LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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dtplqt2.f
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1*> \brief \b 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.
2*
3* =========== DOCUMENTATION ===========
4*
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16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE DTPLQT2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
22*
23* .. Scalar Arguments ..
24* INTEGER INFO, LDA, LDB, LDT, N, M, L
25* ..
26* .. Array Arguments ..
27* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), T( LDT, * )
28* ..
29*
30*
31*> \par Purpose:
32* =============
33*>
34*> \verbatim
35*>
36*> DTPLQT2 computes a LQ a factorization of a real "triangular-pentagonal"
37*> matrix C, which is composed of a triangular block A and pentagonal block B,
38*> using the compact WY representation for Q.
39*> \endverbatim
40*
41* Arguments:
42* ==========
43*
44*> \param[in] M
45*> \verbatim
46*> M is INTEGER
47*> The total number of rows of the matrix B.
48*> M >= 0.
49*> \endverbatim
50*>
51*> \param[in] N
52*> \verbatim
53*> N is INTEGER
54*> The number of columns of the matrix B, and the order of
55*> the triangular matrix A.
56*> N >= 0.
57*> \endverbatim
58*>
59*> \param[in] L
60*> \verbatim
61*> L is INTEGER
62*> The number of rows of the lower trapezoidal part of B.
63*> MIN(M,N) >= L >= 0. See Further Details.
64*> \endverbatim
65*>
66*> \param[in,out] A
67*> \verbatim
68*> A is DOUBLE PRECISION array, dimension (LDA,M)
69*> On entry, the lower triangular M-by-M matrix A.
70*> On exit, the elements on and below the diagonal of the array
71*> contain the lower triangular matrix L.
72*> \endverbatim
73*>
74*> \param[in] LDA
75*> \verbatim
76*> LDA is INTEGER
77*> The leading dimension of the array A. LDA >= max(1,M).
78*> \endverbatim
79*>
80*> \param[in,out] B
81*> \verbatim
82*> B is DOUBLE PRECISION array, dimension (LDB,N)
83*> On entry, the pentagonal M-by-N matrix B. The first N-L columns
84*> are rectangular, and the last L columns are lower trapezoidal.
85*> On exit, B contains the pentagonal matrix V. See Further Details.
86*> \endverbatim
87*>
88*> \param[in] LDB
89*> \verbatim
90*> LDB is INTEGER
91*> The leading dimension of the array B. LDB >= max(1,M).
92*> \endverbatim
93*>
94*> \param[out] T
95*> \verbatim
96*> T is DOUBLE PRECISION array, dimension (LDT,M)
97*> The N-by-N upper triangular factor T of the block reflector.
98*> See Further Details.
99*> \endverbatim
100*>
101*> \param[in] LDT
102*> \verbatim
103*> LDT is INTEGER
104*> The leading dimension of the array T. LDT >= max(1,M)
105*> \endverbatim
106*>
107*> \param[out] INFO
108*> \verbatim
109*> INFO is INTEGER
110*> = 0: successful exit
111*> < 0: if INFO = -i, the i-th argument had an illegal value
112*> \endverbatim
113*
114* Authors:
115* ========
116*
117*> \author Univ. of Tennessee
118*> \author Univ. of California Berkeley
119*> \author Univ. of Colorado Denver
120*> \author NAG Ltd.
121*
122*> \ingroup tplqt2
123*
124*> \par Further Details:
125* =====================
126*>
127*> \verbatim
128*>
129*> The input matrix C is a M-by-(M+N) matrix
130*>
131*> C = [ A ][ B ]
132*>
133*>
134*> where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
135*> matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
136*> upper trapezoidal matrix B2:
137*>
138*> B = [ B1 ][ B2 ]
139*> [ B1 ] <- M-by-(N-L) rectangular
140*> [ B2 ] <- M-by-L lower trapezoidal.
141*>
142*> The lower trapezoidal matrix B2 consists of the first L columns of a
143*> N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N). If L=0,
144*> B is rectangular M-by-N; if M=L=N, B is lower triangular.
145*>
146*> The matrix W stores the elementary reflectors H(i) in the i-th row
147*> above the diagonal (of A) in the M-by-(M+N) input matrix C
148*>
149*> C = [ A ][ B ]
150*> [ A ] <- lower triangular M-by-M
151*> [ B ] <- M-by-N pentagonal
152*>
153*> so that W can be represented as
154*>
155*> W = [ I ][ V ]
156*> [ I ] <- identity, M-by-M
157*> [ V ] <- M-by-N, same form as B.
158*>
159*> Thus, all of information needed for W is contained on exit in B, which
160*> we call V above. Note that V has the same form as B; that is,
161*>
162*> W = [ V1 ][ V2 ]
163*> [ V1 ] <- M-by-(N-L) rectangular
164*> [ V2 ] <- M-by-L lower trapezoidal.
165*>
166*> The rows of V represent the vectors which define the H(i)'s.
167*> The (M+N)-by-(M+N) block reflector H is then given by
168*>
169*> H = I - W**T * T * W
170*>
171*> where W^H is the conjugate transpose of W and T is the upper triangular
172*> factor of the block reflector.
173*> \endverbatim
174*>
175* =====================================================================
176 SUBROUTINE dtplqt2( M, N, L, A, LDA, B, LDB, T, LDT, INFO )
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*
309 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine dgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
DGEMV
Definition dgemv.f:158
subroutine dger(m, n, alpha, x, incx, y, incy, a, lda)
DGER
Definition dger.f:130
subroutine dlarfg(n, alpha, x, incx, tau)
DLARFG generates an elementary reflector (Householder matrix).
Definition dlarfg.f:106
subroutine dtplqt2(m, n, l, a, lda, b, ldb, t, ldt, info)
DTPLQT2 computes a LQ factorization of a real or complex "triangular-pentagonal" matrix,...
Definition dtplqt2.f:177
subroutine dtrmv(uplo, trans, diag, n, a, lda, x, incx)
DTRMV
Definition dtrmv.f:147