LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
slahr2.f
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1 *> \brief \b SLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
2 *
3 * =========== DOCUMENTATION ===========
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER K, LDA, LDT, LDY, N, NB
25 * ..
26 * .. Array Arguments ..
27 * REAL A( LDA, * ), T( LDT, NB ), TAU( NB ),
28 * \$ Y( LDY, NB )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> SLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)
38 *> matrix A so that elements below the k-th subdiagonal are zero. The
39 *> reduction is performed by an orthogonal similarity transformation
40 *> Q**T * A * Q. The routine returns the matrices V and T which determine
41 *> Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
42 *>
43 *> This is an auxiliary routine called by SGEHRD.
44 *> \endverbatim
45 *
46 * Arguments:
47 * ==========
48 *
49 *> \param[in] N
50 *> \verbatim
51 *> N is INTEGER
52 *> The order of the matrix A.
53 *> \endverbatim
54 *>
55 *> \param[in] K
56 *> \verbatim
57 *> K is INTEGER
58 *> The offset for the reduction. Elements below the k-th
59 *> subdiagonal in the first NB columns are reduced to zero.
60 *> K < N.
61 *> \endverbatim
62 *>
63 *> \param[in] NB
64 *> \verbatim
65 *> NB is INTEGER
66 *> The number of columns to be reduced.
67 *> \endverbatim
68 *>
69 *> \param[in,out] A
70 *> \verbatim
71 *> A is REAL array, dimension (LDA,N-K+1)
72 *> On entry, the n-by-(n-k+1) general matrix A.
73 *> On exit, the elements on and above the k-th subdiagonal in
74 *> the first NB columns are overwritten with the corresponding
75 *> elements of the reduced matrix; the elements below the k-th
76 *> subdiagonal, with the array TAU, represent the matrix Q as a
77 *> product of elementary reflectors. The other columns of A are
78 *> unchanged. See Further Details.
79 *> \endverbatim
80 *>
81 *> \param[in] LDA
82 *> \verbatim
83 *> LDA is INTEGER
84 *> The leading dimension of the array A. LDA >= max(1,N).
85 *> \endverbatim
86 *>
87 *> \param[out] TAU
88 *> \verbatim
89 *> TAU is REAL array, dimension (NB)
90 *> The scalar factors of the elementary reflectors. See Further
91 *> Details.
92 *> \endverbatim
93 *>
94 *> \param[out] T
95 *> \verbatim
96 *> T is REAL array, dimension (LDT,NB)
97 *> The upper triangular matrix T.
98 *> \endverbatim
99 *>
100 *> \param[in] LDT
101 *> \verbatim
102 *> LDT is INTEGER
103 *> The leading dimension of the array T. LDT >= NB.
104 *> \endverbatim
105 *>
106 *> \param[out] Y
107 *> \verbatim
108 *> Y is REAL array, dimension (LDY,NB)
109 *> The n-by-nb matrix Y.
110 *> \endverbatim
111 *>
112 *> \param[in] LDY
113 *> \verbatim
114 *> LDY is INTEGER
115 *> The leading dimension of the array Y. LDY >= N.
116 *> \endverbatim
117 *
118 * Authors:
119 * ========
120 *
121 *> \author Univ. of Tennessee
122 *> \author Univ. of California Berkeley
123 *> \author Univ. of Colorado Denver
124 *> \author NAG Ltd.
125 *
126 *> \ingroup realOTHERauxiliary
127 *
128 *> \par Further Details:
129 * =====================
130 *>
131 *> \verbatim
132 *>
133 *> The matrix Q is represented as a product of nb elementary reflectors
134 *>
135 *> Q = H(1) H(2) . . . H(nb).
136 *>
137 *> Each H(i) has the form
138 *>
139 *> H(i) = I - tau * v * v**T
140 *>
141 *> where tau is a real scalar, and v is a real vector with
142 *> v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
143 *> A(i+k+1:n,i), and tau in TAU(i).
144 *>
145 *> The elements of the vectors v together form the (n-k+1)-by-nb matrix
146 *> V which is needed, with T and Y, to apply the transformation to the
147 *> unreduced part of the matrix, using an update of the form:
148 *> A := (I - V*T*V**T) * (A - Y*V**T).
149 *>
150 *> The contents of A on exit are illustrated by the following example
151 *> with n = 7, k = 3 and nb = 2:
152 *>
153 *> ( a a a a a )
154 *> ( a a a a a )
155 *> ( a a a a a )
156 *> ( h h a a a )
157 *> ( v1 h a a a )
158 *> ( v1 v2 a a a )
159 *> ( v1 v2 a a a )
160 *>
161 *> where a denotes an element of the original matrix A, h denotes a
162 *> modified element of the upper Hessenberg matrix H, and vi denotes an
163 *> element of the vector defining H(i).
164 *>
165 *> This subroutine is a slight modification of LAPACK-3.0's SLAHRD
166 *> incorporating improvements proposed by Quintana-Orti and Van de
167 *> Gejin. Note that the entries of A(1:K,2:NB) differ from those
168 *> returned by the original LAPACK-3.0's SLAHRD routine. (This
169 *> subroutine is not backward compatible with LAPACK-3.0's SLAHRD.)
170 *> \endverbatim
171 *
172 *> \par References:
173 * ================
174 *>
175 *> Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
176 *> performance of reduction to Hessenberg form," ACM Transactions on
177 *> Mathematical Software, 32(2):180-194, June 2006.
178 *>
179 * =====================================================================
180  SUBROUTINE slahr2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
181 *
182 * -- LAPACK auxiliary routine --
183 * -- LAPACK is a software package provided by Univ. of Tennessee, --
184 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
185 *
186 * .. Scalar Arguments ..
187  INTEGER K, LDA, LDT, LDY, N, NB
188 * ..
189 * .. Array Arguments ..
190  REAL A( LDA, * ), T( LDT, NB ), TAU( NB ),
191  \$ Y( LDY, NB )
192 * ..
193 *
194 * =====================================================================
195 *
196 * .. Parameters ..
197  REAL ZERO, ONE
198  parameter( zero = 0.0e+0,
199  \$ one = 1.0e+0 )
200 * ..
201 * .. Local Scalars ..
202  INTEGER I
203  REAL EI
204 * ..
205 * .. External Subroutines ..
206  EXTERNAL saxpy, scopy, sgemm, sgemv, slacpy,
207  \$ slarfg, sscal, strmm, strmv
208 * ..
209 * .. Intrinsic Functions ..
210  INTRINSIC min
211 * ..
212 * .. Executable Statements ..
213 *
214 * Quick return if possible
215 *
216  IF( n.LE.1 )
217  \$ RETURN
218 *
219  DO 10 i = 1, nb
220  IF( i.GT.1 ) THEN
221 *
222 * Update A(K+1:N,I)
223 *
224 * Update I-th column of A - Y * V**T
225 *
226  CALL sgemv( 'NO TRANSPOSE', n-k, i-1, -one, y(k+1,1), ldy,
227  \$ a( k+i-1, 1 ), lda, one, a( k+1, i ), 1 )
228 *
229 * Apply I - V * T**T * V**T to this column (call it b) from the
230 * left, using the last column of T as workspace
231 *
232 * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows)
233 * ( V2 ) ( b2 )
234 *
235 * where V1 is unit lower triangular
236 *
237 * w := V1**T * b1
238 *
239  CALL scopy( i-1, a( k+1, i ), 1, t( 1, nb ), 1 )
240  CALL strmv( 'Lower', 'Transpose', 'UNIT',
241  \$ i-1, a( k+1, 1 ),
242  \$ lda, t( 1, nb ), 1 )
243 *
244 * w := w + V2**T * b2
245 *
246  CALL sgemv( 'Transpose', n-k-i+1, i-1,
247  \$ one, a( k+i, 1 ),
248  \$ lda, a( k+i, i ), 1, one, t( 1, nb ), 1 )
249 *
250 * w := T**T * w
251 *
252  CALL strmv( 'Upper', 'Transpose', 'NON-UNIT',
253  \$ i-1, t, ldt,
254  \$ t( 1, nb ), 1 )
255 *
256 * b2 := b2 - V2*w
257 *
258  CALL sgemv( 'NO TRANSPOSE', n-k-i+1, i-1, -one,
259  \$ a( k+i, 1 ),
260  \$ lda, t( 1, nb ), 1, one, a( k+i, i ), 1 )
261 *
262 * b1 := b1 - V1*w
263 *
264  CALL strmv( 'Lower', 'NO TRANSPOSE',
265  \$ 'UNIT', i-1,
266  \$ a( k+1, 1 ), lda, t( 1, nb ), 1 )
267  CALL saxpy( i-1, -one, t( 1, nb ), 1, a( k+1, i ), 1 )
268 *
269  a( k+i-1, i-1 ) = ei
270  END IF
271 *
272 * Generate the elementary reflector H(I) to annihilate
273 * A(K+I+1:N,I)
274 *
275  CALL slarfg( n-k-i+1, a( k+i, i ), a( min( k+i+1, n ), i ), 1,
276  \$ tau( i ) )
277  ei = a( k+i, i )
278  a( k+i, i ) = one
279 *
280 * Compute Y(K+1:N,I)
281 *
282  CALL sgemv( 'NO TRANSPOSE', n-k, n-k-i+1,
283  \$ one, a( k+1, i+1 ),
284  \$ lda, a( k+i, i ), 1, zero, y( k+1, i ), 1 )
285  CALL sgemv( 'Transpose', n-k-i+1, i-1,
286  \$ one, a( k+i, 1 ), lda,
287  \$ a( k+i, i ), 1, zero, t( 1, i ), 1 )
288  CALL sgemv( 'NO TRANSPOSE', n-k, i-1, -one,
289  \$ y( k+1, 1 ), ldy,
290  \$ t( 1, i ), 1, one, y( k+1, i ), 1 )
291  CALL sscal( n-k, tau( i ), y( k+1, i ), 1 )
292 *
293 * Compute T(1:I,I)
294 *
295  CALL sscal( i-1, -tau( i ), t( 1, i ), 1 )
296  CALL strmv( 'Upper', 'No Transpose', 'NON-UNIT',
297  \$ i-1, t, ldt,
298  \$ t( 1, i ), 1 )
299  t( i, i ) = tau( i )
300 *
301  10 CONTINUE
302  a( k+nb, nb ) = ei
303 *
304 * Compute Y(1:K,1:NB)
305 *
306  CALL slacpy( 'ALL', k, nb, a( 1, 2 ), lda, y, ldy )
307  CALL strmm( 'RIGHT', 'Lower', 'NO TRANSPOSE',
308  \$ 'UNIT', k, nb,
309  \$ one, a( k+1, 1 ), lda, y, ldy )
310  IF( n.GT.k+nb )
311  \$ CALL sgemm( 'NO TRANSPOSE', 'NO TRANSPOSE', k,
312  \$ nb, n-k-nb, one,
313  \$ a( 1, 2+nb ), lda, a( k+1+nb, 1 ), lda, one, y,
314  \$ ldy )
315  CALL strmm( 'RIGHT', 'Upper', 'NO TRANSPOSE',
316  \$ 'NON-UNIT', k, nb,
317  \$ one, t, ldt, y, ldy )
318 *
319  RETURN
320 *
321 * End of SLAHR2
322 *
323  END
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
subroutine slahr2(N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
SLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elemen...
Definition: slahr2.f:181
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
subroutine strmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
STRMV
Definition: strmv.f:147
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
subroutine strmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRMM
Definition: strmm.f:177
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187