 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ cgelqt3()

 recursive subroutine cgelqt3 ( integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldt, * ) T, integer LDT, integer INFO )

CGELQT3

Purpose:
``` CGELQT3 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 array, dimension (LDA,N) On entry, the complex 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 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 115 of file cgelqt3.f.

116 *
117 * -- LAPACK computational routine --
118 * -- LAPACK is a software package provided by Univ. of Tennessee, --
119 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
120 *
121 * .. Scalar Arguments ..
122  INTEGER INFO, LDA, M, N, LDT
123 * ..
124 * .. Array Arguments ..
125  COMPLEX A( LDA, * ), T( LDT, * )
126 * ..
127 *
128 * =====================================================================
129 *
130 * .. Parameters ..
131  COMPLEX ONE, ZERO
132  parameter( one = (1.0e+00,0.0e+00) )
133  parameter( zero = (0.0e+00,0.0e+00))
134 * ..
135 * .. Local Scalars ..
136  INTEGER I, I1, J, J1, M1, M2, IINFO
137 * ..
138 * .. External Subroutines ..
139  EXTERNAL clarfg, ctrmm, cgemm, xerbla
140 * ..
141 * .. Executable Statements ..
142 *
143  info = 0
144  IF( m .LT. 0 ) THEN
145  info = -1
146  ELSE IF( n .LT. m ) THEN
147  info = -2
148  ELSE IF( lda .LT. max( 1, m ) ) THEN
149  info = -4
150  ELSE IF( ldt .LT. max( 1, m ) ) THEN
151  info = -6
152  END IF
153  IF( info.NE.0 ) THEN
154  CALL xerbla( 'CGELQT3', -info )
155  RETURN
156  END IF
157 *
158  IF( m.EQ.1 ) THEN
159 *
160 * Compute Householder transform when M=1
161 *
162  CALL clarfg( n, a, a( 1, min( 2, n ) ), lda, t )
163  t(1,1)=conjg(t(1,1))
164 *
165  ELSE
166 *
167 * Otherwise, split A into blocks...
168 *
169  m1 = m/2
170  m2 = m-m1
171  i1 = min( m1+1, m )
172  j1 = min( m+1, n )
173 *
174 * Compute A(1:M1,1:N) <- (Y1,R1,T1), where Q1 = I - Y1 T1 Y1^H
175 *
176  CALL cgelqt3( m1, n, a, lda, t, ldt, iinfo )
177 *
178 * Compute A(J1:M,1:N) = A(J1:M,1:N) Q1^H [workspace: T(1:N1,J1:N)]
179 *
180  DO i=1,m2
181  DO j=1,m1
182  t( i+m1, j ) = a( i+m1, j )
183  END DO
184  END DO
185  CALL ctrmm( 'R', 'U', 'C', 'U', m2, m1, one,
186  & a, lda, t( i1, 1 ), ldt )
187 *
188  CALL cgemm( 'N', 'C', m2, m1, n-m1, one, a( i1, i1 ), lda,
189  & a( 1, i1 ), lda, one, t( i1, 1 ), ldt)
190 *
191  CALL ctrmm( 'R', 'U', 'N', 'N', m2, m1, one,
192  & t, ldt, t( i1, 1 ), ldt )
193 *
194  CALL cgemm( 'N', 'N', m2, n-m1, m1, -one, t( i1, 1 ), ldt,
195  & a( 1, i1 ), lda, one, a( i1, i1 ), lda )
196 *
197  CALL ctrmm( 'R', 'U', 'N', 'U', m2, m1 , one,
198  & a, lda, t( i1, 1 ), ldt )
199 *
200  DO i=1,m2
201  DO j=1,m1
202  a( i+m1, j ) = a( i+m1, j ) - t( i+m1, j )
203  t( i+m1, j )= zero
204  END DO
205  END DO
206 *
207 * Compute A(J1:M,J1:N) <- (Y2,R2,T2) where Q2 = I - Y2 T2 Y2^H
208 *
209  CALL cgelqt3( m2, n-m1, a( i1, i1 ), lda,
210  & t( i1, i1 ), ldt, iinfo )
211 *
212 * Compute T3 = T(J1:N1,1:N) = -T1 Y1^H Y2 T2
213 *
214  DO i=1,m2
215  DO j=1,m1
216  t( j, i+m1 ) = (a( j, i+m1 ))
217  END DO
218  END DO
219 *
220  CALL ctrmm( 'R', 'U', 'C', 'U', m1, m2, one,
221  & a( i1, i1 ), lda, t( 1, i1 ), ldt )
222 *
223  CALL cgemm( 'N', 'C', m1, m2, n-m, one, a( 1, j1 ), lda,
224  & a( i1, j1 ), lda, one, t( 1, i1 ), ldt )
225 *
226  CALL ctrmm( 'L', 'U', 'N', 'N', m1, m2, -one, t, ldt,
227  & t( 1, i1 ), ldt )
228 *
229  CALL ctrmm( 'R', 'U', 'N', 'N', m1, m2, one,
230  & t( i1, i1 ), ldt, t( 1, i1 ), ldt )
231 *
232 *
233 *
234 * Y = (Y1,Y2); L = [ L1 0 ]; T = [T1 T3]
235 * [ A(1:N1,J1:N) L2 ] [ 0 T2]
236 *
237  END IF
238 *
239  RETURN
240 *
241 * End of CGELQT3
242 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CGEMM
Definition: cgemm.f:187
subroutine ctrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
CTRMM
Definition: ctrmm.f:177
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:106
recursive subroutine cgelqt3(M, N, A, LDA, T, LDT, INFO)
CGELQT3
Definition: cgelqt3.f:116
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