LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
ssytrd.f
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1 *> \brief \b SSYTRD
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SSYTRD( 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 * REAL A( LDA, * ), D( * ), E( * ), TAU( * ),
29 * $ WORK( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> SSYTRD 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 realSYcomputational
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 ssytrd( 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  REAL A( LDA, * ), D( * ), E( * ), TAU( * ),
203  $ WORK( * )
204 * ..
205 *
206 * =====================================================================
207 *
208 * .. Parameters ..
209  REAL ONE
210  parameter( one = 1.0e+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 slatrd, ssyr2k, ssytd2, 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, 'SSYTRD', 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( 'SSYTRD', -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, 'SSYTRD', 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, 'SSYTRD', 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 slatrd( 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 ssyr2k( 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 ssytd2( 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 slatrd( 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 ssyr2k( 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 ssytd2( 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 SSYTRD
372 *
373  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slatrd(UPLO, N, NB, A, LDA, E, TAU, W, LDW)
SLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal fo...
Definition: slatrd.f:198
subroutine ssytrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
SSYTRD
Definition: ssytrd.f:192
subroutine ssytd2(UPLO, N, A, LDA, D, E, TAU, INFO)
SSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity tran...
Definition: ssytd2.f:173
subroutine ssyr2k(UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SSYR2K
Definition: ssyr2k.f:192