 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dtplqt2()

 subroutine dtplqt2 ( integer M, integer N, integer L, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldt, * ) T, integer LDT, integer INFO )

DTPLQT2 computes a LQ factorization of a real or complex "triangular-pentagonal" matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

Purpose:
``` DTPLQT2 computes a LQ a factorization of a real "triangular-pentagonal"
matrix C, which is composed of a triangular block A and pentagonal block B,
using the compact WY representation for Q.```
Parameters
 [in] M ``` M is INTEGER The total number of rows of the matrix B. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix B, and the order of the triangular matrix A. N >= 0.``` [in] L ``` L is INTEGER The number of rows of the lower trapezoidal part of B. MIN(M,N) >= L >= 0. See Further Details.``` [in,out] A ``` A is DOUBLE PRECISION array, dimension (LDA,M) On entry, the lower triangular M-by-M matrix A. On exit, the elements on and below the diagonal of the array contain the lower triangular matrix L.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [in,out] B ``` B is DOUBLE PRECISION array, dimension (LDB,N) On entry, the pentagonal M-by-N matrix B. The first N-L columns are rectangular, and the last L columns are lower trapezoidal. On exit, B contains the pentagonal matrix V. See Further Details.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M).``` [out] T ``` T is DOUBLE PRECISION array, dimension (LDT,M) The N-by-N upper triangular factor T of the block reflector. See Further Details.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= max(1,M)``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Further Details:
```  The input matrix C is a M-by-(M+N) matrix

C = [ A ][ B ]

where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
upper trapezoidal matrix B2:

B = [ B1 ][ B2 ]
[ B1 ]  <-     M-by-(N-L) rectangular
[ B2 ]  <-     M-by-L lower trapezoidal.

The lower trapezoidal matrix B2 consists of the first L columns of a
N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.

The matrix W stores the elementary reflectors H(i) in the i-th row
above the diagonal (of A) in the M-by-(M+N) input matrix C

C = [ A ][ B ]
[ A ]  <- lower triangular M-by-M
[ B ]  <- M-by-N pentagonal

so that W can be represented as

W = [ I ][ V ]
[ I ]  <- identity, M-by-M
[ V ]  <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which
we call V above.  Note that V has the same form as B; that is,

W = [ V1 ][ V2 ]
[ V1 ] <-     M-by-(N-L) rectangular
[ V2 ] <-     M-by-L lower trapezoidal.

The rows of V represent the vectors which define the H(i)'s.
The (M+N)-by-(M+N) block reflector H is then given by

H = I - W**T * T * W

where W^H is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.```

Definition at line 176 of file dtplqt2.f.

177*
178* -- LAPACK computational routine --
179* -- LAPACK is a software package provided by Univ. of Tennessee, --
180* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
181*
182* .. Scalar Arguments ..
183 INTEGER INFO, LDA, LDB, LDT, N, M, L
184* ..
185* .. Array Arguments ..
186 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), T( LDT, * )
187* ..
188*
189* =====================================================================
190*
191* .. Parameters ..
192 DOUBLE PRECISION ONE, ZERO
193 parameter( one = 1.0, zero = 0.0 )
194* ..
195* .. Local Scalars ..
196 INTEGER I, J, P, MP, NP
197 DOUBLE PRECISION ALPHA
198* ..
199* .. External Subroutines ..
200 EXTERNAL dlarfg, dgemv, dger, dtrmv, xerbla
201* ..
202* .. Intrinsic Functions ..
203 INTRINSIC max, min
204* ..
205* .. Executable Statements ..
206*
207* Test the input arguments
208*
209 info = 0
210 IF( m.LT.0 ) THEN
211 info = -1
212 ELSE IF( n.LT.0 ) THEN
213 info = -2
214 ELSE IF( l.LT.0 .OR. l.GT.min(m,n) ) THEN
215 info = -3
216 ELSE IF( lda.LT.max( 1, m ) ) THEN
217 info = -5
218 ELSE IF( ldb.LT.max( 1, m ) ) THEN
219 info = -7
220 ELSE IF( ldt.LT.max( 1, m ) ) THEN
221 info = -9
222 END IF
223 IF( info.NE.0 ) THEN
224 CALL xerbla( 'DTPLQT2', -info )
225 RETURN
226 END IF
227*
228* Quick return if possible
229*
230 IF( n.EQ.0 .OR. m.EQ.0 ) RETURN
231*
232 DO i = 1, m
233*
234* Generate elementary reflector H(I) to annihilate B(I,:)
235*
236 p = n-l+min( l, i )
237 CALL dlarfg( p+1, a( i, i ), b( i, 1 ), ldb, t( 1, i ) )
238 IF( i.LT.m ) THEN
239*
240* W(M-I:1) := C(I+1:M,I:N) * C(I,I:N) [use W = T(M,:)]
241*
242 DO j = 1, m-i
243 t( m, j ) = (a( i+j, i ))
244 END DO
245 CALL dgemv( 'N', m-i, p, one, b( i+1, 1 ), ldb,
246 \$ b( i, 1 ), ldb, one, t( m, 1 ), ldt )
247*
248* C(I+1:M,I:N) = C(I+1:M,I:N) + alpha * C(I,I:N)*W(M-1:1)^H
249*
250 alpha = -(t( 1, i ))
251 DO j = 1, m-i
252 a( i+j, i ) = a( i+j, i ) + alpha*(t( m, j ))
253 END DO
254 CALL dger( m-i, p, alpha, t( m, 1 ), ldt,
255 \$ b( i, 1 ), ldb, b( i+1, 1 ), ldb )
256 END IF
257 END DO
258*
259 DO i = 2, m
260*
261* T(I,1:I-1) := C(I:I-1,1:N) * (alpha * C(I,I:N)^H)
262*
263 alpha = -t( 1, i )
264
265 DO j = 1, i-1
266 t( i, j ) = zero
267 END DO
268 p = min( i-1, l )
269 np = min( n-l+1, n )
270 mp = min( p+1, m )
271*
272* Triangular part of B2
273*
274 DO j = 1, p
275 t( i, j ) = alpha*b( i, n-l+j )
276 END DO
277 CALL dtrmv( 'L', 'N', 'N', p, b( 1, np ), ldb,
278 \$ t( i, 1 ), ldt )
279*
280* Rectangular part of B2
281*
282 CALL dgemv( 'N', i-1-p, l, alpha, b( mp, np ), ldb,
283 \$ b( i, np ), ldb, zero, t( i,mp ), ldt )
284*
285* B1
286*
287 CALL dgemv( 'N', i-1, n-l, alpha, b, ldb, b( i, 1 ), ldb,
288 \$ one, t( i, 1 ), ldt )
289*
290* T(1:I-1,I) := T(1:I-1,1:I-1) * T(I,1:I-1)
291*
292 CALL dtrmv( 'L', 'T', 'N', i-1, t, ldt, t( i, 1 ), ldt )
293*
294* T(I,I) = tau(I)
295*
296 t( i, i ) = t( 1, i )
297 t( 1, i ) = zero
298 END DO
299 DO i=1,m
300 DO j= i+1,m
301 t(i,j)=t(j,i)
302 t(j,i)= zero
303 END DO
304 END DO
305
306*
307* End of DTPLQT2
308*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dger(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
DGER
Definition: dger.f:130
subroutine dtrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
DTRMV
Definition: dtrmv.f:147
subroutine dgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DGEMV
Definition: dgemv.f:156
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:106
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