1*> \brief \b DLARFT forms the triangular factor T of a block reflector H = I - vtvH
2*
3*  =========== DOCUMENTATION ===========
4*
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17*
18*  Definition:
19*  ===========
20*
21*       SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
22*
23*       .. Scalar Arguments ..
24*       CHARACTER          DIRECT, STOREV
25*       INTEGER            K, LDT, LDV, N
26*       ..
27*       .. Array Arguments ..
28*       DOUBLE PRECISION   T( LDT, * ), TAU( * ), V( LDV, * )
29*       ..
30*
31*
32*> \par Purpose:
33*  =============
34*>
35*> \verbatim
36*>
37*> DLARFT forms the triangular factor T of a real block reflector H
38*> of order n, which is defined as a product of k elementary reflectors.
39*>
40*> If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
41*>
42*> If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
43*>
44*> If STOREV = 'C', the vector which defines the elementary reflector
45*> H(i) is stored in the i-th column of the array V, and
46*>
47*>    H  =  I - V * T * V**T
48*>
49*> If STOREV = 'R', the vector which defines the elementary reflector
50*> H(i) is stored in the i-th row of the array V, and
51*>
52*>    H  =  I - V**T * T * V
53*> \endverbatim
54*
55*  Arguments:
56*  ==========
57*
58*> \param[in] DIRECT
59*> \verbatim
60*>          DIRECT is CHARACTER*1
61*>          Specifies the order in which the elementary reflectors are
62*>          multiplied to form the block reflector:
63*>          = 'F': H = H(1) H(2) . . . H(k) (Forward)
64*>          = 'B': H = H(k) . . . H(2) H(1) (Backward)
65*> \endverbatim
66*>
67*> \param[in] STOREV
68*> \verbatim
69*>          STOREV is CHARACTER*1
70*>          Specifies how the vectors which define the elementary
71*>          reflectors are stored (see also Further Details):
72*>          = 'C': columnwise
73*>          = 'R': rowwise
74*> \endverbatim
75*>
76*> \param[in] N
77*> \verbatim
78*>          N is INTEGER
79*>          The order of the block reflector H. N >= 0.
80*> \endverbatim
81*>
82*> \param[in] K
83*> \verbatim
84*>          K is INTEGER
85*>          The order of the triangular factor T (= the number of
86*>          elementary reflectors). K >= 1.
87*> \endverbatim
88*>
89*> \param[in] V
90*> \verbatim
91*>          V is DOUBLE PRECISION array, dimension
92*>                               (LDV,K) if STOREV = 'C'
93*>                               (LDV,N) if STOREV = 'R'
94*>          The matrix V. See further details.
95*> \endverbatim
96*>
97*> \param[in] LDV
98*> \verbatim
99*>          LDV is INTEGER
100*>          The leading dimension of the array V.
101*>          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
102*> \endverbatim
103*>
104*> \param[in] TAU
105*> \verbatim
106*>          TAU is DOUBLE PRECISION array, dimension (K)
107*>          TAU(i) must contain the scalar factor of the elementary
108*>          reflector H(i).
109*> \endverbatim
110*>
111*> \param[out] T
112*> \verbatim
113*>          T is DOUBLE PRECISION array, dimension (LDT,K)
114*>          The k by k triangular factor T of the block reflector.
115*>          If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
116*>          lower triangular. The rest of the array is not used.
117*> \endverbatim
118*>
119*> \param[in] LDT
120*> \verbatim
121*>          LDT is INTEGER
122*>          The leading dimension of the array T. LDT >= K.
123*> \endverbatim
124*
125*  Authors:
126*  ========
127*
128*> \author Univ. of Tennessee
129*> \author Univ. of California Berkeley
130*> \author Univ. of Colorado Denver
131*> \author NAG Ltd.
132*
133*> \date September 2012
134*
135*> \ingroup doubleOTHERauxiliary
136*
137*> \par Further Details:
138*  =====================
139*>
140*> \verbatim
141*>
142*>  The shape of the matrix V and the storage of the vectors which define
143*>  the H(i) is best illustrated by the following example with n = 5 and
144*>  k = 3. The elements equal to 1 are not stored.
145*>
146*>  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
147*>
148*>               V = (  1       )                 V = (  1 v1 v1 v1 v1 )
149*>                   ( v1  1    )                     (     1 v2 v2 v2 )
150*>                   ( v1 v2  1 )                     (        1 v3 v3 )
151*>                   ( v1 v2 v3 )
152*>                   ( v1 v2 v3 )
153*>
154*>  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
155*>
156*>               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
157*>                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )
158*>                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
159*>                   (     1 v3 )
160*>                   (        1 )
161*> \endverbatim
162*>
163*  =====================================================================
164      SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
165*
166*  -- LAPACK auxiliary routine (version 3.4.2) --
167*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
168*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
169*     September 2012
170*
171*     .. Scalar Arguments ..
172      CHARACTER          DIRECT, STOREV
173      INTEGER            K, LDT, LDV, N
174*     ..
175*     .. Array Arguments ..
176      DOUBLE PRECISION   T( LDT, * ), TAU( * ), V( LDV, * )
177*     ..
178*
179*  =====================================================================
180*
181*     .. Parameters ..
182      DOUBLE PRECISION   ONE, ZERO
183      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
184*     ..
185*     .. Local Scalars ..
186      INTEGER            I, J, PREVLASTV, LASTV
187*     ..
188*     .. External Subroutines ..
189      EXTERNAL           DGEMV, DTRMV
190*     ..
191*     .. External Functions ..
192      LOGICAL            LSAME
193      EXTERNAL           LSAME
194*     ..
195*     .. Executable Statements ..
196*
197*     Quick return if possible
198*
199      IF( N.EQ.0 )
200     $   RETURN
201*
202      IF( LSAME( DIRECT, 'F' ) ) THEN
203         PREVLASTV = N
204         DO I = 1, K
205            PREVLASTV = MAX( I, PREVLASTV )
206            IF( TAU( I ).EQ.ZERO ) THEN
207*
208*              H(i)  =  I
209*
210               DO J = 1, I
211                  T( J, I ) = ZERO
212               END DO
213            ELSE
214*
215*              general case
216*
217               IF( LSAME( STOREV, 'C' ) ) THEN
218*                 Skip any trailing zeros.
219                  DO LASTV = N, I+1, -1
220                     IF( V( LASTV, I ).NE.ZERO ) EXIT
221                  END DO
222                  DO J = 1, I-1
223                     T( J, I ) = -TAU( I ) * V( I , J )
224                  END DO
225                  J = MIN( LASTV, PREVLASTV )
226*
227*                 T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)**T * V(i:j,i)
228*
229                  CALL DGEMV( 'Transpose', J-I, I-1, -TAU( I ),
230     $                        V( I+1, 1 ), LDV, V( I+1, I ), 1, ONE,
231     $                        T( 1, I ), 1 )
232               ELSE
233*                 Skip any trailing zeros.
234                  DO LASTV = N, I+1, -1
235                     IF( V( I, LASTV ).NE.ZERO ) EXIT
236                  END DO
237                  DO J = 1, I-1
238                     T( J, I ) = -TAU( I ) * V( J , I )
239                  END DO
240                  J = MIN( LASTV, PREVLASTV )
241*
242*                 T(1:i-1,i) := - tau(i) * V(1:i-1,i:j) * V(i,i:j)**T
243*
244                  CALL DGEMV( 'No transpose', I-1, J-I, -TAU( I ),
245     $                        V( 1, I+1 ), LDV, V( I, I+1 ), LDV, ONE,
246     $                        T( 1, I ), 1 )
247               END IF
248*
249*              T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
250*
251               CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T,
252     $                     LDT, T( 1, I ), 1 )
253               T( I, I ) = TAU( I )
254               IF( I.GT.1 ) THEN
255                  PREVLASTV = MAX( PREVLASTV, LASTV )
256               ELSE
257                  PREVLASTV = LASTV
258               END IF
259            END IF
260         END DO
261      ELSE
262         PREVLASTV = 1
263         DO I = K, 1, -1
264            IF( TAU( I ).EQ.ZERO ) THEN
265*
266*              H(i)  =  I
267*
268               DO J = I, K
269                  T( J, I ) = ZERO
270               END DO
271            ELSE
272*
273*              general case
274*
275               IF( I.LT.K ) THEN
276                  IF( LSAME( STOREV, 'C' ) ) THEN
277*                    Skip any leading zeros.
278                     DO LASTV = 1, I-1
279                        IF( V( LASTV, I ).NE.ZERO ) EXIT
280                     END DO
281                     DO J = I+1, K
282                        T( J, I ) = -TAU( I ) * V( N-K+I , J )
283                     END DO
284                     J = MAX( LASTV, PREVLASTV )
285*
286*                    T(i+1:k,i) = -tau(i) * V(j:n-k+i,i+1:k)**T * V(j:n-k+i,i)
287*
288                     CALL DGEMV( 'Transpose', N-K+I-J, K-I, -TAU( I ),
289     $                           V( J, I+1 ), LDV, V( J, I ), 1, ONE,
290     $                           T( I+1, I ), 1 )
291                  ELSE
292*                    Skip any leading zeros.
293                     DO LASTV = 1, I-1
294                        IF( V( I, LASTV ).NE.ZERO ) EXIT
295                     END DO
296                     DO J = I+1, K
297                        T( J, I ) = -TAU( I ) * V( J, N-K+I )
298                     END DO
299                     J = MAX( LASTV, PREVLASTV )
300*
301*                    T(i+1:k,i) = -tau(i) * V(i+1:k,j:n-k+i) * V(i,j:n-k+i)**T
302*
303                     CALL DGEMV( 'No transpose', K-I, N-K+I-J,
304     $                    -TAU( I ), V( I+1, J ), LDV, V( I, J ), LDV,
305     $                    ONE, T( I+1, I ), 1 )
306                  END IF
307*
308*                 T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i)
309*
310                  CALL DTRMV( 'Lower', 'No transpose', 'Non-unit', K-I,
311     $                        T( I+1, I+1 ), LDT, T( I+1, I ), 1 )
312                  IF( I.GT.1 ) THEN
313                     PREVLASTV = MIN( PREVLASTV, LASTV )
314                  ELSE
315                     PREVLASTV = LASTV
316                  END IF
317               END IF
318               T( I, I ) = TAU( I )
319            END IF
320         END DO
321      END IF
322      RETURN
323*
324*     End of DLARFT
325*
326      END
327