
T(3): column block size (NB)
T(6:TSIZE): data structure needed for Q, computed by
ZLASWLQ or ZGELQT

Depending on the matrix dimensions M and N, and row and column
block sizes MB and NB returned by ILAENV, ZGELQ will use either
ZLASWLQ (if the matrix is short-and-wide) or ZGELQT to compute
the LQ factorization.```

Definition at line 170 of file zgelq.f.

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  COMPLEX*16 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 zgelqt, zlaswlq, 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, 'ZGELQ ', ' ', m, n, 1, -1 )
225  nb = ilaenv( 1, 'ZGELQ ', ' ', 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( 'ZGELQ', -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 zgelqt( m, n, mb, a, lda, t( 6 ), mb, work, info )
317  ELSE
318  CALL zlaswlq( m, n, mb, nb, a, lda, t( 6 ), mb, work,
319  \$ lwork, info )
320  END IF
321 *
322  work( 1 ) = lwreq
323 *
324  RETURN
325 *
326 * End of ZGELQ
327 *
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zgelqt(M, N, MB, A, LDA, T, LDT, WORK, INFO)
ZGELQT
Definition: zgelqt.f:139
subroutine zlaswlq(M, N, MB, NB, A, LDA, T, LDT, WORK, LWORK, INFO)
ZLASWLQ
Definition: zlaswlq.f:164
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