 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ zlatsqr()

 subroutine zlatsqr ( integer M, integer N, integer MB, integer NB, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension(ldt, *) T, integer LDT, complex*16, dimension( * ) WORK, integer LWORK, integer INFO )
Purpose:

SLATSQR computes a blocked Tall-Skinny QR factorization of an M-by-N matrix A, where M >= N: A = Q * R .

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 to be used in the blocked QR. MB > N.``` [in] NB ``` NB is INTEGER The column block size to be used in the blocked QR. N >= NB >= 1.``` [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 above the diagonal of the array contain the N-by-N upper triangular matrix R; the elements below the diagonal represent Q by the columns of blocked V (see Further Details).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] T ``` T is COMPLEX*16 array, dimension (LDT, N * Number_of_row_blocks) where Number_of_row_blocks = CEIL((M-N)/(MB-N)) The blocked upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details below.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= NB.``` [out] WORK ` (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK))` [in] LWORK ``` The dimension of the array WORK. LWORK >= 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```
Further Details:
Tall-Skinny QR (TSQR) performs QR by a sequence of orthogonal transformations, representing Q as a product of other orthogonal matrices Q = Q(1) * Q(2) * . . . * Q(k) where each Q(i) zeros out subdiagonal entries of a block of MB rows of A: Q(1) zeros out the subdiagonal entries of rows 1:MB of A Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A . . .

Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors stored under the diagonal of rows 1:MB of A, and by upper triangular block reflectors, stored in array T(1:LDT,1:N). For more information see Further Details in GEQRT.

Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N). The last Q(k) may use fewer rows. For more information see Further Details in TPQRT.

For more details of the overall algorithm, see the description of Sequential TSQR in Section 2.2 of .

 “Communication-Optimal Parallel and Sequential QR and LU Factorizations,” J. Demmel, L. Grigori, M. Hoemmen, J. Langou, SIAM J. Sci. Comput, vol. 34, no. 1, 2012

Definition at line 151 of file zlatsqr.f.

151 *
152 * -- LAPACK computational routine (version 3.7.0) --
153 * -- LAPACK is a software package provided by Univ. of Tennessee, --
154 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd. --
155 * December 2016
156 *
157 * .. Scalar Arguments ..
158  INTEGER info, lda, m, n, mb, nb, ldt, lwork
159 * ..
160 * .. Array Arguments ..
161  COMPLEX*16 a( lda, * ), work( * ), t(ldt, *)
162 * ..
163 *
164 * =====================================================================
165 *
166 * ..
167 * .. Local Scalars ..
168  LOGICAL lquery
169  INTEGER i, ii, kk, ctr
170 * ..
171 * .. EXTERNAL FUNCTIONS ..
172  LOGICAL lsame
173  EXTERNAL lsame
174 * .. EXTERNAL SUBROUTINES ..
175  EXTERNAL zgeqrt, ztpqrt, xerbla
176 * .. INTRINSIC FUNCTIONS ..
177  INTRINSIC max, min, mod
178 * ..
179 * .. EXECUTABLE STATEMENTS ..
180 *
181 * TEST THE INPUT ARGUMENTS
182 *
183  info = 0
184 *
185  lquery = ( lwork.EQ.-1 )
186 *
187  IF( m.LT.0 ) THEN
188  info = -1
189  ELSE IF( n.LT.0 .OR. m.LT.n ) THEN
190  info = -2
191  ELSE IF( mb.LE.n ) THEN
192  info = -3
193  ELSE IF( nb.LT.1 .OR. ( nb.GT.n .AND. n.GT.0 )) THEN
194  info = -4
195  ELSE IF( lda.LT.max( 1, m ) ) THEN
196  info = -5
197  ELSE IF( ldt.LT.nb ) THEN
198  info = -8
199  ELSE IF( lwork.LT.(n*nb) .AND. (.NOT.lquery) ) THEN
200  info = -10
201  END IF
202  IF( info.EQ.0) THEN
203  work(1) = nb*n
204  END IF
205  IF( info.NE.0 ) THEN
206  CALL xerbla( 'ZLATSQR', -info )
207  RETURN
208  ELSE IF (lquery) THEN
209  RETURN
210  END IF
211 *
212 * Quick return if possible
213 *
214  IF( min(m,n).EQ.0 ) THEN
215  RETURN
216  END IF
217 *
218 * The QR Decomposition
219 *
220  IF ((mb.LE.n).OR.(mb.GE.m)) THEN
221  CALL zgeqrt( m, n, nb, a, lda, t, ldt, work, info)
222  RETURN
223  END IF
224  kk = mod((m-n),(mb-n))
225  ii=m-kk+1
226 *
227 * Compute the QR factorization of the first block A(1:MB,1:N)
228 *
229  CALL zgeqrt( mb, n, nb, a(1,1), lda, t, ldt, work, info )
230  ctr = 1
231 *
232  DO i = mb+1, ii-mb+n , (mb-n)
233 *
234 * Compute the QR factorization of the current block A(I:I+MB-N,1:N)
235 *
236  CALL ztpqrt( mb-n, n, 0, nb, a(1,1), lda, a( i, 1 ), lda,
237  \$ t(1, ctr * n + 1),
238  \$ ldt, work, info )
239  ctr = ctr + 1
240  END DO
241 *
242 * Compute the QR factorization of the last block A(II:M,1:N)
243 *
244  IF (ii.LE.m) THEN
245  CALL ztpqrt( kk, n, 0, nb, a(1,1), lda, a( ii, 1 ), lda,
246  \$ t(1,ctr * n + 1), ldt,
247  \$ work, info )
248  END IF
249 *
250  work( 1 ) = n*nb
251  RETURN
252 *
253 * End of ZLATSQR
254 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine zgeqrt(M, N, NB, A, LDA, T, LDT, WORK, INFO)
ZGEQRT
Definition: zgeqrt.f:143
subroutine ztpqrt(M, N, L, NB, A, LDA, B, LDB, T, LDT, WORK, INFO)
ZTPQRT
Definition: ztpqrt.f:191
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