 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

◆ ctpqrt2()

 subroutine ctpqrt2 ( 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 )

CTPQRT2 computes a QR 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:
CTPQRT2 computes a QR 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 upper trapezoidal part of B. MIN(M,N) >= L >= 0. See Further Details. [in,out] A A is COMPLEX array, dimension (LDA,N) On entry, the upper triangular N-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the upper triangular matrix R. [in] LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). [in,out] B B is COMPLEX array, dimension (LDB,N) On entry, the pentagonal M-by-N matrix B. The first M-L rows are rectangular, and the last L rows are upper 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,N) 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,N) [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 (N+M)-by-N matrix

C = [ A ]
[ B ]

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

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

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

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

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

so that W can be represented as

W = [ I ]  <- identity, N-by-N
[ 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,

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

The columns 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 * W**H

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

Definition at line 172 of file ctpqrt2.f.

173 *
174 * -- LAPACK computational routine --
175 * -- LAPACK is a software package provided by Univ. of Tennessee, --
176 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
177 *
178 * .. Scalar Arguments ..
179  INTEGER INFO, LDA, LDB, LDT, N, M, L
180 * ..
181 * .. Array Arguments ..
182  COMPLEX A( LDA, * ), B( LDB, * ), T( LDT, * )
183 * ..
184 *
185 * =====================================================================
186 *
187 * .. Parameters ..
188  COMPLEX ONE, ZERO
189  parameter( one = (1.0,0.0), zero = (0.0,0.0) )
190 * ..
191 * .. Local Scalars ..
192  INTEGER I, J, P, MP, NP
193  COMPLEX ALPHA
194 * ..
195 * .. External Subroutines ..
196  EXTERNAL clarfg, cgemv, cgerc, ctrmv, xerbla
197 * ..
198 * .. Intrinsic Functions ..
199  INTRINSIC max, min
200 * ..
201 * .. Executable Statements ..
202 *
203 * Test the input arguments
204 *
205  info = 0
206  IF( m.LT.0 ) THEN
207  info = -1
208  ELSE IF( n.LT.0 ) THEN
209  info = -2
210  ELSE IF( l.LT.0 .OR. l.GT.min(m,n) ) THEN
211  info = -3
212  ELSE IF( lda.LT.max( 1, n ) ) THEN
213  info = -5
214  ELSE IF( ldb.LT.max( 1, m ) ) THEN
215  info = -7
216  ELSE IF( ldt.LT.max( 1, n ) ) THEN
217  info = -9
218  END IF
219  IF( info.NE.0 ) THEN
220  CALL xerbla( 'CTPQRT2', -info )
221  RETURN
222  END IF
223 *
224 * Quick return if possible
225 *
226  IF( n.EQ.0 .OR. m.EQ.0 ) RETURN
227 *
228  DO i = 1, n
229 *
230 * Generate elementary reflector H(I) to annihilate B(:,I)
231 *
232  p = m-l+min( l, i )
233  CALL clarfg( p+1, a( i, i ), b( 1, i ), 1, t( i, 1 ) )
234  IF( i.LT.n ) THEN
235 *
236 * W(1:N-I) := C(I:M,I+1:N)**H * C(I:M,I) [use W = T(:,N)]
237 *
238  DO j = 1, n-i
239  t( j, n ) = conjg(a( i, i+j ))
240  END DO
241  CALL cgemv( 'C', p, n-i, one, b( 1, i+1 ), ldb,
242  \$ b( 1, i ), 1, one, t( 1, n ), 1 )
243 *
244 * C(I:M,I+1:N) = C(I:m,I+1:N) + alpha*C(I:M,I)*W(1:N-1)**H
245 *
246  alpha = -conjg(t( i, 1 ))
247  DO j = 1, n-i
248  a( i, i+j ) = a( i, i+j ) + alpha*conjg(t( j, n ))
249  END DO
250  CALL cgerc( p, n-i, alpha, b( 1, i ), 1,
251  \$ t( 1, n ), 1, b( 1, i+1 ), ldb )
252  END IF
253  END DO
254 *
255  DO i = 2, n
256 *
257 * T(1:I-1,I) := C(I:M,1:I-1)**H * (alpha * C(I:M,I))
258 *
259  alpha = -t( i, 1 )
260
261  DO j = 1, i-1
262  t( j, i ) = zero
263  END DO
264  p = min( i-1, l )
265  mp = min( m-l+1, m )
266  np = min( p+1, n )
267 *
268 * Triangular part of B2
269 *
270  DO j = 1, p
271  t( j, i ) = alpha*b( m-l+j, i )
272  END DO
273  CALL ctrmv( 'U', 'C', 'N', p, b( mp, 1 ), ldb,
274  \$ t( 1, i ), 1 )
275 *
276 * Rectangular part of B2
277 *
278  CALL cgemv( 'C', l, i-1-p, alpha, b( mp, np ), ldb,
279  \$ b( mp, i ), 1, zero, t( np, i ), 1 )
280 *
281 * B1
282 *
283  CALL cgemv( 'C', m-l, i-1, alpha, b, ldb, b( 1, i ), 1,
284  \$ one, t( 1, i ), 1 )
285 *
286 * T(1:I-1,I) := T(1:I-1,1:I-1) * T(1:I-1,I)
287 *
288  CALL ctrmv( 'U', 'N', 'N', i-1, t, ldt, t( 1, i ), 1 )
289 *
290 * T(I,I) = tau(I)
291 *
292  t( i, i ) = t( i, 1 )
293  t( i, 1 ) = zero
294  END DO
295
296 *
297 * End of CTPQRT2
298 *
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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