 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dgelqt3()

 recursive subroutine dgelqt3 ( integer M, integer N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldt, * ) T, integer LDT, integer INFO )

DGELQT3 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 real 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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```
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 130 of file dgelqt3.f.

131 *
132 * -- LAPACK computational routine --
133 * -- LAPACK is a software package provided by Univ. of Tennessee, --
134 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
135 *
136 * .. Scalar Arguments ..
137  INTEGER INFO, LDA, M, N, LDT
138 * ..
139 * .. Array Arguments ..
140  DOUBLE PRECISION A( LDA, * ), T( LDT, * )
141 * ..
142 *
143 * =====================================================================
144 *
145 * .. Parameters ..
146  DOUBLE PRECISION ONE
147  parameter( one = 1.0d+00 )
148 * ..
149 * .. Local Scalars ..
150  INTEGER I, I1, J, J1, M1, M2, IINFO
151 * ..
152 * .. External Subroutines ..
153  EXTERNAL dlarfg, dtrmm, dgemm, xerbla
154 * ..
155 * .. Executable Statements ..
156 *
157  info = 0
158  IF( m .LT. 0 ) THEN
159  info = -1
160  ELSE IF( n .LT. m ) THEN
161  info = -2
162  ELSE IF( lda .LT. max( 1, m ) ) THEN
163  info = -4
164  ELSE IF( ldt .LT. max( 1, m ) ) THEN
165  info = -6
166  END IF
167  IF( info.NE.0 ) THEN
168  CALL xerbla( 'DGELQT3', -info )
169  RETURN
170  END IF
171 *
172  IF( m.EQ.1 ) THEN
173 *
174 * Compute Householder transform when M=1
175 *
176  CALL dlarfg( n, a, a( 1, min( 2, n ) ), lda, t )
177 *
178  ELSE
179 *
180 * Otherwise, split A into blocks...
181 *
182  m1 = m/2
183  m2 = m-m1
184  i1 = min( m1+1, m )
185  j1 = min( m+1, n )
186 *
187 * Compute A(1:M1,1:N) <- (Y1,R1,T1), where Q1 = I - Y1 T1 Y1^H
188 *
189  CALL dgelqt3( m1, n, a, lda, t, ldt, iinfo )
190 *
191 * Compute A(J1:M,1:N) = Q1^H A(J1:M,1:N) [workspace: T(1:N1,J1:N)]
192 *
193  DO i=1,m2
194  DO j=1,m1
195  t( i+m1, j ) = a( i+m1, j )
196  END DO
197  END DO
198  CALL dtrmm( 'R', 'U', 'T', 'U', m2, m1, one,
199  & a, lda, t( i1, 1 ), ldt )
200 *
201  CALL dgemm( 'N', 'T', m2, m1, n-m1, one, a( i1, i1 ), lda,
202  & a( 1, i1 ), lda, one, t( i1, 1 ), ldt)
203 *
204  CALL dtrmm( 'R', 'U', 'N', 'N', m2, m1, one,
205  & t, ldt, t( i1, 1 ), ldt )
206 *
207  CALL dgemm( 'N', 'N', m2, n-m1, m1, -one, t( i1, 1 ), ldt,
208  & a( 1, i1 ), lda, one, a( i1, i1 ), lda )
209 *
210  CALL dtrmm( 'R', 'U', 'N', 'U', m2, m1 , one,
211  & a, lda, t( i1, 1 ), ldt )
212 *
213  DO i=1,m2
214  DO j=1,m1
215  a( i+m1, j ) = a( i+m1, j ) - t( i+m1, j )
216  t( i+m1, j )=0
217  END DO
218  END DO
219 *
220 * Compute A(J1:M,J1:N) <- (Y2,R2,T2) where Q2 = I - Y2 T2 Y2^H
221 *
222  CALL dgelqt3( m2, n-m1, a( i1, i1 ), lda,
223  & t( i1, i1 ), ldt, iinfo )
224 *
225 * Compute T3 = T(J1:N1,1:N) = -T1 Y1^H Y2 T2
226 *
227  DO i=1,m2
228  DO j=1,m1
229  t( j, i+m1 ) = (a( j, i+m1 ))
230  END DO
231  END DO
232 *
233  CALL dtrmm( 'R', 'U', 'T', 'U', m1, m2, one,
234  & a( i1, i1 ), lda, t( 1, i1 ), ldt )
235 *
236  CALL dgemm( 'N', 'T', m1, m2, n-m, one, a( 1, j1 ), lda,
237  & a( i1, j1 ), lda, one, t( 1, i1 ), ldt )
238 *
239  CALL dtrmm( 'L', 'U', 'N', 'N', m1, m2, -one, t, ldt,
240  & t( 1, i1 ), ldt )
241 *
242  CALL dtrmm( 'R', 'U', 'N', 'N', m1, m2, one,
243  & t( i1, i1 ), ldt, t( 1, i1 ), ldt )
244 *
245 *
246 *
247 * Y = (Y1,Y2); L = [ L1 0 ]; T = [T1 T3]
248 * [ A(1:N1,J1:N) L2 ] [ 0 T2]
249 *
250  END IF
251 *
252  RETURN
253 *
254 * End of DGELQT3
255 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187
subroutine dtrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
DTRMM
Definition: dtrmm.f:177
recursive subroutine dgelqt3(M, N, A, LDA, T, LDT, INFO)
DGELQT3 recursively computes a LQ factorization of a general real or complex matrix using the compact...
Definition: dgelqt3.f:131
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:106
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