LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ sorgtsqr()

 subroutine sorgtsqr ( integer M, integer N, integer MB, integer NB, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) WORK, integer LWORK, integer INFO )

SORGTSQR

Purpose:
``` SORGTSQR generates an M-by-N real matrix Q_out with orthonormal columns,
which are the first N columns of a product of real orthogonal
matrices of order M which are returned by SLATSQR

Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ).

See the documentation for SLATSQR.```
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. M >= N >= 0.``` [in] MB ``` MB is INTEGER The row block size used by SLATSQR to return arrays A and T. MB > N. (Note that if MB > M, then M is used instead of MB as the row block size).``` [in] NB ``` NB is INTEGER The column block size used by SLATSQR to return arrays A and T. NB >= 1. (Note that if NB > N, then N is used instead of NB as the column block size).``` [in,out] A ``` A is REAL array, dimension (LDA,N) On entry: The elements on and above the diagonal are not accessed. The elements below the diagonal represent the unit lower-trapezoidal blocked matrix V computed by SLATSQR that defines the input matrices Q_in(k) (ones on the diagonal are not stored) (same format as the output A below the diagonal in SLATSQR). On exit: The array A contains an M-by-N orthonormal matrix Q_out, i.e the columns of A are orthogonal unit vectors.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [in] T ``` T is REAL array, dimension (LDT, N * NIRB) where NIRB = Number_of_input_row_blocks = MAX( 1, CEIL((M-N)/(MB-N)) ) Let NICB = Number_of_input_col_blocks = CEIL(N/NB) The upper-triangular block reflectors used to define the input matrices Q_in(k), k=(1:NIRB*NICB). The block reflectors are stored in compact form in NIRB block reflector sequences. Each of NIRB block reflector sequences is stored in a larger NB-by-N column block of T and consists of NICB smaller NB-by-NB upper-triangular column blocks. (same format as the output T in SLATSQR).``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= max(1,min(NB1,N)).``` [out] WORK ``` (workspace) REAL array, dimension (MAX(2,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` The dimension of the array WORK. LWORK >= (M+NB)*N. 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 had an illegal value```
Contributors:
``` November 2019, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley```

Definition at line 173 of file sorgtsqr.f.

175  IMPLICIT NONE
176 *
177 * -- LAPACK computational routine --
178 * -- LAPACK is a software package provided by Univ. of Tennessee, --
179 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
180 *
181 * .. Scalar Arguments ..
182  INTEGER INFO, LDA, LDT, LWORK, M, N, MB, NB
183 * ..
184 * .. Array Arguments ..
185  REAL A( LDA, * ), T( LDT, * ), WORK( * )
186 * ..
187 *
188 * =====================================================================
189 *
190 * .. Parameters ..
191  REAL ONE, ZERO
192  parameter( one = 1.0e+0, zero = 0.0e+0 )
193 * ..
194 * .. Local Scalars ..
195  LOGICAL LQUERY
196  INTEGER IINFO, LDC, LWORKOPT, LC, LW, NBLOCAL, J
197 * ..
198 * .. External Subroutines ..
199  EXTERNAL scopy, slamtsqr, slaset, xerbla
200 * ..
201 * .. Intrinsic Functions ..
202  INTRINSIC real, max, min
203 * ..
204 * .. Executable Statements ..
205 *
206 * Test the input parameters
207 *
208  lquery = lwork.EQ.-1
209  info = 0
210  IF( m.LT.0 ) THEN
211  info = -1
212  ELSE IF( n.LT.0 .OR. m.LT.n ) THEN
213  info = -2
214  ELSE IF( mb.LE.n ) THEN
215  info = -3
216  ELSE IF( nb.LT.1 ) THEN
217  info = -4
218  ELSE IF( lda.LT.max( 1, m ) ) THEN
219  info = -6
220  ELSE IF( ldt.LT.max( 1, min( nb, n ) ) ) THEN
221  info = -8
222  ELSE
223 *
224 * Test the input LWORK for the dimension of the array WORK.
225 * This workspace is used to store array C(LDC, N) and WORK(LWORK)
226 * in the call to SLAMTSQR. See the documentation for SLAMTSQR.
227 *
228  IF( lwork.LT.2 .AND. (.NOT.lquery) ) THEN
229  info = -10
230  ELSE
231 *
232 * Set block size for column blocks
233 *
234  nblocal = min( nb, n )
235 *
236 * LWORK = -1, then set the size for the array C(LDC,N)
237 * in SLAMTSQR call and set the optimal size of the work array
238 * WORK(LWORK) in SLAMTSQR call.
239 *
240  ldc = m
241  lc = ldc*n
242  lw = n * nblocal
243 *
244  lworkopt = lc+lw
245 *
246  IF( ( lwork.LT.max( 1, lworkopt ) ).AND.(.NOT.lquery) ) THEN
247  info = -10
248  END IF
249  END IF
250 *
251  END IF
252 *
253 * Handle error in the input parameters and return workspace query.
254 *
255  IF( info.NE.0 ) THEN
256  CALL xerbla( 'SORGTSQR', -info )
257  RETURN
258  ELSE IF ( lquery ) THEN
259  work( 1 ) = real( lworkopt )
260  RETURN
261  END IF
262 *
263 * Quick return if possible
264 *
265  IF( min( m, n ).EQ.0 ) THEN
266  work( 1 ) = real( lworkopt )
267  RETURN
268  END IF
269 *
270 * (1) Form explicitly the tall-skinny M-by-N left submatrix Q1_in
271 * of M-by-M orthogonal matrix Q_in, which is implicitly stored in
272 * the subdiagonal part of input array A and in the input array T.
273 * Perform by the following operation using the routine SLAMTSQR.
274 *
275 * Q1_in = Q_in * ( I ), where I is a N-by-N identity matrix,
276 * ( 0 ) 0 is a (M-N)-by-N zero matrix.
277 *
278 * (1a) Form M-by-N matrix in the array WORK(1:LDC*N) with ones
279 * on the diagonal and zeros elsewhere.
280 *
281  CALL slaset( 'F', m, n, zero, one, work, ldc )
282 *
283 * (1b) On input, WORK(1:LDC*N) stores ( I );
284 * ( 0 )
285 *
286 * On output, WORK(1:LDC*N) stores Q1_in.
287 *
288  CALL slamtsqr( 'L', 'N', m, n, n, mb, nblocal, a, lda, t, ldt,
289  \$ work, ldc, work( lc+1 ), lw, iinfo )
290 *
291 * (2) Copy the result from the part of the work array (1:M,1:N)
292 * with the leading dimension LDC that starts at WORK(1) into
293 * the output array A(1:M,1:N) column-by-column.
294 *
295  DO j = 1, n
296  CALL scopy( m, work( (j-1)*ldc + 1 ), 1, a( 1, j ), 1 )
297  END DO
298 *
299  work( 1 ) = real( lworkopt )
300  RETURN
301 *
302 * End of SORGTSQR
303 *
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: slaset.f:110
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine slamtsqr(SIDE, TRANS, M, N, K, MB, NB, A, LDA, T, LDT, C, LDC, WORK, LWORK, INFO)
SLAMTSQR
Definition: slamtsqr.f:197
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