LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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dsytrd.f
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1*> \brief \b DSYTRD
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytrd.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
22*
23* .. Scalar Arguments ..
24* CHARACTER UPLO
25* INTEGER INFO, LDA, LWORK, N
26* ..
27* .. Array Arguments ..
28* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ),
29* \$ WORK( * )
30* ..
31*
32*
33*> \par Purpose:
34* =============
35*>
36*> \verbatim
37*>
38*> DSYTRD reduces a real symmetric matrix A to real symmetric
39*> tridiagonal form T by an orthogonal similarity transformation:
40*> Q**T * A * Q = T.
41*> \endverbatim
42*
43* Arguments:
44* ==========
45*
46*> \param[in] UPLO
47*> \verbatim
48*> UPLO is CHARACTER*1
49*> = 'U': Upper triangle of A is stored;
50*> = 'L': Lower triangle of A is stored.
51*> \endverbatim
52*>
53*> \param[in] N
54*> \verbatim
55*> N is INTEGER
56*> The order of the matrix A. N >= 0.
57*> \endverbatim
58*>
59*> \param[in,out] A
60*> \verbatim
61*> A is DOUBLE PRECISION array, dimension (LDA,N)
62*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
63*> N-by-N upper triangular part of A contains the upper
64*> triangular part of the matrix A, and the strictly lower
65*> triangular part of A is not referenced. If UPLO = 'L', the
66*> leading N-by-N lower triangular part of A contains the lower
67*> triangular part of the matrix A, and the strictly upper
68*> triangular part of A is not referenced.
69*> On exit, if UPLO = 'U', the diagonal and first superdiagonal
70*> of A are overwritten by the corresponding elements of the
71*> tridiagonal matrix T, and the elements above the first
72*> superdiagonal, with the array TAU, represent the orthogonal
73*> matrix Q as a product of elementary reflectors; if UPLO
74*> = 'L', the diagonal and first subdiagonal of A are over-
75*> written by the corresponding elements of the tridiagonal
76*> matrix T, and the elements below the first subdiagonal, with
77*> the array TAU, represent the orthogonal matrix Q as a product
78*> of elementary reflectors. 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] D
88*> \verbatim
89*> D is DOUBLE PRECISION array, dimension (N)
90*> The diagonal elements of the tridiagonal matrix T:
91*> D(i) = A(i,i).
92*> \endverbatim
93*>
94*> \param[out] E
95*> \verbatim
96*> E is DOUBLE PRECISION array, dimension (N-1)
97*> The off-diagonal elements of the tridiagonal matrix T:
98*> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
99*> \endverbatim
100*>
101*> \param[out] TAU
102*> \verbatim
103*> TAU is DOUBLE PRECISION array, dimension (N-1)
104*> The scalar factors of the elementary reflectors (see Further
105*> Details).
106*> \endverbatim
107*>
108*> \param[out] WORK
109*> \verbatim
110*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
111*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
112*> \endverbatim
113*>
114*> \param[in] LWORK
115*> \verbatim
116*> LWORK is INTEGER
117*> The dimension of the array WORK. LWORK >= 1.
118*> For optimum performance LWORK >= N*NB, where NB is the
119*> optimal blocksize.
120*>
121*> If LWORK = -1, then a workspace query is assumed; the routine
122*> only calculates the optimal size of the WORK array, returns
123*> this value as the first entry of the WORK array, and no error
124*> message related to LWORK is issued by XERBLA.
125*> \endverbatim
126*>
127*> \param[out] INFO
128*> \verbatim
129*> INFO is INTEGER
130*> = 0: successful exit
131*> < 0: if INFO = -i, the i-th argument had an illegal value
132*> \endverbatim
133*
134* Authors:
135* ========
136*
137*> \author Univ. of Tennessee
138*> \author Univ. of California Berkeley
139*> \author Univ. of Colorado Denver
140*> \author NAG Ltd.
141*
142*> \ingroup doubleSYcomputational
143*
144*> \par Further Details:
145* =====================
146*>
147*> \verbatim
148*>
149*> If UPLO = 'U', the matrix Q is represented as a product of elementary
150*> reflectors
151*>
152*> Q = H(n-1) . . . H(2) H(1).
153*>
154*> Each H(i) has the form
155*>
156*> H(i) = I - tau * v * v**T
157*>
158*> where tau is a real scalar, and v is a real vector with
159*> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
160*> A(1:i-1,i+1), and tau in TAU(i).
161*>
162*> If UPLO = 'L', the matrix Q is represented as a product of elementary
163*> reflectors
164*>
165*> Q = H(1) H(2) . . . H(n-1).
166*>
167*> Each H(i) has the form
168*>
169*> H(i) = I - tau * v * v**T
170*>
171*> where tau is a real scalar, and v is a real vector with
172*> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
173*> and tau in TAU(i).
174*>
175*> The contents of A on exit are illustrated by the following examples
176*> with n = 5:
177*>
178*> if UPLO = 'U': if UPLO = 'L':
179*>
180*> ( d e v2 v3 v4 ) ( d )
181*> ( d e v3 v4 ) ( e d )
182*> ( d e v4 ) ( v1 e d )
183*> ( d e ) ( v1 v2 e d )
184*> ( d ) ( v1 v2 v3 e d )
185*>
186*> where d and e denote diagonal and off-diagonal elements of T, and vi
187*> denotes an element of the vector defining H(i).
188*> \endverbatim
189*>
190* =====================================================================
191 SUBROUTINE dsytrd( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
192*
193* -- LAPACK computational routine --
194* -- LAPACK is a software package provided by Univ. of Tennessee, --
195* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
196*
197* .. Scalar Arguments ..
198 CHARACTER UPLO
199 INTEGER INFO, LDA, LWORK, N
200* ..
201* .. Array Arguments ..
202 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ),
203 \$ WORK( * )
204* ..
205*
206* =====================================================================
207*
208* .. Parameters ..
209 DOUBLE PRECISION ONE
210 parameter( one = 1.0d+0 )
211* ..
212* .. Local Scalars ..
213 LOGICAL LQUERY, UPPER
214 INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
215 \$ NBMIN, NX
216* ..
217* .. External Subroutines ..
218 EXTERNAL dlatrd, dsyr2k, dsytd2, xerbla
219* ..
220* .. Intrinsic Functions ..
221 INTRINSIC max
222* ..
223* .. External Functions ..
224 LOGICAL LSAME
225 INTEGER ILAENV
226 EXTERNAL lsame, ilaenv
227* ..
228* .. Executable Statements ..
229*
230* Test the input parameters
231*
232 info = 0
233 upper = lsame( uplo, 'U' )
234 lquery = ( lwork.EQ.-1 )
235 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
236 info = -1
237 ELSE IF( n.LT.0 ) THEN
238 info = -2
239 ELSE IF( lda.LT.max( 1, n ) ) THEN
240 info = -4
241 ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
242 info = -9
243 END IF
244*
245 IF( info.EQ.0 ) THEN
246*
247* Determine the block size.
248*
249 nb = ilaenv( 1, 'DSYTRD', uplo, n, -1, -1, -1 )
250 lwkopt = n*nb
251 work( 1 ) = lwkopt
252 END IF
253*
254 IF( info.NE.0 ) THEN
255 CALL xerbla( 'DSYTRD', -info )
256 RETURN
257 ELSE IF( lquery ) THEN
258 RETURN
259 END IF
260*
261* Quick return if possible
262*
263 IF( n.EQ.0 ) THEN
264 work( 1 ) = 1
265 RETURN
266 END IF
267*
268 nx = n
269 iws = 1
270 IF( nb.GT.1 .AND. nb.LT.n ) THEN
271*
272* Determine when to cross over from blocked to unblocked code
273* (last block is always handled by unblocked code).
274*
275 nx = max( nb, ilaenv( 3, 'DSYTRD', uplo, n, -1, -1, -1 ) )
276 IF( nx.LT.n ) THEN
277*
278* Determine if workspace is large enough for blocked code.
279*
280 ldwork = n
281 iws = ldwork*nb
282 IF( lwork.LT.iws ) THEN
283*
284* Not enough workspace to use optimal NB: determine the
285* minimum value of NB, and reduce NB or force use of
286* unblocked code by setting NX = N.
287*
288 nb = max( lwork / ldwork, 1 )
289 nbmin = ilaenv( 2, 'DSYTRD', uplo, n, -1, -1, -1 )
290 IF( nb.LT.nbmin )
291 \$ nx = n
292 END IF
293 ELSE
294 nx = n
295 END IF
296 ELSE
297 nb = 1
298 END IF
299*
300 IF( upper ) THEN
301*
302* Reduce the upper triangle of A.
303* Columns 1:kk are handled by the unblocked method.
304*
305 kk = n - ( ( n-nx+nb-1 ) / nb )*nb
306 DO 20 i = n - nb + 1, kk + 1, -nb
307*
308* Reduce columns i:i+nb-1 to tridiagonal form and form the
309* matrix W which is needed to update the unreduced part of
310* the matrix
311*
312 CALL dlatrd( uplo, i+nb-1, nb, a, lda, e, tau, work,
313 \$ ldwork )
314*
315* Update the unreduced submatrix A(1:i-1,1:i-1), using an
316* update of the form: A := A - V*W**T - W*V**T
317*
318 CALL dsyr2k( uplo, 'No transpose', i-1, nb, -one, a( 1, i ),
319 \$ lda, work, ldwork, one, a, lda )
320*
321* Copy superdiagonal elements back into A, and diagonal
322* elements into D
323*
324 DO 10 j = i, i + nb - 1
325 a( j-1, j ) = e( j-1 )
326 d( j ) = a( j, j )
327 10 CONTINUE
328 20 CONTINUE
329*
330* Use unblocked code to reduce the last or only block
331*
332 CALL dsytd2( uplo, kk, a, lda, d, e, tau, iinfo )
333 ELSE
334*
335* Reduce the lower triangle of A
336*
337 DO 40 i = 1, n - nx, nb
338*
339* Reduce columns i:i+nb-1 to tridiagonal form and form the
340* matrix W which is needed to update the unreduced part of
341* the matrix
342*
343 CALL dlatrd( uplo, n-i+1, nb, a( i, i ), lda, e( i ),
344 \$ tau( i ), work, ldwork )
345*
346* Update the unreduced submatrix A(i+ib:n,i+ib:n), using
347* an update of the form: A := A - V*W**T - W*V**T
348*
349 CALL dsyr2k( uplo, 'No transpose', n-i-nb+1, nb, -one,
350 \$ a( i+nb, i ), lda, work( nb+1 ), ldwork, one,
351 \$ a( i+nb, i+nb ), lda )
352*
353* Copy subdiagonal elements back into A, and diagonal
354* elements into D
355*
356 DO 30 j = i, i + nb - 1
357 a( j+1, j ) = e( j )
358 d( j ) = a( j, j )
359 30 CONTINUE
360 40 CONTINUE
361*
362* Use unblocked code to reduce the last or only block
363*
364 CALL dsytd2( uplo, n-i+1, a( i, i ), lda, d( i ), e( i ),
365 \$ tau( i ), iinfo )
366 END IF
367*
368 work( 1 ) = lwkopt
369 RETURN
370*
371* End of DSYTRD
372*
373 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dsyr2k(UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DSYR2K
Definition: dsyr2k.f:192
subroutine dlatrd(UPLO, N, NB, A, LDA, E, TAU, W, LDW)
DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal fo...
Definition: dlatrd.f:198
subroutine dsytd2(UPLO, N, A, LDA, D, E, TAU, INFO)
DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity tran...
Definition: dsytd2.f:173
subroutine dsytrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
DSYTRD
Definition: dsytrd.f:192