 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ stplqt2()

 subroutine stplqt2 ( integer M, integer N, integer L, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldt, * ) T, integer LDT, integer INFO )

STPLQT2 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:
``` STPLQT2 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 REAL 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 REAL 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 REAL 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 stplqt2.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  REAL A( LDA, * ), B( LDB, * ), T( LDT, * )
187 * ..
188 *
189 * =====================================================================
190 *
191 * .. Parameters ..
192  REAL ONE, ZERO
193  parameter( one = 1.0, zero = 0.0 )
194 * ..
195 * .. Local Scalars ..
196  INTEGER I, J, P, MP, NP
197  REAL ALPHA
198 * ..
199 * .. External Subroutines ..
200  EXTERNAL slarfg, sgemv, sger, strmv, 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( 'STPLQT2', -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 slarfg( 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 sgemv( '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 sger( 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 strmv( 'L', 'N', 'N', p, b( 1, np ), ldb,
278  \$ t( i, 1 ), ldt )
279 *
280 * Rectangular part of B2
281 *
282  CALL sgemv( '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 sgemv( '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 strmv( '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 STPLQT2
308 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
subroutine strmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
STRMV
Definition: strmv.f:147
subroutine sger(M, N, ALPHA, X, INCX, Y, INCY, A, LDA)
SGER
Definition: sger.f:130
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
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