1 *> \brief \b SLARFT forms the triangular factor T of a block reflector H = I - vtvH
3 * =========== DOCUMENTATION ===========
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21 * SUBROUTINE SLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
23 * .. Scalar Arguments ..
24 * CHARACTER DIRECT, STOREV
25 * INTEGER K, LDT, LDV, N
27 * .. Array Arguments ..
28 * REAL T( LDT, * ), TAU( * ), V( LDV, * )
37 *> SLARFT 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.
40 *> If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
42 *> If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
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
47 *> H = I - V * T * V**T
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
52 *> H = I - V**T * T * V
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)
69 *> STOREV is CHARACTER*1
70 *> Specifies how the vectors which define the elementary
71 *> reflectors are stored (see also Further Details):
79 *> The order of the block reflector H. N >= 0.
85 *> The order of the triangular factor T (= the number of
86 *> elementary reflectors). K >= 1.
91 *> V is REAL array, dimension
92 *> (LDV,K) if STOREV = 'C'
93 *> (LDV,N) if STOREV = 'R'
94 *> The matrix V. See further details.
100 *> The leading dimension of the array V.
101 *> If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
106 *> TAU is REAL array, dimension (K)
107 *> TAU(i) must contain the scalar factor of the elementary
113 *> T is REAL 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.
122 *> The leading dimension of the array T. LDT >= K.
128 *> \author Univ. of Tennessee
129 *> \author Univ. of California Berkeley
130 *> \author Univ. of Colorado Denver
133 *> \date September 2012
135 *> \ingroup realOTHERauxiliary
137 *> \par Further Details:
138 * =====================
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.
146 *> DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
148 *> V = ( 1 ) V = ( 1 v1 v1 v1 v1 )
149 *> ( v1 1 ) ( 1 v2 v2 v2 )
150 *> ( v1 v2 1 ) ( 1 v3 v3 )
154 *> DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
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 )
163 * =====================================================================
164 SUBROUTINE SLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
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..--
171 * .. Scalar Arguments ..
172 CHARACTER DIRECT, STOREV
173 INTEGER K, LDT, LDV, N
175 * .. Array Arguments ..
176 REAL T( LDT, * ), TAU( * ), V( LDV, * )
179 * =====================================================================
183 PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
185 * .. Local Scalars ..
186 INTEGER I, J, PREVLASTV, LASTV
188 * .. External Subroutines ..
189 EXTERNAL SGEMV, STRMV
191 * .. External Functions ..
195 * .. Executable Statements ..
197 * Quick return if possible
202 IF( LSAME( DIRECT, 'F' ) ) THEN
205 PREVLASTV = MAX( I, PREVLASTV )
206 IF( TAU( I ).EQ.ZERO ) THEN
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
223 T( J, I ) = -TAU( I ) * V( I , J )
225 J = MIN( LASTV, PREVLASTV )
227 * T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)**T * V(i:j,i)
229 CALL SGEMV( 'Transpose', J-I, I-1, -TAU( I ),
230 $ V( I+1, 1 ), LDV, V( I+1, I ), 1, ONE,
233 * Skip any trailing zeros.
234 DO LASTV = N, I+1, -1
235 IF( V( I, LASTV ).NE.ZERO ) EXIT
238 T( J, I ) = -TAU( I ) * V( J , I )
240 J = MIN( LASTV, PREVLASTV )
242 * T(1:i-1,i) := - tau(i) * V(1:i-1,i:j) * V(i,i:j)**T
244 CALL SGEMV( 'No transpose', I-1, J-I, -TAU( I ),
245 $ V( 1, I+1 ), LDV, V( I, I+1 ), LDV,
246 $ ONE, T( 1, I ), 1 )
249 * T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
251 CALL STRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T,
252 $ LDT, T( 1, I ), 1 )
255 PREVLASTV = MAX( PREVLASTV, LASTV )
264 IF( TAU( I ).EQ.ZERO ) THEN
276 IF( LSAME( STOREV, 'C' ) ) THEN
277 * Skip any leading zeros.
279 IF( V( LASTV, I ).NE.ZERO ) EXIT
282 T( J, I ) = -TAU( I ) * V( N-K+I , J )
284 J = MAX( LASTV, PREVLASTV )
286 * T(i+1:k,i) = -tau(i) * V(j:n-k+i,i+1:k)**T * V(j:n-k+i,i)
288 CALL SGEMV( 'Transpose', N-K+I-J, K-I, -TAU( I ),
289 $ V( J, I+1 ), LDV, V( J, I ), 1, ONE,
292 * Skip any leading zeros.
294 IF( V( I, LASTV ).NE.ZERO ) EXIT
297 T( J, I ) = -TAU( I ) * V( J, N-K+I )
299 J = MAX( LASTV, PREVLASTV )
301 * T(i+1:k,i) = -tau(i) * V(i+1:k,j:n-k+i) * V(i,j:n-k+i)**T
303 CALL SGEMV( '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 )
308 * T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i)
310 CALL STRMV( 'Lower', 'No transpose', 'Non-unit', K-I,
311 $ T( I+1, I+1 ), LDT, T( I+1, I ), 1 )
313 PREVLASTV = MIN( PREVLASTV, LASTV )