 LAPACK  3.10.0 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 )

ZLATSQR

Purpose:
ZLATSQR computes a blocked Tall-Skinny QR factorization of
a complex M-by-N matrix A for M >= N:

A = Q * ( R ),
( 0 )

where:

Q is a M-by-M orthogonal matrix, stored on exit in an implicit
form in the elements below the diagonal of the array A and in
the elements of the array T;

R is an upper-triangular N-by-N matrix, stored on exit in
the elements on and above the diagonal of the array A.

0 is a (M-N)-by-N zero matrix, and is not stored.
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).

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 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 164 of file zlatsqr.f.

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*16 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 zgeqrt, ztpqrt, 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( 'ZLATSQR', -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 zgeqrt( 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 zgeqrt( 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 ztpqrt( 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 ztpqrt( 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 ZLATSQR
268 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zgeqrt(M, N, NB, A, LDA, T, LDT, WORK, INFO)
ZGEQRT
Definition: zgeqrt.f:141
subroutine ztpqrt(M, N, L, NB, A, LDA, B, LDB, T, LDT, WORK, INFO)
ZTPQRT
Definition: ztpqrt.f:189
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