 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ ctplqt2()

 subroutine ctplqt2 ( integer M, integer N, integer L, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldt, * ) T, integer LDT, integer INFO )

CTPLQT2

Purpose:
``` CTPLQT2 computes a LQ a factorization of a complex "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 COMPLEX 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 COMPLEX 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 COMPLEX 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 161 of file ctplqt2.f.

162 *
163 * -- LAPACK computational routine --
164 * -- LAPACK is a software package provided by Univ. of Tennessee, --
165 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
166 *
167 * .. Scalar Arguments ..
168  INTEGER INFO, LDA, LDB, LDT, N, M, L
169 * ..
170 * .. Array Arguments ..
171  COMPLEX A( LDA, * ), B( LDB, * ), T( LDT, * )
172 * ..
173 *
174 * =====================================================================
175 *
176 * .. Parameters ..
177  COMPLEX ONE, ZERO
178  parameter( zero = ( 0.0e+0, 0.0e+0 ),one = ( 1.0e+0, 0.0e+0 ) )
179 * ..
180 * .. Local Scalars ..
181  INTEGER I, J, P, MP, NP
182  COMPLEX ALPHA
183 * ..
184 * .. External Subroutines ..
185  EXTERNAL clarfg, cgemv, cgerc, ctrmv, xerbla
186 * ..
187 * .. Intrinsic Functions ..
188  INTRINSIC max, min
189 * ..
190 * .. Executable Statements ..
191 *
192 * Test the input arguments
193 *
194  info = 0
195  IF( m.LT.0 ) THEN
196  info = -1
197  ELSE IF( n.LT.0 ) THEN
198  info = -2
199  ELSE IF( l.LT.0 .OR. l.GT.min(m,n) ) THEN
200  info = -3
201  ELSE IF( lda.LT.max( 1, m ) ) THEN
202  info = -5
203  ELSE IF( ldb.LT.max( 1, m ) ) THEN
204  info = -7
205  ELSE IF( ldt.LT.max( 1, m ) ) THEN
206  info = -9
207  END IF
208  IF( info.NE.0 ) THEN
209  CALL xerbla( 'CTPLQT2', -info )
210  RETURN
211  END IF
212 *
213 * Quick return if possible
214 *
215  IF( n.EQ.0 .OR. m.EQ.0 ) RETURN
216 *
217  DO i = 1, m
218 *
219 * Generate elementary reflector H(I) to annihilate B(I,:)
220 *
221  p = n-l+min( l, i )
222  CALL clarfg( p+1, a( i, i ), b( i, 1 ), ldb, t( 1, i ) )
223  t(1,i)=conjg(t(1,i))
224  IF( i.LT.m ) THEN
225  DO j = 1, p
226  b( i, j ) = conjg(b(i,j))
227  END DO
228 *
229 * W(M-I:1) := C(I+1:M,I:N) * C(I,I:N) [use W = T(M,:)]
230 *
231  DO j = 1, m-i
232  t( m, j ) = (a( i+j, i ))
233  END DO
234  CALL cgemv( 'N', m-i, p, one, b( i+1, 1 ), ldb,
235  \$ b( i, 1 ), ldb, one, t( m, 1 ), ldt )
236 *
237 * C(I+1:M,I:N) = C(I+1:M,I:N) + alpha * C(I,I:N)*W(M-1:1)^H
238 *
239  alpha = -(t( 1, i ))
240  DO j = 1, m-i
241  a( i+j, i ) = a( i+j, i ) + alpha*(t( m, j ))
242  END DO
243  CALL cgerc( m-i, p, (alpha), t( m, 1 ), ldt,
244  \$ b( i, 1 ), ldb, b( i+1, 1 ), ldb )
245  DO j = 1, p
246  b( i, j ) = conjg(b(i,j))
247  END DO
248  END IF
249  END DO
250 *
251  DO i = 2, m
252 *
253 * T(I,1:I-1) := C(I:I-1,1:N)**H * (alpha * C(I,I:N))
254 *
255  alpha = -(t( 1, i ))
256  DO j = 1, i-1
257  t( i, j ) = zero
258  END DO
259  p = min( i-1, l )
260  np = min( n-l+1, n )
261  mp = min( p+1, m )
262  DO j = 1, n-l+p
263  b(i,j)=conjg(b(i,j))
264  END DO
265 *
266 * Triangular part of B2
267 *
268  DO j = 1, p
269  t( i, j ) = (alpha*b( i, n-l+j ))
270  END DO
271  CALL ctrmv( 'L', 'N', 'N', p, b( 1, np ), ldb,
272  \$ t( i, 1 ), ldt )
273 *
274 * Rectangular part of B2
275 *
276  CALL cgemv( 'N', i-1-p, l, alpha, b( mp, np ), ldb,
277  \$ b( i, np ), ldb, zero, t( i,mp ), ldt )
278 *
279 * B1
280
281 *
282  CALL cgemv( 'N', i-1, n-l, alpha, b, ldb, b( i, 1 ), ldb,
283  \$ one, t( i, 1 ), ldt )
284 *
285
286 *
287 * T(1:I-1,I) := T(1:I-1,1:I-1) * T(I,1:I-1)
288 *
289  DO j = 1, i-1
290  t(i,j)=conjg(t(i,j))
291  END DO
292  CALL ctrmv( 'L', 'C', 'N', i-1, t, ldt, t( i, 1 ), ldt )
293  DO j = 1, i-1
294  t(i,j)=conjg(t(i,j))
295  END DO
296  DO j = 1, n-l+p
297  b(i,j)=conjg(b(i,j))
298  END DO
299 *
300 * T(I,I) = tau(I)
301 *
302  t( i, i ) = t( 1, i )
303  t( 1, i ) = zero
304  END DO
305  DO i=1,m
306  DO j= i+1,m
307  t(i,j)=(t(j,i))
308  t(j,i)=zero
309  END DO
310  END DO
311
312 *
313 * End of CTPLQT2
314 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGEMV
Definition: cgemv.f:158
subroutine cgerc(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
CGERC
Definition: cgerc.f:130
subroutine ctrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
CTRMV
Definition: ctrmv.f:147
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:106
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