LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ sgelqt3()

 recursive subroutine sgelqt3 ( integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, integer INFO )

SGELQT3

Purpose:
``` SGELQT3 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 REAL 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 REAL 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 sgelqt3.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 REAL A( LDA, * ), T( LDT, * )
126* ..
127*
128* =====================================================================
129*
130* .. Parameters ..
131 REAL ONE
132 parameter( one = 1.0e+00 )
133* ..
134* .. Local Scalars ..
135 INTEGER I, I1, J, J1, M1, M2, IINFO
136* ..
137* .. External Subroutines ..
138 EXTERNAL slarfg, strmm, sgemm, xerbla
139* ..
140* .. Executable Statements ..
141*
142 info = 0
143 IF( m .LT. 0 ) THEN
144 info = -1
145 ELSE IF( n .LT. m ) THEN
146 info = -2
147 ELSE IF( lda .LT. max( 1, m ) ) THEN
148 info = -4
149 ELSE IF( ldt .LT. max( 1, m ) ) THEN
150 info = -6
151 END IF
152 IF( info.NE.0 ) THEN
153 CALL xerbla( 'SGELQT3', -info )
154 RETURN
155 END IF
156*
157 IF( m.EQ.1 ) THEN
158*
159* Compute Householder transform when M=1
160*
161 CALL slarfg( n, a, a( 1, min( 2, n ) ), lda, t )
162*
163 ELSE
164*
165* Otherwise, split A into blocks...
166*
167 m1 = m/2
168 m2 = m-m1
169 i1 = min( m1+1, m )
170 j1 = min( m+1, n )
171*
172* Compute A(1:M1,1:N) <- (Y1,R1,T1), where Q1 = I - Y1 T1 Y1^H
173*
174 CALL sgelqt3( m1, n, a, lda, t, ldt, iinfo )
175*
176* Compute A(J1:M,1:N) = Q1^H A(J1:M,1:N) [workspace: T(1:N1,J1:N)]
177*
178 DO i=1,m2
179 DO j=1,m1
180 t( i+m1, j ) = a( i+m1, j )
181 END DO
182 END DO
183 CALL strmm( 'R', 'U', 'T', 'U', m2, m1, one,
184 & a, lda, t( i1, 1 ), ldt )
185*
186 CALL sgemm( 'N', 'T', m2, m1, n-m1, one, a( i1, i1 ), lda,
187 & a( 1, i1 ), lda, one, t( i1, 1 ), ldt)
188*
189 CALL strmm( 'R', 'U', 'N', 'N', m2, m1, one,
190 & t, ldt, t( i1, 1 ), ldt )
191*
192 CALL sgemm( 'N', 'N', m2, n-m1, m1, -one, t( i1, 1 ), ldt,
193 & a( 1, i1 ), lda, one, a( i1, i1 ), lda )
194*
195 CALL strmm( 'R', 'U', 'N', 'U', m2, m1 , one,
196 & a, lda, t( i1, 1 ), ldt )
197*
198 DO i=1,m2
199 DO j=1,m1
200 a( i+m1, j ) = a( i+m1, j ) - t( i+m1, j )
201 t( i+m1, j )=0
202 END DO
203 END DO
204*
205* Compute A(J1:M,J1:N) <- (Y2,R2,T2) where Q2 = I - Y2 T2 Y2^H
206*
207 CALL sgelqt3( m2, n-m1, a( i1, i1 ), lda,
208 & t( i1, i1 ), ldt, iinfo )
209*
210* Compute T3 = T(J1:N1,1:N) = -T1 Y1^H Y2 T2
211*
212 DO i=1,m2
213 DO j=1,m1
214 t( j, i+m1 ) = (a( j, i+m1 ))
215 END DO
216 END DO
217*
218 CALL strmm( 'R', 'U', 'T', 'U', m1, m2, one,
219 & a( i1, i1 ), lda, t( 1, i1 ), ldt )
220*
221 CALL sgemm( 'N', 'T', m1, m2, n-m, one, a( 1, j1 ), lda,
222 & a( i1, j1 ), lda, one, t( 1, i1 ), ldt )
223*
224 CALL strmm( 'L', 'U', 'N', 'N', m1, m2, -one, t, ldt,
225 & t( 1, i1 ), ldt )
226*
227 CALL strmm( 'R', 'U', 'N', 'N', m1, m2, one,
228 & t( i1, i1 ), ldt, t( 1, i1 ), ldt )
229*
230*
231*
232* Y = (Y1,Y2); L = [ L1 0 ]; T = [T1 T3]
233* [ A(1:N1,J1:N) L2 ] [ 0 T2]
234*
235 END IF
236*
237 RETURN
238*
239* End of SGELQT3
240*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
recursive subroutine sgelqt3(M, N, A, LDA, T, LDT, INFO)
SGELQT3
Definition: sgelqt3.f:116
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
subroutine strmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRMM
Definition: strmm.f:177
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
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