LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ sgelqt3()

 recursive subroutine sgelqt3 ( integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, integer INFO )
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 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```
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 116 of file sgelqt3.f.

116 *
117 * -- LAPACK computational routine (version 3.8.0) --
118 * -- LAPACK is a software package provided by Univ. of Tennessee, --
119 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
120 * November 2017
121 *
122 * .. Scalar Arguments ..
123  INTEGER info, lda, m, n, ldt
124 * ..
125 * .. Array Arguments ..
126  REAL a( lda, * ), t( ldt, * )
127 * ..
128 *
129 * =====================================================================
130 *
131 * .. Parameters ..
132  REAL one
133  parameter( one = 1.0e+00 )
134 * ..
135 * .. Local Scalars ..
136  INTEGER i, i1, j, j1, m1, m2, iinfo
137 * ..
138 * .. External Subroutines ..
139  EXTERNAL slarfg, strmm, sgemm, 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( 'SGELQT3', -info )
155  RETURN
156  END IF
157 *
158  IF( m.EQ.1 ) THEN
159 *
160 * Compute Householder transform when N=1
161 *
162  CALL slarfg( n, a, a( 1, min( 2, n ) ), lda, t )
163 *
164  ELSE
165 *
166 * Otherwise, split A into blocks...
167 *
168  m1 = m/2
169  m2 = m-m1
170  i1 = min( m1+1, m )
171  j1 = min( m+1, n )
172 *
173 * Compute A(1:M1,1:N) <- (Y1,R1,T1), where Q1 = I - Y1 T1 Y1^H
174 *
175  CALL sgelqt3( m1, n, a, lda, t, ldt, iinfo )
176 *
177 * Compute A(J1:M,1:N) = Q1^H A(J1:M,1:N) [workspace: T(1:N1,J1:N)]
178 *
179  DO i=1,m2
180  DO j=1,m1
181  t( i+m1, j ) = a( i+m1, j )
182  END DO
183  END DO
184  CALL strmm( 'R', 'U', 'T', 'U', m2, m1, one,
185  & a, lda, t( i1, 1 ), ldt )
186 *
187  CALL sgemm( 'N', 'T', m2, m1, n-m1, one, a( i1, i1 ), lda,
188  & a( 1, i1 ), lda, one, t( i1, 1 ), ldt)
189 *
190  CALL strmm( 'R', 'U', 'N', 'N', m2, m1, one,
191  & t, ldt, t( i1, 1 ), ldt )
192 *
193  CALL sgemm( 'N', 'N', m2, n-m1, m1, -one, t( i1, 1 ), ldt,
194  & a( 1, i1 ), lda, one, a( i1, i1 ), lda )
195 *
196  CALL strmm( 'R', 'U', 'N', 'U', m2, m1 , one,
197  & a, lda, t( i1, 1 ), ldt )
198 *
199  DO i=1,m2
200  DO j=1,m1
201  a( i+m1, j ) = a( i+m1, j ) - t( i+m1, j )
202  t( i+m1, j )=0
203  END DO
204  END DO
205 *
206 * Compute A(J1:M,J1:N) <- (Y2,R2,T2) where Q2 = I - Y2 T2 Y2^H
207 *
208  CALL sgelqt3( m2, n-m1, a( i1, i1 ), lda,
209  & t( i1, i1 ), ldt, iinfo )
210 *
211 * Compute T3 = T(J1:N1,1:N) = -T1 Y1^H Y2 T2
212 *
213  DO i=1,m2
214  DO j=1,m1
215  t( j, i+m1 ) = (a( j, i+m1 ))
216  END DO
217  END DO
218 *
219  CALL strmm( 'R', 'U', 'T', 'U', m1, m2, one,
220  & a( i1, i1 ), lda, t( 1, i1 ), ldt )
221 *
222  CALL sgemm( 'N', 'T', m1, m2, n-m, one, a( 1, j1 ), lda,
223  & a( i1, j1 ), lda, one, t( 1, i1 ), ldt )
224 *
225  CALL strmm( 'L', 'U', 'N', 'N', m1, m2, -one, t, ldt,
226  & t( 1, i1 ), ldt )
227 *
228  CALL strmm( 'R', 'U', 'N', 'N', m1, m2, one,
229  & t( i1, i1 ), ldt, t( 1, i1 ), ldt )
230 *
231 *
232 *
233 * Y = (Y1,Y2); L = [ L1 0 ]; T = [T1 T3]
234 * [ A(1:N1,J1:N) L2 ] [ 0 T2]
235 *
236  END IF
237 *
238  RETURN
239 *
240 * End of SGELQT3
241 *
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:189
recursive subroutine sgelqt3(M, N, A, LDA, T, LDT, INFO)
Definition: sgelqt3.f:116
subroutine strmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRMM
Definition: strmm.f:179
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:108
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