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

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

STPLQT

Purpose:
``` STPLQT computes a blocked LQ 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 number of rows of the matrix B, and the order of the triangular matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix B. 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] MB ``` MB is INTEGER The block size to be used in the blocked QR. M >= MB >= 1.``` [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,N) The lower triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= MB.``` [out] WORK ` WORK is REAL array, dimension (MB*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 on 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
M-by-M 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,
[ V ] = [ 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 number of blocks is B = ceiling(M/MB), where each
block is of order MB except for the last block, which is of order
IB = M - (M-1)*MB.  For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
for the last block) T's are stored in the MB-by-N matrix T as

T = [T1 T2 ... TB].```

Purpose:
``` STPLQT computes a blocked LQ 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 number of rows of the matrix B, and the order of the triangular matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix B. 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] MB ``` MB is INTEGER The block size to be used in the blocked QR. M >= MB >= 1.``` [in,out] A ``` A is REAL array, dimension (LDA,N) On entry, the lower triangular N-by-N 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,N).``` [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,N) The lower triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= MB.``` [out] WORK ` WORK is REAL array, dimension (MB*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 N-by-N matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 on left of a M-by-L
upper trapezoidal matrix B2:
[ B ] = [ B1 ] [ B2 ]
[ B1 ]  <- M-by-(N-L) rectangular
[ B2 ]  <-     M-by-L upper 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 N-by-N
[ B ]  <- M-by-N pentagonal

so that W can be represented as
[ W ] = [ I ] [ V ]
[ 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 ] [ 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 number of blocks is B = ceiling(M/MB), where each
block is of order MB except for the last block, which is of order
IB = M - (M-1)*MB.  For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
for the last block) T's are stored in the MB-by-N matrix T as

T = [T1 T2 ... TB].```

Definition at line 187 of file stplqt.f.

189*
190* -- LAPACK computational routine --
191* -- LAPACK is a software package provided by Univ. of Tennessee, --
192* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
193*
194* .. Scalar Arguments ..
195 INTEGER INFO, LDA, LDB, LDT, N, M, L, MB
196* ..
197* .. Array Arguments ..
198 REAL A( LDA, * ), B( LDB, * ), T( LDT, * ), WORK( * )
199* ..
200*
201* =====================================================================
202*
203* ..
204* .. Local Scalars ..
205 INTEGER I, IB, LB, NB, IINFO
206* ..
207* .. External Subroutines ..
208 EXTERNAL stplqt2, stprfb, xerbla
209* ..
210* .. Executable Statements ..
211*
212* Test the input arguments
213*
214 info = 0
215 IF( m.LT.0 ) THEN
216 info = -1
217 ELSE IF( n.LT.0 ) THEN
218 info = -2
219 ELSE IF( l.LT.0 .OR. (l.GT.min(m,n) .AND. min(m,n).GE.0)) THEN
220 info = -3
221 ELSE IF( mb.LT.1 .OR. (mb.GT.m .AND. m.GT.0)) THEN
222 info = -4
223 ELSE IF( lda.LT.max( 1, m ) ) THEN
224 info = -6
225 ELSE IF( ldb.LT.max( 1, m ) ) THEN
226 info = -8
227 ELSE IF( ldt.LT.mb ) THEN
228 info = -10
229 END IF
230 IF( info.NE.0 ) THEN
231 CALL xerbla( 'STPLQT', -info )
232 RETURN
233 END IF
234*
235* Quick return if possible
236*
237 IF( m.EQ.0 .OR. n.EQ.0 ) RETURN
238*
239 DO i = 1, m, mb
240*
241* Compute the QR factorization of the current block
242*
243 ib = min( m-i+1, mb )
244 nb = min( n-l+i+ib-1, n )
245 IF( i.GE.l ) THEN
246 lb = 0
247 ELSE
248 lb = nb-n+l-i+1
249 END IF
250*
251 CALL stplqt2( ib, nb, lb, a(i,i), lda, b( i, 1 ), ldb,
252 \$ t(1, i ), ldt, iinfo )
253*
254* Update by applying H**T to B(I+IB:M,:) from the right
255*
256 IF( i+ib.LE.m ) THEN
257 CALL stprfb( 'R', 'N', 'F', 'R', m-i-ib+1, nb, ib, lb,
258 \$ b( i, 1 ), ldb, t( 1, i ), ldt,
259 \$ a( i+ib, i ), lda, b( i+ib, 1 ), ldb,
260 \$ work, m-i-ib+1)
261 END IF
262 END DO
263 RETURN
264*
265* End of STPLQT
266*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine stplqt2(M, N, L, A, LDA, B, LDB, T, LDT, INFO)
STPLQT2 computes a LQ factorization of a real or complex "triangular-pentagonal" matrix,...
Definition: stplqt2.f:177
subroutine stprfb(SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, A, LDA, B, LDB, WORK, LDWORK)
STPRFB applies a real "triangular-pentagonal" block reflector to a real matrix, which is composed of ...
Definition: stprfb.f:251
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