LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
Loading...
Searching...
No Matches
dlahr2.f
Go to the documentation of this file.
1*> \brief \b DLAHR2 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 ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download DLAHR2 + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlahr2.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlahr2.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahr2.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE DLAHR2( 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* DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
28* $ Y( LDY, NB )
29* ..
30*
31*
32*> \par Purpose:
33* =============
34*>
35*> \verbatim
36*>
37*> DLAHR2 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 DGEHRD.
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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 doubleOTHERauxiliary
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 DLAHRD
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 DLAHRD routine. (This
169*> subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
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 dlahr2( 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 DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ),
191 $ Y( LDY, NB )
192* ..
193*
194* =====================================================================
195*
196* .. Parameters ..
197 DOUBLE PRECISION ZERO, ONE
198 parameter( zero = 0.0d+0,
199 $ one = 1.0d+0 )
200* ..
201* .. Local Scalars ..
202 INTEGER I
203 DOUBLE PRECISION EI
204* ..
205* .. External Subroutines ..
206 EXTERNAL daxpy, dcopy, dgemm, dgemv, dlacpy,
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 dgemv( '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 dcopy( i-1, a( k+1, i ), 1, t( 1, nb ), 1 )
240 CALL dtrmv( '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 dgemv( '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 dtrmv( 'Upper', 'Transpose', 'NON-UNIT',
253 $ i-1, t, ldt,
254 $ t( 1, nb ), 1 )
255*
256* b2 := b2 - V2*w
257*
258 CALL dgemv( '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 dtrmv( 'Lower', 'NO TRANSPOSE',
265 $ 'UNIT', i-1,
266 $ a( k+1, 1 ), lda, t( 1, nb ), 1 )
267 CALL daxpy( 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 dlarfg( 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 dgemv( '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 dgemv( '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 dgemv( '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 dscal( n-k, tau( i ), y( k+1, i ), 1 )
292*
293* Compute T(1:I,I)
294*
295 CALL dscal( i-1, -tau( i ), t( 1, i ), 1 )
296 CALL dtrmv( '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 dlacpy( 'ALL', k, nb, a( 1, 2 ), lda, y, ldy )
307 CALL dtrmm( '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 dgemm( '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 dtrmm( 'RIGHT', 'Upper', 'NO TRANSPOSE',
316 $ 'NON-UNIT', k, nb,
317 $ one, t, ldt, y, ldy )
318*
319 RETURN
320*
321* End of DLAHR2
322*
323 END
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dtrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
DTRMV
Definition: dtrmv.f:147
subroutine dgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DGEMV
Definition: dgemv.f:156
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187
subroutine dtrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
DTRMM
Definition: dtrmm.f:177
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:106
subroutine dlahr2(N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
DLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elemen...
Definition: dlahr2.f:181