LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ zgelqt3()

 recursive subroutine zgelqt3 ( integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldt, * ) T, integer LDT, integer INFO )

ZGELQT3 recursively computes a LQ factorization of a general real or complex matrix using the compact WY representation of Q.

Purpose:
``` DGELQT3 recursively computes a LQ factorization of a complex M-by-N
matrix A, using the compact WY representation of Q.

Based on the algorithm of Elmroth and Gustavson,
IBM J. Res. Develop. Vol 44 No. 4 July 2000.```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix A. M =< N.``` [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 real M-by-N matrix A. On exit, the elements on and below the diagonal contain the N-by-N lower triangular matrix L; the elements above the diagonal are the rows of V. See below for 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) The N-by-N upper triangular factor of the block reflector. The elements on and above the diagonal contain the block reflector T; the elements below the diagonal are not used. See below for further details.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N).``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Date
November 2017
Further Details:
```  The matrix V stores the elementary reflectors H(i) in the i-th row
above the diagonal. For example, if M=5 and N=3, the matrix V is

V = (  1  v1 v1 v1 v1 )
(     1  v2 v2 v2 )
(     1  v3 v3 v3 )

where the vi's represent the vectors which define H(i), which are returned
in the matrix A.  The 1's along the diagonal of V are not stored in A.  The
block reflector H is then given by

H = I - V * T * V**T

where V**T is the transpose of V.

For details of the algorithm, see Elmroth and Gustavson (cited above).```

Definition at line 133 of file zgelqt3.f.

133 *
134 * -- LAPACK computational routine (version 3.8.0) --
135 * -- LAPACK is a software package provided by Univ. of Tennessee, --
136 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
137 * November 2017
138 *
139 * .. Scalar Arguments ..
140  INTEGER info, lda, m, n, ldt
141 * ..
142 * .. Array Arguments ..
143  COMPLEX*16 a( lda, * ), t( ldt, * )
144 * ..
145 *
146 * =====================================================================
147 *
148 * .. Parameters ..
149  COMPLEX*16 one, zero
150  parameter( one = (1.0d+00,0.0d+00) )
151  parameter( zero = (0.0d+00,0.0d+00))
152 * ..
153 * .. Local Scalars ..
154  INTEGER i, i1, j, j1, m1, m2, iinfo
155 * ..
156 * .. External Subroutines ..
157  EXTERNAL zlarfg, ztrmm, zgemm, xerbla
158 * ..
159 * .. Executable Statements ..
160 *
161  info = 0
162  IF( m .LT. 0 ) THEN
163  info = -1
164  ELSE IF( n .LT. m ) THEN
165  info = -2
166  ELSE IF( lda .LT. max( 1, m ) ) THEN
167  info = -4
168  ELSE IF( ldt .LT. max( 1, m ) ) THEN
169  info = -6
170  END IF
171  IF( info.NE.0 ) THEN
172  CALL xerbla( 'ZGELQT3', -info )
173  RETURN
174  END IF
175 *
176  IF( m.EQ.1 ) THEN
177 *
178 * Compute Householder transform when N=1
179 *
180  CALL zlarfg( n, a, a( 1, min( 2, n ) ), lda, t )
181  t(1,1)=conjg(t(1,1))
182 *
183  ELSE
184 *
185 * Otherwise, split A into blocks...
186 *
187  m1 = m/2
188  m2 = m-m1
189  i1 = min( m1+1, m )
190  j1 = min( m+1, n )
191 *
192 * Compute A(1:M1,1:N) <- (Y1,R1,T1), where Q1 = I - Y1 T1 Y1^H
193 *
194  CALL zgelqt3( m1, n, a, lda, t, ldt, iinfo )
195 *
196 * Compute A(J1:M,1:N) = A(J1:M,1:N) Q1^H [workspace: T(1:N1,J1:N)]
197 *
198  DO i=1,m2
199  DO j=1,m1
200  t( i+m1, j ) = a( i+m1, j )
201  END DO
202  END DO
203  CALL ztrmm( 'R', 'U', 'C', 'U', m2, m1, one,
204  & a, lda, t( i1, 1 ), ldt )
205 *
206  CALL zgemm( 'N', 'C', m2, m1, n-m1, one, a( i1, i1 ), lda,
207  & a( 1, i1 ), lda, one, t( i1, 1 ), ldt)
208 *
209  CALL ztrmm( 'R', 'U', 'N', 'N', m2, m1, one,
210  & t, ldt, t( i1, 1 ), ldt )
211 *
212  CALL zgemm( 'N', 'N', m2, n-m1, m1, -one, t( i1, 1 ), ldt,
213  & a( 1, i1 ), lda, one, a( i1, i1 ), lda )
214 *
215  CALL ztrmm( 'R', 'U', 'N', 'U', m2, m1 , one,
216  & a, lda, t( i1, 1 ), ldt )
217 *
218  DO i=1,m2
219  DO j=1,m1
220  a( i+m1, j ) = a( i+m1, j ) - t( i+m1, j )
221  t( i+m1, j )= zero
222  END DO
223  END DO
224 *
225 * Compute A(J1:M,J1:N) <- (Y2,R2,T2) where Q2 = I - Y2 T2 Y2^H
226 *
227  CALL zgelqt3( m2, n-m1, a( i1, i1 ), lda,
228  & t( i1, i1 ), ldt, iinfo )
229 *
230 * Compute T3 = T(J1:N1,1:N) = -T1 Y1^H Y2 T2
231 *
232  DO i=1,m2
233  DO j=1,m1
234  t( j, i+m1 ) = (a( j, i+m1 ))
235  END DO
236  END DO
237 *
238  CALL ztrmm( 'R', 'U', 'C', 'U', m1, m2, one,
239  & a( i1, i1 ), lda, t( 1, i1 ), ldt )
240 *
241  CALL zgemm( 'N', 'C', m1, m2, n-m, one, a( 1, j1 ), lda,
242  & a( i1, j1 ), lda, one, t( 1, i1 ), ldt )
243 *
244  CALL ztrmm( 'L', 'U', 'N', 'N', m1, m2, -one, t, ldt,
245  & t( 1, i1 ), ldt )
246 *
247  CALL ztrmm( 'R', 'U', 'N', 'N', m1, m2, one,
248  & t( i1, i1 ), ldt, t( 1, i1 ), ldt )
249 *
250 *
251 *
252 * Y = (Y1,Y2); L = [ L1 0 ]; T = [T1 T3]
253 * [ A(1:N1,J1:N) L2 ] [ 0 T2]
254 *
255  END IF
256 *
257  RETURN
258 *
259 * End of ZGELQT3
260 *
recursive subroutine zgelqt3(M, N, A, LDA, T, LDT, INFO)
ZGELQT3 recursively computes a LQ factorization of a general real or complex matrix using the compact...
Definition: zgelqt3.f:133
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:189
subroutine ztrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
ZTRMM
Definition: ztrmm.f:179
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:108
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