LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
slarzt.f
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1 *> \brief \b SLARZT 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 SLARZT( 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 * REAL T( LDT, * ), TAU( * ), V( LDV, * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> SLARZT forms the triangular factor T of a real block reflector
38 *> H of order > n, which is defined as a product of k elementary
39 *> reflectors.
40 *>
41 *> If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
42 *>
43 *> If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
44 *>
45 *> If STOREV = 'C', the vector which defines the elementary reflector
46 *> H(i) is stored in the i-th column of the array V, and
47 *>
48 *> H = I - V * T * V**T
49 *>
50 *> If STOREV = 'R', the vector which defines the elementary reflector
51 *> H(i) is stored in the i-th row of the array V, and
52 *>
53 *> H = I - V**T * T * V
54 *>
55 *> Currently, only STOREV = 'R' and DIRECT = 'B' are supported.
56 *> \endverbatim
57 *
58 * Arguments:
59 * ==========
60 *
61 *> \param[in] DIRECT
62 *> \verbatim
63 *> DIRECT is CHARACTER*1
64 *> Specifies the order in which the elementary reflectors are
65 *> multiplied to form the block reflector:
66 *> = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
67 *> = 'B': H = H(k) . . . H(2) H(1) (Backward)
68 *> \endverbatim
69 *>
70 *> \param[in] STOREV
71 *> \verbatim
72 *> STOREV is CHARACTER*1
73 *> Specifies how the vectors which define the elementary
75 *> = 'C': columnwise (not supported yet)
76 *> = 'R': rowwise
77 *> \endverbatim
78 *>
79 *> \param[in] N
80 *> \verbatim
81 *> N is INTEGER
82 *> The order of the block reflector H. N >= 0.
83 *> \endverbatim
84 *>
85 *> \param[in] K
86 *> \verbatim
87 *> K is INTEGER
88 *> The order of the triangular factor T (= the number of
89 *> elementary reflectors). K >= 1.
90 *> \endverbatim
91 *>
92 *> \param[in,out] V
93 *> \verbatim
94 *> V is REAL array, dimension
95 *> (LDV,K) if STOREV = 'C'
96 *> (LDV,N) if STOREV = 'R'
97 *> The matrix V. See further details.
98 *> \endverbatim
99 *>
100 *> \param[in] LDV
101 *> \verbatim
102 *> LDV is INTEGER
103 *> The leading dimension of the array V.
104 *> If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
105 *> \endverbatim
106 *>
107 *> \param[in] TAU
108 *> \verbatim
109 *> TAU is REAL array, dimension (K)
110 *> TAU(i) must contain the scalar factor of the elementary
111 *> reflector H(i).
112 *> \endverbatim
113 *>
114 *> \param[out] T
115 *> \verbatim
116 *> T is REAL array, dimension (LDT,K)
117 *> The k by k triangular factor T of the block reflector.
118 *> If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
119 *> lower triangular. The rest of the array is not used.
120 *> \endverbatim
121 *>
122 *> \param[in] LDT
123 *> \verbatim
124 *> LDT is INTEGER
125 *> The leading dimension of the array T. LDT >= K.
126 *> \endverbatim
127 *
128 * Authors:
129 * ========
130 *
131 *> \author Univ. of Tennessee
132 *> \author Univ. of California Berkeley
133 *> \author Univ. of Colorado Denver
134 *> \author NAG Ltd.
135 *
136 *> \date September 2012
137 *
138 *> \ingroup realOTHERcomputational
139 *
140 *> \par Contributors:
141 * ==================
142 *>
143 *> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
144 *
145 *> \par Further Details:
146 * =====================
147 *>
148 *> \verbatim
149 *>
150 *> The shape of the matrix V and the storage of the vectors which define
151 *> the H(i) is best illustrated by the following example with n = 5 and
152 *> k = 3. The elements equal to 1 are not stored; the corresponding
153 *> array elements are modified but restored on exit. The rest of the
154 *> array is not used.
155 *>
156 *> DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
157 *>
158 *> ______V_____
159 *> ( v1 v2 v3 ) / \
160 *> ( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 )
161 *> V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 )
162 *> ( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 )
163 *> ( v1 v2 v3 )
164 *> . . .
165 *> . . .
166 *> 1 . .
167 *> 1 .
168 *> 1
169 *>
170 *> DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
171 *>
172 *> ______V_____
173 *> 1 / \
174 *> . 1 ( 1 . . . . v1 v1 v1 v1 v1 )
175 *> . . 1 ( . 1 . . . v2 v2 v2 v2 v2 )
176 *> . . . ( . . 1 . . v3 v3 v3 v3 v3 )
177 *> . . .
178 *> ( v1 v2 v3 )
179 *> ( v1 v2 v3 )
180 *> V = ( v1 v2 v3 )
181 *> ( v1 v2 v3 )
182 *> ( v1 v2 v3 )
183 *> \endverbatim
184 *>
185 * =====================================================================
186  SUBROUTINE slarzt( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
187 *
188 * -- LAPACK computational routine (version 3.4.2) --
189 * -- LAPACK is a software package provided by Univ. of Tennessee, --
190 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
191 * September 2012
192 *
193 * .. Scalar Arguments ..
194  CHARACTER DIRECT, STOREV
195  INTEGER K, LDT, LDV, N
196 * ..
197 * .. Array Arguments ..
198  REAL T( ldt, * ), TAU( * ), V( ldv, * )
199 * ..
200 *
201 * =====================================================================
202 *
203 * .. Parameters ..
204  REAL ZERO
205  parameter ( zero = 0.0e+0 )
206 * ..
207 * .. Local Scalars ..
208  INTEGER I, INFO, J
209 * ..
210 * .. External Subroutines ..
211  EXTERNAL sgemv, strmv, xerbla
212 * ..
213 * .. External Functions ..
214  LOGICAL LSAME
215  EXTERNAL lsame
216 * ..
217 * .. Executable Statements ..
218 *
219 * Check for currently supported options
220 *
221  info = 0
222  IF( .NOT.lsame( direct, 'B' ) ) THEN
223  info = -1
224  ELSE IF( .NOT.lsame( storev, 'R' ) ) THEN
225  info = -2
226  END IF
227  IF( info.NE.0 ) THEN
228  CALL xerbla( 'SLARZT', -info )
229  RETURN
230  END IF
231 *
232  DO 20 i = k, 1, -1
233  IF( tau( i ).EQ.zero ) THEN
234 *
235 * H(i) = I
236 *
237  DO 10 j = i, k
238  t( j, i ) = zero
239  10 CONTINUE
240  ELSE
241 *
242 * general case
243 *
244  IF( i.LT.k ) THEN
245 *
246 * T(i+1:k,i) = - tau(i) * V(i+1:k,1:n) * V(i,1:n)**T
247 *
248  CALL sgemv( 'No transpose', k-i, n, -tau( i ),
249  \$ v( i+1, 1 ), ldv, v( i, 1 ), ldv, zero,
250  \$ t( i+1, i ), 1 )
251 *
252 * T(i+1:k,i) = T(i+1:k,i+1:k) * T(i+1:k,i)
253 *
254  CALL strmv( 'Lower', 'No transpose', 'Non-unit', k-i,
255  \$ t( i+1, i+1 ), ldt, t( i+1, i ), 1 )
256  END IF
257  t( i, i ) = tau( i )
258  END IF
259  20 CONTINUE
260  RETURN
261 *
262 * End of SLARZT
263 *
264  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:158
subroutine slarzt(DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
SLARZT forms the triangular factor T of a block reflector H = I - vtvH.
Definition: slarzt.f:187
subroutine strmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
STRMV
Definition: strmv.f:149